In physics, motion is a change in the position of a body over time with respect to a reference system.
The study of movement can be done through kinematics or through dynamics. Depending on the choice of the reference system, the equations of motion will be defined, equations that will determine the position, speed and acceleration of the body at each instant of time. All movement can be represented and studied using graphs. The most common are those that represent space, speed or acceleration as a function of time, their measurement is through mileage or meters per second.
Velocity and acceleration are always relative since they depend on the reference system chosen to measure or calculate them. Once a reference system has been chosen and the equations of motion have been defined, the speed and acceleration of the body at each instant of time.
A real physical system is characterized by at least three important properties:
Have a position in space-time.
Have a defined physical state subject to temporal evolution.
Being able to associate a physical magnitude called energy.
Motion refers to the change over time of a property in space, such as location, orientation, geometric shape or size, as measured by a physical observer. A little more generally, the change of property in space can be influenced by the internal properties of a body or physical system, or even the study of movement in all its generality leads to considering the change of said physical state.
The description of the movement of physical bodies without considering the causes that originate it is called kinematics (from the Greek κινεω, kineo, movement) (which would only deal with properties 1 and 2 above). It is limited to the study of trajectory and displacement based on geometric elements that evolve over time. This discipline aims to describe the way in which a certain body moves. Classical physics was born studying the kinematics of rigid bodies.
Later, the study of the evolution over time of a physical system in relation to the causes that cause or preserve movement led to the development of dynamics. The most important dynamic principles are inertia, momentum, force, and mechanical energy.
Vibration in light elements
Introduction
In physics, motion is a change in the position of a body over time with respect to a reference system.
The study of movement can be done through kinematics or through dynamics. Depending on the choice of the reference system, the equations of motion will be defined, equations that will determine the position, speed and acceleration of the body at each instant of time. All movement can be represented and studied using graphs. The most common are those that represent space, speed or acceleration as a function of time, their measurement is through mileage or meters per second.
Velocity and acceleration are always relative since they depend on the reference system chosen to measure or calculate them. Once a reference system has been chosen and the equations of motion have been defined, the speed and acceleration of the body at each instant of time.
A real physical system is characterized by at least three important properties:
Have a position in space-time.
Have a defined physical state subject to temporal evolution.
Being able to associate a physical magnitude called energy.
Motion refers to the change over time of a property in space, such as location, orientation, geometric shape or size, as measured by a physical observer. A little more generally, the change of property in space can be influenced by the internal properties of a body or physical system, or even the study of movement in all its generality leads to considering the change of said physical state.
The description of the movement of physical bodies without considering the causes that originate it is called kinematics (from the Greek κινεω, kineo, movement) (which would only deal with properties 1 and 2 above). It is limited to the study of trajectory and displacement based on geometric elements that evolve over time. This discipline aims to describe the way in which a certain body moves. Classical physics was born studying the kinematics of rigid bodies.
The integration of kinematics and dynamics develops the general discipline called mechanics (Greek Μηχανική and Latin mechanica or 'art of building machines'), which is the branch of physics that studies and analyzes the movement and rest of bodies. Theoretical mechanics was, during the 17th, 18th, and early 20th centuries, the discipline of physics that reached greater mathematical abstraction and was a source of improvement in scientific knowledge of the world. Applied mechanics is usually related to engineering. Both points of view are partially justified since, although mechanics is the basis for most of the classical engineering sciences, it is not as empirical in nature as these and, on the other hand, due to its rigor and deductive reasoning, it is more similar to mathematics.
During the century the appearance of new physical facts, both the consideration of physical bodies moving at speeds close to the speed of light and the movement of subatomic particles, led to the formulation of more abstract theories such as relativistic mechanics and quantum mechanics that continued to be interested in the evolution over time of physical systems, although in a more abstract and general way than classical mechanics had done, whose objective was basically to quantify the change in position in space of particles over time and the agents responsible. of such changes.
Kinematic characteristics of movement
Mobile
Mobile is understood as a moving object whose kinematics and dynamics are to be studied. According to the mobile studied, different perspectives of the movement can be identified:
• - Point mobile: the mobile is reduced to a theoretical point to simplify its study. From a kinematic point of view, the only type of movement admissible for a point mobile is that in which a change in location is observed with respect to a reference coordinate system, defined as translational movement "Translation (physics)").
• - Rigid solid mobile: the mobile is a non-deformable three-dimensional object. The term "rigid" refers, from a mathematical idealization, to the fact that the distance between any two material points of the body remains unchanged over time.
The most general motion of a rigid solid mobile can be considered as the superposition of two basic types of motion:
• - Deformable solid mobile: the mobile is a deformable three-dimensional object, that is, there exists throughout the temporal evolution a state in which the distance between any two material points of the body can vary, which can be evidenced as a change in the size or shape of the mobile, called deformation. The deformation can be:
The global movement of a deformable solid mobile is decomposed into its translational, rotational and deformation movements.
• - Fluid: the mobile is described as a continuous medium without a defined shape, infinitely deformable, in which the displacements that a material point can reach within the fluid are not determined (this contrasts with deformable solids, where the displacements are much more limited). It presents an absence of shape memory, that is, it takes the shape of the container that contains it, without elastic recovery forces as in solid mobiles.
Moment and duration
Time is a physical quantity with which we measure the duration or separation of events. Time allows events to be ordered in sequences. Given two point events E and E, which occur respectively in two temporal coordinates t and t, and at different points in space P and P, all physical theories admit that these can satisfy one and only one of the following three conditions:[1].
It is possible for an observer to be present at event E, and then be at event E, and in that case it is stated that E is an event prior to E. Furthermore, if that happens, that observer will not be able to verify E2.
It is possible for an observer to be present at event E and then be at event E, and in that case it is stated that E is an event after E. Furthermore, if that happens, that observer will not be able to verify E1.
It is impossible, for a specific observer, to be simultaneously present in both events E and E.
For motion description purposes, a duration between time coordinates t and t can be defined as Δt. If said duration is infinitesimal (dt) it is called instant.
Position and displacement
Physical space is the place where material entities are found. Physical space is usually conceived of as having three linear dimensions, although modern physicists usually consider it, over time, as one part of an infinite four-dimensional continuum known as spacetime, which in the presence of matter is curved. The position of a mobile is defined as the state variable "State variable (dynamic system)") that provides a geometric description defined at a given instant dt with respect to a geometric location described by the viewer. Thus, an orthogonal, cylindrical or spherical coordinate system can be used to describe the position of a body.
Displacement is the vector that defines the position of a point or particle relative to an origin A with respect to a position B. The vector extends from the reference point to the final position. When talking about displacement in space, only the initial position and the final position matter, since the trajectory described is not important.
Path
The trajectory is the geometric locus of the successive positions through which a body passes in its movement. The trajectory depends on the reference system in which the movement is described, with respect to the type of mobile and the point of view of the observer.
The trajectory of a translational movement is defined by the parameterized equation of the curve described in geometric space by a point mobile or the center of mass of a solid mobile.
The trajectories of a rotation are described in terms of the Euler angles and surfaces of revolution.
The trajectory of a deformation is described by geometric measurements of deformation.
For fluids in motion, the analogue to the path is the type of flow: a laminar flow is the movement of a fluid when it is orderly, stratified, smooth. In a laminar flow the fluid moves in parallel sheets without intermingling and each fluid particle follows a smooth path, called a streamline. Turbulent flow is the movement of a fluid that occurs chaotically, in which the particles move disorderly and the trajectories of the particles form small periodic eddies (not coordinated).
Speed
In a generic way, a speed is defined as the rate of variation of a certain physical quantity with respect to time.
In the case of translational movements, the speed is a physical magnitude of vector character "Vector (physics)") that expresses the displacement "Displacement (vector)") of an object "Mobile (physics)") per unit of time.
In everyday language the words speed and speed are used interchangeably. In physics a distinction is made between them. Very simply, the difference is that velocity is the speed in a given direction. When we say that a car is traveling at 60 km/h we are indicating its speed. But by saying that a car is traveling at 60 km/h towards the north, we are specifying its speed. Speed describes how fast an object is moving; Velocity describes how fast you do it and in what direction.
The speed of movement at a given instant depends on the observer both in classical mechanics and in the theory of relativity. In quantum mechanics, the speed of a mobile phone, like its trajectory, does not have to be defined at a given instant, according to some interpretations of the theory. The Zitterbewegung phenomenon suggests that an electron could have a transverse oscillatory motion around its classical "path" (i.e. the path it should follow if the classical description were correct).
Speed or also called celerity is the relationship between the distance traveled and the time spent traveling it. A car, for example, travels a certain number of kilometers in an hour, which may be 110km/h. Speed is a measure of how fast a moving object (physics) moves. It is the rate of change "Action (physical)") at which the distance is traveled, since the expression rate of change indicates that we are dividing some quantity by time, therefore, speed is always measured in terms of a unit of distance divided by a unit of time.
In rotational movements, angular velocity is defined as the rate of variation between the angle rotated per unit of time and is designated by the Greek letter ω. Its unit in the International System is the radian per second (rad/s). It is used as a measure of rotation speed.
The strain rate is a magnitude that measures the change in strain with respect to time. For uniaxial problems it is simply the temporal derivative of the longitudinal deformation, while for three-dimensional problems or situations it is represented by a second rank tensor.
In periodic movements, frequency is additionally used, which is a magnitude that measures the number of repetitions per unit of time of any periodic phenomenon or event, such as rotations, oscillations and vibrations. Its unit is Hertz.
In the movement of a fluid, the volumetric flow is a kinematic variable that is defined as the volume of fluid that passes through a given surface in a given time. Given an area A, over which a fluid flows at an angle from the direction perpendicular to with a volumetric flow , the of flow can be defined as:.
Acceleration
In physics, the term acceleration is a vector magnitude that applies to both increases and decreases in speed in a unit of time. The term acceleration applies to both changes in speed and changes in direction.
In translational movements, the velocity vector v is tangent to the trajectory, while the acceleration vector a can be decomposed into two mutually perpendicular components (called intrinsic components): a tangential component a (in the direction of the tangent to the trajectory), called tangential acceleration, and a normal component a (in the direction of the main normal to the trajectory), called normal or centripetal acceleration (this last name because it is always directed towards the center of curvature).
If you travel through a curve at a constant speed of 50 km/h, the effects of acceleration will be felt as a tendency to lean towards the outside of the curve (inertia). You can travel the curve with a constant speed, but the speed is not constant since the direction changes every moment, therefore, the state of movement changes, that is, it is accelerating. For example, the brakes of a car can produce large retarding accelerations, that is, they can produce a large decrease per second in its speed. This is often called deceleration or negative acceleration.
Normal acceleration is a measure of the curvature of the trajectory; different observers in non-uniform motion with respect to them will observe different forces and accelerations and therefore different trajectories. If an inertial observer examines the trajectory of a particle moving in a straight line and with uniform speed (zero curvature trajectory), any other inertial observer will see the particle moving in a straight line and with uniform speed (although not the same straight line), in the case of arbitrary observers in accelerated motion between them the shapes of the trajectories can differ noticeably, since when the two observers measure completely different accelerations, the trajectory of the particle will curve in very different ways for each observer.
In rotational movements, the concept of angular acceleration is used as the rate of variation between angular velocity with respect to time.
Dynamic characteristics of movement
Contenido
Todas las teorías físicas del movimiento atribuyen al movimiento una serie de características o atributos dinámicos como:.
• - Inercia.
• - La cantidad de movimiento.
• - El sistema de fuerzas ejercidos sobre el móvil.
• - La energía mecánica.
En mecánica clásica y mecánica relativista todos ellos son valores numéricos medibles, mientras que en mecánica cuántica esas magnitudes son en general variables aleatorias para las que es posible predecir sus valores medios, pero no el valor exacto en todo momento.
Inertia
In physics, inertia (from the Latin inertĭa) is the property that bodies have of remaining in their state of relative rest or relative motion. Generally speaking, it is the resistance that matter opposes when modifying its state of motion, including changes in speed or direction of motion. As a consequence, a body maintains its state of relative rest or relative uniform rectilinear motion if there is no force that, acting on it, manages to change its state of motion.
In translational movements, the measure of inertia is provided by the mass. In a point mobile it is assumed that all the mass is concentrated at the point that describes the mobile, while in solid mobiles its translation can be simplified by describing a point called Center of mass in a manner analogous to a point mobile.
In rotational movements, the rotational moment of inertia (symbol I) is a measure of resistance to rotation of a body that reflects the distribution of mass of a body with respect to an axis of rotation. The moment of inertia only depends on the geometry of the body and the position of the axis of rotation.
On the other hand, the measures of resistance to deformation are mainly represented by the measures of stiffness, such as Hooke's constant.
In the movement of a fluid, the inertia of a flow is represented by both its density and its viscosity, its friction with the container and its adhesion to the walls.
Momentum
The quantity of motion, linear momentum, impetus or momentum is a fundamental physical magnitude of vector type that describes the motion of a body in any mechanical theory defined as the product of a unit of inertia and a rate of spatial variation with respect to time at a given instant.
According to the concept of momentum, mechanical rest is defined as the mechanical state in which for any given instant, any measurement of momentum is equal to zero.
According to the concept of momentum, uniform motion is defined as that where the momentum remains constant over time.
The vector magnitude defined as the variation in the momentum experienced by a physical object in a closed system is called impulse.
In translational movements of a single mobile, the quantity of linear movement (or linear momentum) is defined as the product of the mass with its linear velocity. The intuitive idea behind this definition is that the "momentum of motion" depended on both mass and speed: if you imagine a fly and a truck, both moving at 40 km/h, everyday experience says that the fly is easy to stop with your hand while the truck is not, even if they are both going at the same speed. This intuition led to defining a magnitude that was proportional to both the mass of the moving object and its speed.
In rotational movements, the angular momentum or kinetic moment is a physical quantity that is related to the vector product between the moment of inertia and the angular velocity. This magnitude plays a role analogous to the linear moment in translations with respect to rotations. The angular momentum for a rigid body rotating with respect to an axis is the resistance offered by said body to the variation of angular velocity. However, this does not imply that it is an exclusive magnitude of rotations; For example, the angular momentum of a particle moving freely with constant velocity (in magnitude and direction) is also conserved.
If we are interested in finding out the momentum of, for example, a fluid that moves according to a velocity field, it is necessary to add the momentum of each particle of the fluid, that is, of each mass differential or infinitesimal element:
In fluid motions, the conservation of the linear momentum of a fluid in motion is generalized by the Navier-Stokes equations.
Force
In physics, force is a physical quantity that measures the rate of variation of the exchange of momentum between two mobiles with respect to the duration of said exchange. According to a classical definition, force is any agent capable of modifying the momentum or shape of material bodies. In the International System of Units, force is measured in “Newtons (N)”.
When several forces act on a single mobile, all of them will be added vector-wise to constitute a single force called resultant force. In translational movements of point or solid mobiles, when the resulting force is zero and the momentum is zero, mechanical equilibrium will be observed as a rest. When the resultant force is zero and the momentum is constant with a value other than zero, a uniform rectilinear motion will be observed. If the resulting force is different from zero, it will be equivalent to the product of the instantaneous mass and the instantaneous acceleration. In terms of momentum, an impulse was applied. This is stated in Newton's first and second laws. Curvilinear translational movements involve the application of a normal force called centripetal force.
In rotational movements, the analogue of force is called the moment of a force (with respect to a given point) or torque, a magnitude obtained as a vector product of the position vector of the point of application of the force (with respect to the point at which the moment is taken) by the force vector, in that order. Thus, the sum of all the torques in a rotating system will lead to a resulting torque. If the resulting torque is zero with zero angular momentum, the body will not rotate. If the resulting torque is zero with constant non-zero angular momentum, uniform rotation or uniform circular motion will be observed. If the resulting torque is different from zero, a net angular acceleration will be seen and therefore the state of rotation will change. The Euler equations "Euler equations (solids)") describe the motion of a rotating rigid solid in a frame of reference with the solid.
In deformations, the analogue to forces are mechanical stresses, mechanical tensions and mechanical torsions. The dynamics of the deformations are described by:.
In fluid mechanics, the equivalent of force is pressure. In a fluid there can be the following types of pressure:
Energy
In physics, energy is defined as the ability to effect a transformation in a physical system, for example by lifting an object, transporting it (moving it), deforming it, or heating it. Energy is not a real physical state, nor an "intangible substance" but a scalar magnitude that is assigned to the state of the physical system, that is, energy is a tool or mathematical abstraction of a property of physical systems. For example, a system with zero kinetic energy can be said to be at rest. Energy is measured with the unit «joule "Joule (unit)") (J)».
In this way, every movement at a specific instant is assigned an amount of energy associated kinematically with its speed and dynamically with its momentum. This magnitude is called kinetic energy.
For any translational motion, its instantaneous kinetic energy is defined as a function of half the mass and the square of the magnitude of the instantaneous linear velocity: .
Thus, a body at rest has instantaneous kinetic energy equal to zero.
In a uniform rectilinear motion, the kinetic energy is constant for any given instant.
The work performed by a force is defined as the product of it by the path taken by its point of application and by the cosine of the angle they form with each other.[2] Work is a scalar physical quantity "Scalar (physics)") that is represented by the letter (from English Work) and is expressed in units of energy, this is in joules "Joule (unit)") or joules (J) in the International System of Units.
Mathematically, the work for a particle moving along a curve C is expressed as:.
For the case of a constant force the previous equation reduces to:.
Where is the mechanical work, is the magnitude of the force, is the displacement "Displacement (vector)") and is the angle between the force vector "Vector (Euclidean space)") and the displacement vector (see drawing).
When the force vector is perpendicular to the displacement vector of the body on which it is applied, said force does not do any work. Likewise, if there is no displacement, the work will also be null.
For any rotational motion, the rotational kinetic energy is described as a function of half the rotational moment of inertia and the square of the magnitude of the instantaneous angular velocity: . Thus, a rotation in uniform circular motion presents a constant value of rotational kinetic energy.
For harmonic movements and any type of deformations, the kinetic energy can be described as a consequence of the forces involved being central and, therefore, conservative. Consequently, a scalar field called potential energy (E) associated with the force can be defined. To find the expression of the potential energy, it is enough to integrate the expression of the force (this is extendable to all conservative forces) and change its sign, obtaining:
History of the physical concept of movement
Las cuestiones acerca de las causas del movimiento surgieron en la mente del hombre hace más de 25 siglos, pero las respuestas que hoy conocemos no se desarrollaron hasta los tiempos de Galileo Galilei (1564–1642) e Isaac Newton (1642–1727).
Movement studies in classical times
• - Anaximander thought that nature came from the separation, through eternal movement, of opposite elements (for example, cold-heat), which were locked in something called primordial matter.
• - Democritus said that nature is made up of indivisible pieces of matter called atoms, and that movement was their main characteristic, movement being a change of place in space.
• - Zeno's paradoxes are a series of paradoxes or aporias devised by Zeno of Elea. Devoted mainly to the problem of the continuum and the relationships between space "Space (physics)"), time and movement, Zeno would have posed - according to Proclus - a total of 40 paradoxes, of which nine or ten complete descriptions have been preserved (in Aristotle's Physics "Physics (Aristotle)")[3][4] and Simplicio's commentary on this work).
• - Aristotle rejects the task of returning to the concept of the atom, from Democritus, and of energy, from Aristotle, defining energy as the absolute indetermination of matter, what we understand as non-mass matter, and bodies as the absolute determination of matter, what we understand as mass matter. Let us remember that Epicurus is the first absolute physicist, hence two important features arise, that the perceived bodies are material and that the energy, which causes movement in them, is also material.
The importance of this thesis, Epicurean, is immeasurable in the history of physics, because it resolves the problems of the theses presented before it, and subsequently has influence on physics, especially since the centuries and, thanks to the rediscovery of Poggio Bracciolini and Pierre Gassendi of the works of Epicurus. A clear example of influence is in Newton, who in fact distorted the theory, thus leading to errors in his law of universal gravitation, a clear error is the foundation he gives to movement in gravity, analogically compared to the mechanistic determinism of Democritus. Those who definitively confirmed, with their works, Epicurus's thesis were Max Planck and Albert Einstein, after twenty-one centuries of doubt about Epicurus's thesis.
• - Lucretius: to avoid mechanistic determinism, already criticized by Aristotle, he takes the thought of Epicurus and introduces the thesis that atoms fall into a vacuum and experience a decline by themselves that allows them to find themselves. In this way it is about imposing a certain order to the original idea that assumed that things were formed with a chaotic movement of atoms.
• - The great Greek philosopher Aristotle (384 BC-322 BC) proposed explanations for what was happening in nature, considering the observations he made of everyday experiences and his reasoning, although he did not worry about verifying his statements.
Aristotle formulated his theory about the fall of bodies stating that the heavier ones fell faster than the lighter ones, that is, the more weight the bodies have, the faster they fall.
This theory was accepted for almost two thousand years until in the century Galileo carried out a more careful study on the movement of bodies and their fall, about which he stated: "any velocity, once imparted to a body, will be maintained constantly, as long as there are no causes of acceleration or retardation, a phenomenon that will be observed in horizontal planes where friction has been reduced to a minimum." This statement carries with it Galileo's principle of inertia which briefly says: "If no force is exerted on a body, it will remain at rest or move in a straight line with constant speed."
He was studying the movements of various objects on an inclined plane and observed that in the case of planes with a downward slope there is a cause of acceleration, while in planes with an upward slope there is a cause of retardation. From this experience he reasoned that when the slopes of the planes are neither downward nor upward there should be no acceleration or retardation, which is why he came to the conclusion that when the movement is along a horizontal plane it must be permanent. Galileo did a study to verify what Aristotle had said about the fall of bodies. To do so, he climbed to the top of the Tower of Pisa and dropped two objects of different weight; and he observed that bodies fall at the same speed regardless of their weight, thus ruling out Aristotle's theory of the fall of bodies.
Movement according to classical mechanics
Classical mechanics is a formulation of mechanics to describe through laws the behavior of macroscopic physical bodies at rest and at small speeds compared to the speed of light. Starting with Galileo, scientists began to develop analysis techniques that allowed a quantifiable description of the phenomenon.
In classical mechanics, the trajectory is the geometric locus of the successive positions through which a body passes in its movement. The trajectory depends on the reference system in which the movement is described; that is, the observer's point of view. The description of the motion of point particles or corpuscles (whose internal structure is not required to describe the general position of the particle) is similar in classical mechanics and relativistic mechanics. In both, the movement trajectory is a curve parameterized by a scalar parameter. In the description of classical mechanics the parameter is universal time, while in relativity the relativistic interval is used since the proper time perceived by the particle and the time measured by different observers do not coincide.
In classical mechanics it is perfectly possible to univocally define the length L of the trajectory or path traveled by a body. The distance d between a starting point and the end of its trajectory can also be defined unambiguously; It is represented by the length of the straight line that joins the start point to the end point. Both magnitudes are related by the following inequality:
There are several different formulations of classical mechanics to describe the same natural phenomenon, which regardless of the formal and methodological aspects they use, reach the same conclusion.
• - Vector mechanics comes directly from Newton's laws, which is why it is also known as Newtonian. It is applicable to bodies that move relative to an observer at speeds small compared to the speed of light. It was originally built for a single particle moving in a gravitational field. It is based on the treatment of two vector magnitudes under a causal relationship: the force and the action of the force, measured by the variation of the momentum (amount of movement). The analysis and synthesis of forces and moments constitutes the basic method of vector mechanics. It requires the privileged use of inertial reference systems.
• - Analytical mechanics (analytical in the mathematical and not philosophical sense of the word). His methods are powerful and transcend Mechanics to other fields of physics. The germ of analytical mechanics can be found in the work of Leibniz who proposes other basic magnitudes to solve mechanical problems (less obscure according to Leibniz than Newton's force and moment), but now scalars "Scalar (physics)"), which are: kinetic energy and work "Work (physics)"). These magnitudes are differentially related. The essential characteristic is that, in the formulation, first general principles (differentials and integrals) are taken as foundations, and that the equations of motion are obtained analytically from these principles.
Equations of motion in classical mechanics
Historically, the first example of the equation of motion that was introduced in physics was Newton's second law for physical systems composed of aggregates of point material particles. In these systems, the dynamic state of a system was set by the position and speed of all the particles at a given instant. Towards the end of the century, analytical or rational mechanics was introduced, as a generalization of Newton's laws applicable to inertial reference systems. Two basically equivalent approaches known as Lagrangian mechanics and Hamiltonian mechanics were conceived, which can reach a high degree of abstraction and formalization. The best-known classic examples of the equation of motion are:
Newton's second law used in Newtonian mechanics:.
The Euler-Lagrange equations that appear in Lagrangian mechanics:.
Hamilton's equations that appear in Hamiltonian mechanics:
Historically, the concept of momentum arose in the context of Newtonian mechanics in close relationship with the concept of velocity and mass. In Newtonian mechanics, the quantity of linear motion is defined as the product of mass and velocity:.
The intuitive idea behind this definition is that the "momentum of motion" depended on both mass and speed: if you imagine a fly and a truck, both moving at 40 km/h, everyday experience says that the fly is easy to stop with your hand while the truck is not, even if they are both going at the same speed. This intuition led to defining a magnitude that was proportional to both the mass of the moving object and its speed.
Lagrangian and Hamiltonian mechanics
In the most abstract formulations of classical mechanics, such as Lagrangian mechanics and Hamiltonian mechanics, in addition to linear momentum and angular momentum, other moments can be defined, called generalized moments or conjugate moments, associated with any type of generalized coordinate. The notion of moment is thus generalized.
If we have a mechanical system defined by its Lagrangian L defined in terms of the generalized coordinates (q,q,...,q) and the generalized velocities, then the conjugate moment of the coordinate q is given by:.
When the coordinate q is one of the coordinates of a Cartesian coordinate system, the conjugate moment coincides with one of the components of the linear momentum, and, when the generalized coordinate represents an angular coordinate or the measure of an angle, the corresponding conjugate moment turns out to be one of the components of the angular momentum.
Movement according to Relativistic Mechanics
To describe the position of a material particle, relativistic mechanics makes use of a system of four coordinates defined over a four-dimensional space-time. The movement of a material particle is given by a curve in a 4-Lorentzian manifold, whose tangent vector is of a temporal type. Furthermore, instantaneous actions at a distance are excluded since by propagating faster than the speed of light they give rise to contractions in the principle of causality. Therefore, a system of point particles in interaction must be described with the help of "delayed fields", that is, those that do not act instantaneously, whose variation must be determined as propagation from the position of the particle. This reasonably complicates the number of equations necessary to describe a set of interacting particles.
Another added difficulty is that there is no universal time for all observers, so relating the measurements of different observers in relative motion is slightly more complex than in classical mechanics. A convenient way is to define the relativistic invariant interval and parameterize the trajectories in space-time as a function of said parameter. The description of force or fluid fields requires defining certain tensor magnitudes on the vector space tangent to space-time.
In relativistic mechanics, the trajectory is the geometric locus of the successive positions through which a body passes in its movement. The trajectory depends on the reference system in which the movement is described; that is, the observer's point of view. The description of the motion of point particles or corpuscles (whose internal structure is not required to describe the general position of the particle) is similar in classical mechanics and relativistic mechanics. In both, the movement trajectory is a curve parameterized by a scalar parameter. In the description of classical mechanics the parameter is universal time, while in relativity the relativistic interval is used since the proper time perceived by the particle and the time measured by different observers do not coincide.
Movement according to quantum mechanics
Quantum mechanics[5][6] is one of the main branches of physics, and one of the greatest advances of the century for human knowledge, which explains the behavior of matter and energy. The quantum description of movement is more complex since the quantum description of movement does not necessarily assume that the particles follow a classical type trajectory (some interpretations of quantum mechanics do assume that there is a single trajectory, but other formulations completely dispense with the concept of trajectory), so in these formulations it does not make sense to talk about position or speed.
The application of quantum mechanics has made possible the discovery and development of many technologies, such as transistors that are used more than anything else in computing. Likewise, quantum mechanics accounted for the properties of the atomic structure and many other problems for which classical mechanics gives completely incorrect predictions. The quantum mechanical description of particles completely abandons the notion of trajectory, since due to the uncertainty principle there cannot exist a conventional quantum state where position and momentum have perfectly defined values. Instead, the fundamental object in the quantum description of particles is not states defined by position and momentum, that is, a point in a phase space, but rather distributions over a phase space. These distributions can be provided with a Hilbert space structure.
Quantum mechanics as it was originally formulated did not incorporate the theory of relativity in its formalism, which initially could only be taken into account through perturbation theory.[7] The part of quantum mechanics that does incorporate relativistic elements in a formal way and with various problems, is relativistic quantum mechanics or, more accurately and powerfully, quantum field theory (which in turn includes quantum electrodynamics, quantum chromodynamics and electroweak theory within the standard model)[8] and more generally, quantum field theory in curved space-time. The only interaction that could not be quantified was the gravitational interaction.
Quantum mechanics is the basis of studies of the atom, nuclei and elementary particles (the relativistic treatment being already necessary), but also in information theory, cryptography and chemistry.
Translation movements
Para un cuerpo clásico (y, por tanto, moviéndose en un espacio euclídeo), una traslación es la operación que modifica las posiciones de todos los cuerpos según la fórmula:.
donde es un vector constante. Dicha operación puede ser generalizada a otras coordenadas, por ejemplo la coordenada temporal. Obviamente una traslación matemática es una isometría del espacio euclídeo.
En cinemática clásica se utiliza un sistema de coordenadas para describir las trayectorias de traslación, denominado sistema de referencia. El estudio de la cinemática usualmente empieza con la consideración de casos particulares de movimientos de traslación con características particulares. Usualmente se empieza el estudio cinemático considerando el movimiento de un móvil puntual o cuerpo sólido cuya estructura y propiedades internas pueden ignorarse para explicar su movimiento global. Entre los movimientos típicos que puede ejecutar un móvil puntual son particularmente interesantes los siguientes:.
rectilinear movements
A movement is rectilinear when it describes a straight path. The rectilinear trajectory is defined when the normal acceleration component is equal to zero. Three particular cases of rectilinear motion are usually studied:
• - Uniform rectilinear movement. The mobile travels the trajectory at a constant speed, that is, with zero acceleration. This implies that the average speed between any two instants will always have the same value. Furthermore, the instantaneous and average speed of this movement will coincide. Dynamically, the mobile phone has constant momentum and kinetic energy, and according to Newton's first law, the resulting force is zero.
• - Uniformly accelerated rectilinear movement is that in which a mobile moves on a straight line with constant acceleration and collinear with the speed. This implies that in any time interval, the acceleration of the body will always have the same value, and that the net value of the resulting acceleration corresponds to the tangential acceleration, while the value of the normal acceleration is zero. For example, the free fall of a body, with constant acceleration due to gravity.
• - Simple harmonic motion is a particular case of a conservative periodic reciprocating rectilinear system, in which a body oscillates from one side to the other of its equilibrium position, in a given direction, and in equal time intervals. The mobile moves oscillating around the equilibrium position when it is separated from it and returns to the origin. Velocity and acceleration vary periodically. The mathematical model is:
where:.
Curvilinear movements
A movement is curvilinear when it describes a curved path. A curvilinear translation is generated when there is an acceleration component normal to the trajectory. When a body performs a uniform circular motion, the direction of the velocity vector changes at every instant. This variation is experienced by the linear vector, due to a force called centripetal, directed towards the center of the circle that gives rise to centripetal acceleration.
When a particle moves in a curvilinear path, even if it moves with a constant speed (for example the MCU), its speed changes direction, since this is a tangent vector to the path, and on curves said tangent is not constant.
Centripetal acceleration, unlike centrifugal acceleration, is caused by a real force required for any inertial observer to be able to account for how the trajectory of a particle that does not perform a rectilinear movement is curved.
• - Circular movement. Circular movement is one that is based on an axis of rotation and a constant radius: the trajectory will be a circle. If, in addition, the speed of rotation is constant, uniform circular motion is produced, which is a particular case of circular motion, with a fixed radius and reference angular velocity. In this case the vector speed is not constant, although the celerity (or speed module) can be constant. Centripetal acceleration constantly changes the direction of the tangential velocity and always remains perpendicular to it.
• - Pendulum movement. Pendular motion is a form of displacement that some physical systems present as a practical application of quasi-harmonic motion. There are several variants of pendulum movement: simple pendulum, torsion pendulum and physical pendulum.
The first three are of interest both in classical mechanics, as well as in relativistic mechanics and quantum mechanics. While parabolic motion and pendulum motion are of interest almost exclusively in classical mechanics. Simple harmonic motion is also interesting in quantum mechanics to approximate certain properties of solids at the atomic level.
• - Elliptical movement. Elliptical motion is a case of bounded motion in which a particle describes an elliptical path. There are various physical systems where this happens, including planetary motion in a Newtonian gravitational potential.
• - Spiral movements"). Those in which a circular movement and a normal speed or acceleration additional to the centripetal are combined. The classic example is the uniform spiral movement, in which the trajectory corresponds to an Archimedes spiral in which the punctual mobile moves at a constant speed on a straight line that rotates about a fixed point of origin at a constant angular speed.
• - Helical movement "Helix (geometry)") is one that presents a trajectory whose tangents form a constant angle (α), following a fixed direction in space.
• - Trochoid movements"). Those that describe a curved trajectory of the plane, determined by a fixed point of a circle called the generatrix, the same that rolls, tangentially, without slipping on a straight line called the directrix. The name cycloid is given to the curve described by a point on the circumference, when this wheel runs along a straight line without slipping:.
A brachistochronous curve or curve of the fastest descent, is the curve between two points that is traveled in the shortest time, by a body that begins at the initial point with zero speed, and that must move along the curve until reaching the second point, under the action of a constant force of gravity and assuming that there is no friction.[9].
A hypotrochoid, in geometry, is the plane curve that describes a point linked to a generating circle that rolls within a directing circle, tangentially, without sliding.
The epicycloid is the curve generated by the trajectory of a point belonging to a circle (generatrix) that rolls, without slipping, along the outside of another circle (directrix). It is a type of cycloidal roulette.
• - Parabolic movement. Parabolic motion is the motion performed by an object whose trajectory describes a parabola. It is generated when a mobile phone with a velocity of constant magnitude and direction has a net normal acceleration and the tangential acceleration is zero. In classical mechanics it corresponds to the ideal trajectory of a projectile that moves in a medium that offers no resistance to advancement and that is subject to a uniform gravitational field. It is also possible to demonstrate that it can be analyzed as the composition of two rectilinear motions, a uniform horizontal rectilinear motion and a uniformly accelerated vertical rectilinear motion.
• - Hyperbolic movement. Movement in which the mobile follows a trajectory in the shape of a hyperbola. In celestial mechanics, a mobile with a speed greater than that necessary to escape the gravitational attraction of a central body describes a hyperbolic trajectory.
In more technical terms, this can be expressed by the condition that the orbital eccentricity be greater than one. In a hyperbolic trajectory, the true anomaly is linked to the distance between the orbiting bodies () by the orbital equation"):.
• - Pursuit movement is one that describes an object (located at P) that is dragged by another (located at A), that remains at a constant distance d and that moves in a straight line. Its trajectory is a tractor.[10].
• - Wave motion or simple sinusoidal motion, is the movement that can be analyzed as the composition of two rectilinear motions, a horizontal uniform rectilinear motion and a simple vertical harmonic rectilinear motion. This movement occurs in a continuous medium in which a disturbance propagates from one particle to neighboring particles unless there is a net flow of mass, even when there is energy transport in the medium. It corresponds to the ideal trajectory of a body that moves in a medium that offers no resistance to progress and that is subject to a uniform gravitational field.
Rotary movements
Three-dimensional rotations are of special practical interest because they correspond to the geometry of the physical space in which we live (naturally as long as medium-scale regions are considered, since for large distances the geometry is not strictly Euclidean). In three dimensions it is convenient to distinguish between plane or rectangular rotations, which are those in which the rotated vector and the one that determines the axis of rotation form a right angle, and conical rotations, in which the angle between these vectors is not right. Plane rotations have a simpler mathematical treatment, as they can be reduced to the two-dimensional case described above, while conic rotations are much more complex and are generally treated as a combination of plane rotations (especially the Euler angles and the Euler-Rodrigues parameters).
When studying the motion of a rigid solid, it is convenient to decompose it into a translational motion plus a rotational motion:
To describe the translation we only need to calculate the resulting forces and apply Newton's laws as if they were material points.
On the other hand, the description of rotation is more complex, since we need some magnitude that accounts for how the mass is distributed around a certain point or axis of rotation (for example an axis that passes through the center of mass). This magnitude is the inertia tensor that characterizes the rotational inertia of the solid.
That rigid solid inertia tensor is defined as a second-order symmetric tensor such that the quadratic form constructed from the tensor and the angular velocity ω gives the kinetic energy of rotation, that is:.
Not only can kinetic energy be expressed simply in terms of the inertia tensor, if we rewrite expression (3) for angular momentum by introducing into it the definition of the inertia tensor, we have that this tensor is the linear application that relates angular velocity and angular momentum:
According to the mechanics of the rigid solid, in addition to rotation around its axis of symmetry, a gyroscope generally presents two main movements: precession and nutation.
In a gyroscope we must take into account that the change in the angular momentum of the wheel must occur in the direction of the moment of force acting on the wheel.
Wave and vibrational movement
Wave movement is the movement performed by an object whose trajectory describes an undulation.
A type of frequent wave motion, sound that involves propagation in the form of longitudinal elastic waves (whether audible or not), usually through a fluid (or other elastic medium) that is generating the vibratory motion of a body.
A complex harmonic motion is a linear superposition motion of simple harmonic motions. Although a simple harmonic motion is always periodic, a complex harmonic motion is not necessarily periodic, although it can be analyzed by Fourier harmonic analysis. A complex harmonic motion is periodic only if it is the combination of simple harmonic motions whose frequencies are all rational multiples of a base frequency. In classical mechanics, the trajectory of a two-dimensional complex harmonic motion is a Lissajous curve.
Movement record
Technology today offers us many ways to record the movement made by a body. Thus, to measure speed there is traffic radar whose operation is based on the Doppler effect. The tachymeter is an indicator of a vehicle's speed based on the frequency of rotation of the wheels. Walkers have pedometers that detect the characteristic vibrations of the step and, assuming a characteristic average distance for each step, allow the distance traveled to be calculated. The video, together with the computer analysis of the images, also allows the position and speed of the vehicles to be determined.
molecular motion
Molecular dynamics (MD) is a simulation technique in which atoms and molecules are allowed to interact for a period, allowing a visualization of the movement of the particles. It was originally conceived within theoretical physics, although today it is mainly used in biophysics and materials science. Its field of application ranges from catalytic surfaces to biological systems such as proteins. Although X-ray crystallography experiments allow us to take “static photographs” and the NMR technique gives us clues to molecular motion, no experiment is capable of accessing all the time scales involved. It is tempting, although not entirely correct, to describe DM as a “virtual microscope” with high spatial and temporal resolution.
In general, molecular systems are complex and consist of a large number of particles, so it would be impossible to find their properties analytically. To avoid this problem, the DM uses numerical methods. DM represents an intermediate point between experiments and theory. It can be understood as an experiment on the computer.
We know that matter is made up of particles in motion and interaction at least since Boltzmann's time in the 20th century. But many still imagine molecules as static models in a museum. Richard Feynman said in 1963 that "everything living things do can be understood through the leaps and contortions of atoms."
One of the most important contributions of molecular dynamics is to raise awareness that DNA and proteins are machines in motion. It is used to explore the relationship between structure, movement and function.
Molecular dynamics is a multidisciplinary field. Its laws and theories come from Mathematics, Physics and Chemistry. It uses algorithms from Computer Science and Information Theory. It allows materials and molecules to be understood not as rigid entities, but as animated bodies. It has also been called “numerical mechanical statistics” or “Laplace's view of Newtonian mechanics,” in the sense of predicting the future by animating the forces of nature.
To use this technique correctly, it is important to understand the approximations used and avoid falling into the conceptual error that we are simulating the real and exact behavior of a molecular system. The integration of the equations of motion are poorly conditioned, which generates cumulative numerical errors, which can be minimized by appropriately selecting the algorithms, but not completely eliminated. On the other hand, the interactions between particles are modeled with an approximate force field "Force field (physics)"), which may or may not be suitable depending on the problem we want to solve. In any case, molecular dynamics allows us to explore their representative behavior in phase space.
In DM, we must balance the computational cost and the reliability of the results. In classical DM, Newton's Equations are used, the computational cost of which is much lower than that of quantum mechanics. This is why many properties that may be of interest, such as bond formation or breaking, cannot be studied using this method since it does not contemplate excited states or reactivity.
There are hybrid methods called QM/MM(Quantum Mechanics/Molecular Mechanics) in which a reactive center is treated in a quantum way while the environment around it is treated in a classical way. The challenge in this type of methods results in precisely defining the interaction between the two ways of describing the system...
The result of a molecular dynamics simulation is the positions and velocities of each atom of the molecule, for each instant in discretized time. This is called trajectory.
Brownian motion is the random motion observed in particles found in a fluid medium (liquid or gas), as a result of collisions with the molecules of said fluid. This phenomenon is named in honor of the Scottish biologist and botanist Robert Brown. In 1827, while looking through a microscope at particles trapped in cavities within a grain of pollen in water, he noted that the particles moved through the liquid; but he was not able to determine the mechanisms that caused this movement. Atoms and molecules had been theorized as components of matter, and Albert Einstein published a paper in 1905 explaining in detail how the movement Brown had observed was the result of pollen being moved by individual water molecules. The direction of the atomic bombardment force is constantly changing, and at different times, the particle is hit more on one side than the other, leading to the random nature of the motion.
Brownian motion is among the simplest stochastic processes, and is akin to two other simpler and more complex stochastic processes: the random walk and Donsker's theorem).
• - Physics.
• - Quantity of movement.
• - Kinematics of the rigid solid.
• - Mechanics.
• - Thrust force.
• - Relative speed.
• - Centripetal acceleration.
• - A small part of this article corresponds to the information acquired by the encyclopedic book Nature Studies, Yaditzha Irausquin (2008)..
• - Physics – Physical Science Study Committee (1966). ISBN 978-0-669-97451-5.
[8] ↑ Halzen, Francis; D.Martin, Alan (1984). Universidad de Wisconsin, ed. Quarks and Lepons: An Introducory Course in Modern Particle Physics. Universidad de Durham (1ª edición). Canadá: Wiley. p. 396. ISBN QC793.5.Q2522H34 |isbn= incorrecto (ayuda).
[9] ↑ Hofmann: Historia de la matemática ISBN 968-18-6286-4.
[10] ↑ I. Bronshtein, K Semendiaev Manual para ingenieros y estudiantes Editorial Mir Moscú (1973).
Later, the study of the evolution over time of a physical system in relation to the causes that cause or preserve movement led to the development of dynamics. The most important dynamic principles are inertia, momentum, force, and mechanical energy.
The integration of kinematics and dynamics develops the general discipline called mechanics (Greek Μηχανική and Latin mechanica or 'art of building machines'), which is the branch of physics that studies and analyzes the movement and rest of bodies. Theoretical mechanics was, during the 17th, 18th, and early 20th centuries, the discipline of physics that reached greater mathematical abstraction and was a source of improvement in scientific knowledge of the world. Applied mechanics is usually related to engineering. Both points of view are partially justified since, although mechanics is the basis for most of the classical engineering sciences, it is not as empirical in nature as these and, on the other hand, due to its rigor and deductive reasoning, it is more similar to mathematics.
During the century the appearance of new physical facts, both the consideration of physical bodies moving at speeds close to the speed of light and the movement of subatomic particles, led to the formulation of more abstract theories such as relativistic mechanics and quantum mechanics that continued to be interested in the evolution over time of physical systems, although in a more abstract and general way than classical mechanics had done, whose objective was basically to quantify the change in position in space of particles over time and the agents responsible. of such changes.
Kinematic characteristics of movement
Mobile
Mobile is understood as a moving object whose kinematics and dynamics are to be studied. According to the mobile studied, different perspectives of the movement can be identified:
• - Point mobile: the mobile is reduced to a theoretical point to simplify its study. From a kinematic point of view, the only type of movement admissible for a point mobile is that in which a change in location is observed with respect to a reference coordinate system, defined as translational movement "Translation (physics)").
• - Rigid solid mobile: the mobile is a non-deformable three-dimensional object. The term "rigid" refers, from a mathematical idealization, to the fact that the distance between any two material points of the body remains unchanged over time.
The most general motion of a rigid solid mobile can be considered as the superposition of two basic types of motion:
• - Deformable solid mobile: the mobile is a deformable three-dimensional object, that is, there exists throughout the temporal evolution a state in which the distance between any two material points of the body can vary, which can be evidenced as a change in the size or shape of the mobile, called deformation. The deformation can be:
The global movement of a deformable solid mobile is decomposed into its translational, rotational and deformation movements.
• - Fluid: the mobile is described as a continuous medium without a defined shape, infinitely deformable, in which the displacements that a material point can reach within the fluid are not determined (this contrasts with deformable solids, where the displacements are much more limited). It presents an absence of shape memory, that is, it takes the shape of the container that contains it, without elastic recovery forces as in solid mobiles.
Moment and duration
Time is a physical quantity with which we measure the duration or separation of events. Time allows events to be ordered in sequences. Given two point events E and E, which occur respectively in two temporal coordinates t and t, and at different points in space P and P, all physical theories admit that these can satisfy one and only one of the following three conditions:[1].
It is possible for an observer to be present at event E, and then be at event E, and in that case it is stated that E is an event prior to E. Furthermore, if that happens, that observer will not be able to verify E2.
It is possible for an observer to be present at event E and then be at event E, and in that case it is stated that E is an event after E. Furthermore, if that happens, that observer will not be able to verify E1.
It is impossible, for a specific observer, to be simultaneously present in both events E and E.
For motion description purposes, a duration between time coordinates t and t can be defined as Δt. If said duration is infinitesimal (dt) it is called instant.
Position and displacement
Physical space is the place where material entities are found. Physical space is usually conceived of as having three linear dimensions, although modern physicists usually consider it, over time, as one part of an infinite four-dimensional continuum known as spacetime, which in the presence of matter is curved. The position of a mobile is defined as the state variable "State variable (dynamic system)") that provides a geometric description defined at a given instant dt with respect to a geometric location described by the viewer. Thus, an orthogonal, cylindrical or spherical coordinate system can be used to describe the position of a body.
Displacement is the vector that defines the position of a point or particle relative to an origin A with respect to a position B. The vector extends from the reference point to the final position. When talking about displacement in space, only the initial position and the final position matter, since the trajectory described is not important.
Path
The trajectory is the geometric locus of the successive positions through which a body passes in its movement. The trajectory depends on the reference system in which the movement is described, with respect to the type of mobile and the point of view of the observer.
The trajectory of a translational movement is defined by the parameterized equation of the curve described in geometric space by a point mobile or the center of mass of a solid mobile.
The trajectories of a rotation are described in terms of the Euler angles and surfaces of revolution.
The trajectory of a deformation is described by geometric measurements of deformation.
For fluids in motion, the analogue to the path is the type of flow: a laminar flow is the movement of a fluid when it is orderly, stratified, smooth. In a laminar flow the fluid moves in parallel sheets without intermingling and each fluid particle follows a smooth path, called a streamline. Turbulent flow is the movement of a fluid that occurs chaotically, in which the particles move disorderly and the trajectories of the particles form small periodic eddies (not coordinated).
Speed
In a generic way, a speed is defined as the rate of variation of a certain physical quantity with respect to time.
In the case of translational movements, the speed is a physical magnitude of vector character "Vector (physics)") that expresses the displacement "Displacement (vector)") of an object "Mobile (physics)") per unit of time.
In everyday language the words speed and speed are used interchangeably. In physics a distinction is made between them. Very simply, the difference is that velocity is the speed in a given direction. When we say that a car is traveling at 60 km/h we are indicating its speed. But by saying that a car is traveling at 60 km/h towards the north, we are specifying its speed. Speed describes how fast an object is moving; Velocity describes how fast you do it and in what direction.
The speed of movement at a given instant depends on the observer both in classical mechanics and in the theory of relativity. In quantum mechanics, the speed of a mobile phone, like its trajectory, does not have to be defined at a given instant, according to some interpretations of the theory. The Zitterbewegung phenomenon suggests that an electron could have a transverse oscillatory motion around its classical "path" (i.e. the path it should follow if the classical description were correct).
Speed or also called celerity is the relationship between the distance traveled and the time spent traveling it. A car, for example, travels a certain number of kilometers in an hour, which may be 110km/h. Speed is a measure of how fast a moving object (physics) moves. It is the rate of change "Action (physical)") at which the distance is traveled, since the expression rate of change indicates that we are dividing some quantity by time, therefore, speed is always measured in terms of a unit of distance divided by a unit of time.
In rotational movements, angular velocity is defined as the rate of variation between the angle rotated per unit of time and is designated by the Greek letter ω. Its unit in the International System is the radian per second (rad/s). It is used as a measure of rotation speed.
The strain rate is a magnitude that measures the change in strain with respect to time. For uniaxial problems it is simply the temporal derivative of the longitudinal deformation, while for three-dimensional problems or situations it is represented by a second rank tensor.
In periodic movements, frequency is additionally used, which is a magnitude that measures the number of repetitions per unit of time of any periodic phenomenon or event, such as rotations, oscillations and vibrations. Its unit is Hertz.
In the movement of a fluid, the volumetric flow is a kinematic variable that is defined as the volume of fluid that passes through a given surface in a given time. Given an area A, over which a fluid flows at an angle from the direction perpendicular to with a volumetric flow , the of flow can be defined as:.
Acceleration
In physics, the term acceleration is a vector magnitude that applies to both increases and decreases in speed in a unit of time. The term acceleration applies to both changes in speed and changes in direction.
In translational movements, the velocity vector v is tangent to the trajectory, while the acceleration vector a can be decomposed into two mutually perpendicular components (called intrinsic components): a tangential component a (in the direction of the tangent to the trajectory), called tangential acceleration, and a normal component a (in the direction of the main normal to the trajectory), called normal or centripetal acceleration (this last name because it is always directed towards the center of curvature).
If you travel through a curve at a constant speed of 50 km/h, the effects of acceleration will be felt as a tendency to lean towards the outside of the curve (inertia). You can travel the curve with a constant speed, but the speed is not constant since the direction changes every moment, therefore, the state of movement changes, that is, it is accelerating. For example, the brakes of a car can produce large retarding accelerations, that is, they can produce a large decrease per second in its speed. This is often called deceleration or negative acceleration.
Normal acceleration is a measure of the curvature of the trajectory; different observers in non-uniform motion with respect to them will observe different forces and accelerations and therefore different trajectories. If an inertial observer examines the trajectory of a particle moving in a straight line and with uniform speed (zero curvature trajectory), any other inertial observer will see the particle moving in a straight line and with uniform speed (although not the same straight line), in the case of arbitrary observers in accelerated motion between them the shapes of the trajectories can differ noticeably, since when the two observers measure completely different accelerations, the trajectory of the particle will curve in very different ways for each observer.
In rotational movements, the concept of angular acceleration is used as the rate of variation between angular velocity with respect to time.
Dynamic characteristics of movement
Contenido
Todas las teorías físicas del movimiento atribuyen al movimiento una serie de características o atributos dinámicos como:.
• - Inercia.
• - La cantidad de movimiento.
• - El sistema de fuerzas ejercidos sobre el móvil.
• - La energía mecánica.
En mecánica clásica y mecánica relativista todos ellos son valores numéricos medibles, mientras que en mecánica cuántica esas magnitudes son en general variables aleatorias para las que es posible predecir sus valores medios, pero no el valor exacto en todo momento.
Inertia
In physics, inertia (from the Latin inertĭa) is the property that bodies have of remaining in their state of relative rest or relative motion. Generally speaking, it is the resistance that matter opposes when modifying its state of motion, including changes in speed or direction of motion. As a consequence, a body maintains its state of relative rest or relative uniform rectilinear motion if there is no force that, acting on it, manages to change its state of motion.
In translational movements, the measure of inertia is provided by the mass. In a point mobile it is assumed that all the mass is concentrated at the point that describes the mobile, while in solid mobiles its translation can be simplified by describing a point called Center of mass in a manner analogous to a point mobile.
In rotational movements, the rotational moment of inertia (symbol I) is a measure of resistance to rotation of a body that reflects the distribution of mass of a body with respect to an axis of rotation. The moment of inertia only depends on the geometry of the body and the position of the axis of rotation.
On the other hand, the measures of resistance to deformation are mainly represented by the measures of stiffness, such as Hooke's constant.
In the movement of a fluid, the inertia of a flow is represented by both its density and its viscosity, its friction with the container and its adhesion to the walls.
Momentum
The quantity of motion, linear momentum, impetus or momentum is a fundamental physical magnitude of vector type that describes the motion of a body in any mechanical theory defined as the product of a unit of inertia and a rate of spatial variation with respect to time at a given instant.
According to the concept of momentum, mechanical rest is defined as the mechanical state in which for any given instant, any measurement of momentum is equal to zero.
According to the concept of momentum, uniform motion is defined as that where the momentum remains constant over time.
The vector magnitude defined as the variation in the momentum experienced by a physical object in a closed system is called impulse.
In translational movements of a single mobile, the quantity of linear movement (or linear momentum) is defined as the product of the mass with its linear velocity. The intuitive idea behind this definition is that the "momentum of motion" depended on both mass and speed: if you imagine a fly and a truck, both moving at 40 km/h, everyday experience says that the fly is easy to stop with your hand while the truck is not, even if they are both going at the same speed. This intuition led to defining a magnitude that was proportional to both the mass of the moving object and its speed.
In rotational movements, the angular momentum or kinetic moment is a physical quantity that is related to the vector product between the moment of inertia and the angular velocity. This magnitude plays a role analogous to the linear moment in translations with respect to rotations. The angular momentum for a rigid body rotating with respect to an axis is the resistance offered by said body to the variation of angular velocity. However, this does not imply that it is an exclusive magnitude of rotations; For example, the angular momentum of a particle moving freely with constant velocity (in magnitude and direction) is also conserved.
If we are interested in finding out the momentum of, for example, a fluid that moves according to a velocity field, it is necessary to add the momentum of each particle of the fluid, that is, of each mass differential or infinitesimal element:
In fluid motions, the conservation of the linear momentum of a fluid in motion is generalized by the Navier-Stokes equations.
Force
In physics, force is a physical quantity that measures the rate of variation of the exchange of momentum between two mobiles with respect to the duration of said exchange. According to a classical definition, force is any agent capable of modifying the momentum or shape of material bodies. In the International System of Units, force is measured in “Newtons (N)”.
When several forces act on a single mobile, all of them will be added vector-wise to constitute a single force called resultant force. In translational movements of point or solid mobiles, when the resulting force is zero and the momentum is zero, mechanical equilibrium will be observed as a rest. When the resultant force is zero and the momentum is constant with a value other than zero, a uniform rectilinear motion will be observed. If the resulting force is different from zero, it will be equivalent to the product of the instantaneous mass and the instantaneous acceleration. In terms of momentum, an impulse was applied. This is stated in Newton's first and second laws. Curvilinear translational movements involve the application of a normal force called centripetal force.
In rotational movements, the analogue of force is called the moment of a force (with respect to a given point) or torque, a magnitude obtained as a vector product of the position vector of the point of application of the force (with respect to the point at which the moment is taken) by the force vector, in that order. Thus, the sum of all the torques in a rotating system will lead to a resulting torque. If the resulting torque is zero with zero angular momentum, the body will not rotate. If the resulting torque is zero with constant non-zero angular momentum, uniform rotation or uniform circular motion will be observed. If the resulting torque is different from zero, a net angular acceleration will be seen and therefore the state of rotation will change. The Euler equations "Euler equations (solids)") describe the motion of a rotating rigid solid in a frame of reference with the solid.
In deformations, the analogue to forces are mechanical stresses, mechanical tensions and mechanical torsions. The dynamics of the deformations are described by:.
In fluid mechanics, the equivalent of force is pressure. In a fluid there can be the following types of pressure:
Energy
In physics, energy is defined as the ability to effect a transformation in a physical system, for example by lifting an object, transporting it (moving it), deforming it, or heating it. Energy is not a real physical state, nor an "intangible substance" but a scalar magnitude that is assigned to the state of the physical system, that is, energy is a tool or mathematical abstraction of a property of physical systems. For example, a system with zero kinetic energy can be said to be at rest. Energy is measured with the unit «joule "Joule (unit)") (J)».
In this way, every movement at a specific instant is assigned an amount of energy associated kinematically with its speed and dynamically with its momentum. This magnitude is called kinetic energy.
For any translational motion, its instantaneous kinetic energy is defined as a function of half the mass and the square of the magnitude of the instantaneous linear velocity: .
Thus, a body at rest has instantaneous kinetic energy equal to zero.
In a uniform rectilinear motion, the kinetic energy is constant for any given instant.
The work performed by a force is defined as the product of it by the path taken by its point of application and by the cosine of the angle they form with each other.[2] Work is a scalar physical quantity "Scalar (physics)") that is represented by the letter (from English Work) and is expressed in units of energy, this is in joules "Joule (unit)") or joules (J) in the International System of Units.
Mathematically, the work for a particle moving along a curve C is expressed as:.
For the case of a constant force the previous equation reduces to:.
Where is the mechanical work, is the magnitude of the force, is the displacement "Displacement (vector)") and is the angle between the force vector "Vector (Euclidean space)") and the displacement vector (see drawing).
When the force vector is perpendicular to the displacement vector of the body on which it is applied, said force does not do any work. Likewise, if there is no displacement, the work will also be null.
For any rotational motion, the rotational kinetic energy is described as a function of half the rotational moment of inertia and the square of the magnitude of the instantaneous angular velocity: . Thus, a rotation in uniform circular motion presents a constant value of rotational kinetic energy.
For harmonic movements and any type of deformations, the kinetic energy can be described as a consequence of the forces involved being central and, therefore, conservative. Consequently, a scalar field called potential energy (E) associated with the force can be defined. To find the expression of the potential energy, it is enough to integrate the expression of the force (this is extendable to all conservative forces) and change its sign, obtaining:
History of the physical concept of movement
Las cuestiones acerca de las causas del movimiento surgieron en la mente del hombre hace más de 25 siglos, pero las respuestas que hoy conocemos no se desarrollaron hasta los tiempos de Galileo Galilei (1564–1642) e Isaac Newton (1642–1727).
Movement studies in classical times
• - Anaximander thought that nature came from the separation, through eternal movement, of opposite elements (for example, cold-heat), which were locked in something called primordial matter.
• - Democritus said that nature is made up of indivisible pieces of matter called atoms, and that movement was their main characteristic, movement being a change of place in space.
• - Zeno's paradoxes are a series of paradoxes or aporias devised by Zeno of Elea. Devoted mainly to the problem of the continuum and the relationships between space "Space (physics)"), time and movement, Zeno would have posed - according to Proclus - a total of 40 paradoxes, of which nine or ten complete descriptions have been preserved (in Aristotle's Physics "Physics (Aristotle)")[3][4] and Simplicio's commentary on this work).
• - Aristotle rejects the task of returning to the concept of the atom, from Democritus, and of energy, from Aristotle, defining energy as the absolute indetermination of matter, what we understand as non-mass matter, and bodies as the absolute determination of matter, what we understand as mass matter. Let us remember that Epicurus is the first absolute physicist, hence two important features arise, that the perceived bodies are material and that the energy, which causes movement in them, is also material.
The importance of this thesis, Epicurean, is immeasurable in the history of physics, because it resolves the problems of the theses presented before it, and subsequently has influence on physics, especially since the centuries and, thanks to the rediscovery of Poggio Bracciolini and Pierre Gassendi of the works of Epicurus. A clear example of influence is in Newton, who in fact distorted the theory, thus leading to errors in his law of universal gravitation, a clear error is the foundation he gives to movement in gravity, analogically compared to the mechanistic determinism of Democritus. Those who definitively confirmed, with their works, Epicurus's thesis were Max Planck and Albert Einstein, after twenty-one centuries of doubt about Epicurus's thesis.
• - Lucretius: to avoid mechanistic determinism, already criticized by Aristotle, he takes the thought of Epicurus and introduces the thesis that atoms fall into a vacuum and experience a decline by themselves that allows them to find themselves. In this way it is about imposing a certain order to the original idea that assumed that things were formed with a chaotic movement of atoms.
• - The great Greek philosopher Aristotle (384 BC-322 BC) proposed explanations for what was happening in nature, considering the observations he made of everyday experiences and his reasoning, although he did not worry about verifying his statements.
Aristotle formulated his theory about the fall of bodies stating that the heavier ones fell faster than the lighter ones, that is, the more weight the bodies have, the faster they fall.
This theory was accepted for almost two thousand years until in the century Galileo carried out a more careful study on the movement of bodies and their fall, about which he stated: "any velocity, once imparted to a body, will be maintained constantly, as long as there are no causes of acceleration or retardation, a phenomenon that will be observed in horizontal planes where friction has been reduced to a minimum." This statement carries with it Galileo's principle of inertia which briefly says: "If no force is exerted on a body, it will remain at rest or move in a straight line with constant speed."
He was studying the movements of various objects on an inclined plane and observed that in the case of planes with a downward slope there is a cause of acceleration, while in planes with an upward slope there is a cause of retardation. From this experience he reasoned that when the slopes of the planes are neither downward nor upward there should be no acceleration or retardation, which is why he came to the conclusion that when the movement is along a horizontal plane it must be permanent. Galileo did a study to verify what Aristotle had said about the fall of bodies. To do so, he climbed to the top of the Tower of Pisa and dropped two objects of different weight; and he observed that bodies fall at the same speed regardless of their weight, thus ruling out Aristotle's theory of the fall of bodies.
Movement according to classical mechanics
Classical mechanics is a formulation of mechanics to describe through laws the behavior of macroscopic physical bodies at rest and at small speeds compared to the speed of light. Starting with Galileo, scientists began to develop analysis techniques that allowed a quantifiable description of the phenomenon.
In classical mechanics, the trajectory is the geometric locus of the successive positions through which a body passes in its movement. The trajectory depends on the reference system in which the movement is described; that is, the observer's point of view. The description of the motion of point particles or corpuscles (whose internal structure is not required to describe the general position of the particle) is similar in classical mechanics and relativistic mechanics. In both, the movement trajectory is a curve parameterized by a scalar parameter. In the description of classical mechanics the parameter is universal time, while in relativity the relativistic interval is used since the proper time perceived by the particle and the time measured by different observers do not coincide.
In classical mechanics it is perfectly possible to univocally define the length L of the trajectory or path traveled by a body. The distance d between a starting point and the end of its trajectory can also be defined unambiguously; It is represented by the length of the straight line that joins the start point to the end point. Both magnitudes are related by the following inequality:
There are several different formulations of classical mechanics to describe the same natural phenomenon, which regardless of the formal and methodological aspects they use, reach the same conclusion.
• - Vector mechanics comes directly from Newton's laws, which is why it is also known as Newtonian. It is applicable to bodies that move relative to an observer at speeds small compared to the speed of light. It was originally built for a single particle moving in a gravitational field. It is based on the treatment of two vector magnitudes under a causal relationship: the force and the action of the force, measured by the variation of the momentum (amount of movement). The analysis and synthesis of forces and moments constitutes the basic method of vector mechanics. It requires the privileged use of inertial reference systems.
• - Analytical mechanics (analytical in the mathematical and not philosophical sense of the word). His methods are powerful and transcend Mechanics to other fields of physics. The germ of analytical mechanics can be found in the work of Leibniz who proposes other basic magnitudes to solve mechanical problems (less obscure according to Leibniz than Newton's force and moment), but now scalars "Scalar (physics)"), which are: kinetic energy and work "Work (physics)"). These magnitudes are differentially related. The essential characteristic is that, in the formulation, first general principles (differentials and integrals) are taken as foundations, and that the equations of motion are obtained analytically from these principles.
Equations of motion in classical mechanics
Historically, the first example of the equation of motion that was introduced in physics was Newton's second law for physical systems composed of aggregates of point material particles. In these systems, the dynamic state of a system was set by the position and speed of all the particles at a given instant. Towards the end of the century, analytical or rational mechanics was introduced, as a generalization of Newton's laws applicable to inertial reference systems. Two basically equivalent approaches known as Lagrangian mechanics and Hamiltonian mechanics were conceived, which can reach a high degree of abstraction and formalization. The best-known classic examples of the equation of motion are:
Newton's second law used in Newtonian mechanics:.
The Euler-Lagrange equations that appear in Lagrangian mechanics:.
Hamilton's equations that appear in Hamiltonian mechanics:
Historically, the concept of momentum arose in the context of Newtonian mechanics in close relationship with the concept of velocity and mass. In Newtonian mechanics, the quantity of linear motion is defined as the product of mass and velocity:.
The intuitive idea behind this definition is that the "momentum of motion" depended on both mass and speed: if you imagine a fly and a truck, both moving at 40 km/h, everyday experience says that the fly is easy to stop with your hand while the truck is not, even if they are both going at the same speed. This intuition led to defining a magnitude that was proportional to both the mass of the moving object and its speed.
Lagrangian and Hamiltonian mechanics
In the most abstract formulations of classical mechanics, such as Lagrangian mechanics and Hamiltonian mechanics, in addition to linear momentum and angular momentum, other moments can be defined, called generalized moments or conjugate moments, associated with any type of generalized coordinate. The notion of moment is thus generalized.
If we have a mechanical system defined by its Lagrangian L defined in terms of the generalized coordinates (q,q,...,q) and the generalized velocities, then the conjugate moment of the coordinate q is given by:.
When the coordinate q is one of the coordinates of a Cartesian coordinate system, the conjugate moment coincides with one of the components of the linear momentum, and, when the generalized coordinate represents an angular coordinate or the measure of an angle, the corresponding conjugate moment turns out to be one of the components of the angular momentum.
Movement according to Relativistic Mechanics
To describe the position of a material particle, relativistic mechanics makes use of a system of four coordinates defined over a four-dimensional space-time. The movement of a material particle is given by a curve in a 4-Lorentzian manifold, whose tangent vector is of a temporal type. Furthermore, instantaneous actions at a distance are excluded since by propagating faster than the speed of light they give rise to contractions in the principle of causality. Therefore, a system of point particles in interaction must be described with the help of "delayed fields", that is, those that do not act instantaneously, whose variation must be determined as propagation from the position of the particle. This reasonably complicates the number of equations necessary to describe a set of interacting particles.
Another added difficulty is that there is no universal time for all observers, so relating the measurements of different observers in relative motion is slightly more complex than in classical mechanics. A convenient way is to define the relativistic invariant interval and parameterize the trajectories in space-time as a function of said parameter. The description of force or fluid fields requires defining certain tensor magnitudes on the vector space tangent to space-time.
In relativistic mechanics, the trajectory is the geometric locus of the successive positions through which a body passes in its movement. The trajectory depends on the reference system in which the movement is described; that is, the observer's point of view. The description of the motion of point particles or corpuscles (whose internal structure is not required to describe the general position of the particle) is similar in classical mechanics and relativistic mechanics. In both, the movement trajectory is a curve parameterized by a scalar parameter. In the description of classical mechanics the parameter is universal time, while in relativity the relativistic interval is used since the proper time perceived by the particle and the time measured by different observers do not coincide.
Movement according to quantum mechanics
Quantum mechanics[5][6] is one of the main branches of physics, and one of the greatest advances of the century for human knowledge, which explains the behavior of matter and energy. The quantum description of movement is more complex since the quantum description of movement does not necessarily assume that the particles follow a classical type trajectory (some interpretations of quantum mechanics do assume that there is a single trajectory, but other formulations completely dispense with the concept of trajectory), so in these formulations it does not make sense to talk about position or speed.
The application of quantum mechanics has made possible the discovery and development of many technologies, such as transistors that are used more than anything else in computing. Likewise, quantum mechanics accounted for the properties of the atomic structure and many other problems for which classical mechanics gives completely incorrect predictions. The quantum mechanical description of particles completely abandons the notion of trajectory, since due to the uncertainty principle there cannot exist a conventional quantum state where position and momentum have perfectly defined values. Instead, the fundamental object in the quantum description of particles is not states defined by position and momentum, that is, a point in a phase space, but rather distributions over a phase space. These distributions can be provided with a Hilbert space structure.
Quantum mechanics as it was originally formulated did not incorporate the theory of relativity in its formalism, which initially could only be taken into account through perturbation theory.[7] The part of quantum mechanics that does incorporate relativistic elements in a formal way and with various problems, is relativistic quantum mechanics or, more accurately and powerfully, quantum field theory (which in turn includes quantum electrodynamics, quantum chromodynamics and electroweak theory within the standard model)[8] and more generally, quantum field theory in curved space-time. The only interaction that could not be quantified was the gravitational interaction.
Quantum mechanics is the basis of studies of the atom, nuclei and elementary particles (the relativistic treatment being already necessary), but also in information theory, cryptography and chemistry.
Translation movements
Para un cuerpo clásico (y, por tanto, moviéndose en un espacio euclídeo), una traslación es la operación que modifica las posiciones de todos los cuerpos según la fórmula:.
donde es un vector constante. Dicha operación puede ser generalizada a otras coordenadas, por ejemplo la coordenada temporal. Obviamente una traslación matemática es una isometría del espacio euclídeo.
En cinemática clásica se utiliza un sistema de coordenadas para describir las trayectorias de traslación, denominado sistema de referencia. El estudio de la cinemática usualmente empieza con la consideración de casos particulares de movimientos de traslación con características particulares. Usualmente se empieza el estudio cinemático considerando el movimiento de un móvil puntual o cuerpo sólido cuya estructura y propiedades internas pueden ignorarse para explicar su movimiento global. Entre los movimientos típicos que puede ejecutar un móvil puntual son particularmente interesantes los siguientes:.
rectilinear movements
A movement is rectilinear when it describes a straight path. The rectilinear trajectory is defined when the normal acceleration component is equal to zero. Three particular cases of rectilinear motion are usually studied:
• - Uniform rectilinear movement. The mobile travels the trajectory at a constant speed, that is, with zero acceleration. This implies that the average speed between any two instants will always have the same value. Furthermore, the instantaneous and average speed of this movement will coincide. Dynamically, the mobile phone has constant momentum and kinetic energy, and according to Newton's first law, the resulting force is zero.
• - Uniformly accelerated rectilinear movement is that in which a mobile moves on a straight line with constant acceleration and collinear with the speed. This implies that in any time interval, the acceleration of the body will always have the same value, and that the net value of the resulting acceleration corresponds to the tangential acceleration, while the value of the normal acceleration is zero. For example, the free fall of a body, with constant acceleration due to gravity.
• - Simple harmonic motion is a particular case of a conservative periodic reciprocating rectilinear system, in which a body oscillates from one side to the other of its equilibrium position, in a given direction, and in equal time intervals. The mobile moves oscillating around the equilibrium position when it is separated from it and returns to the origin. Velocity and acceleration vary periodically. The mathematical model is:
where:.
Curvilinear movements
A movement is curvilinear when it describes a curved path. A curvilinear translation is generated when there is an acceleration component normal to the trajectory. When a body performs a uniform circular motion, the direction of the velocity vector changes at every instant. This variation is experienced by the linear vector, due to a force called centripetal, directed towards the center of the circle that gives rise to centripetal acceleration.
When a particle moves in a curvilinear path, even if it moves with a constant speed (for example the MCU), its speed changes direction, since this is a tangent vector to the path, and on curves said tangent is not constant.
Centripetal acceleration, unlike centrifugal acceleration, is caused by a real force required for any inertial observer to be able to account for how the trajectory of a particle that does not perform a rectilinear movement is curved.
• - Circular movement. Circular movement is one that is based on an axis of rotation and a constant radius: the trajectory will be a circle. If, in addition, the speed of rotation is constant, uniform circular motion is produced, which is a particular case of circular motion, with a fixed radius and reference angular velocity. In this case the vector speed is not constant, although the celerity (or speed module) can be constant. Centripetal acceleration constantly changes the direction of the tangential velocity and always remains perpendicular to it.
• - Pendulum movement. Pendular motion is a form of displacement that some physical systems present as a practical application of quasi-harmonic motion. There are several variants of pendulum movement: simple pendulum, torsion pendulum and physical pendulum.
The first three are of interest both in classical mechanics, as well as in relativistic mechanics and quantum mechanics. While parabolic motion and pendulum motion are of interest almost exclusively in classical mechanics. Simple harmonic motion is also interesting in quantum mechanics to approximate certain properties of solids at the atomic level.
• - Elliptical movement. Elliptical motion is a case of bounded motion in which a particle describes an elliptical path. There are various physical systems where this happens, including planetary motion in a Newtonian gravitational potential.
• - Spiral movements"). Those in which a circular movement and a normal speed or acceleration additional to the centripetal are combined. The classic example is the uniform spiral movement, in which the trajectory corresponds to an Archimedes spiral in which the punctual mobile moves at a constant speed on a straight line that rotates about a fixed point of origin at a constant angular speed.
• - Helical movement "Helix (geometry)") is one that presents a trajectory whose tangents form a constant angle (α), following a fixed direction in space.
• - Trochoid movements"). Those that describe a curved trajectory of the plane, determined by a fixed point of a circle called the generatrix, the same that rolls, tangentially, without slipping on a straight line called the directrix. The name cycloid is given to the curve described by a point on the circumference, when this wheel runs along a straight line without slipping:.
A brachistochronous curve or curve of the fastest descent, is the curve between two points that is traveled in the shortest time, by a body that begins at the initial point with zero speed, and that must move along the curve until reaching the second point, under the action of a constant force of gravity and assuming that there is no friction.[9].
A hypotrochoid, in geometry, is the plane curve that describes a point linked to a generating circle that rolls within a directing circle, tangentially, without sliding.
The epicycloid is the curve generated by the trajectory of a point belonging to a circle (generatrix) that rolls, without slipping, along the outside of another circle (directrix). It is a type of cycloidal roulette.
• - Parabolic movement. Parabolic motion is the motion performed by an object whose trajectory describes a parabola. It is generated when a mobile phone with a velocity of constant magnitude and direction has a net normal acceleration and the tangential acceleration is zero. In classical mechanics it corresponds to the ideal trajectory of a projectile that moves in a medium that offers no resistance to advancement and that is subject to a uniform gravitational field. It is also possible to demonstrate that it can be analyzed as the composition of two rectilinear motions, a uniform horizontal rectilinear motion and a uniformly accelerated vertical rectilinear motion.
• - Hyperbolic movement. Movement in which the mobile follows a trajectory in the shape of a hyperbola. In celestial mechanics, a mobile with a speed greater than that necessary to escape the gravitational attraction of a central body describes a hyperbolic trajectory.
In more technical terms, this can be expressed by the condition that the orbital eccentricity be greater than one. In a hyperbolic trajectory, the true anomaly is linked to the distance between the orbiting bodies () by the orbital equation"):.
• - Pursuit movement is one that describes an object (located at P) that is dragged by another (located at A), that remains at a constant distance d and that moves in a straight line. Its trajectory is a tractor.[10].
• - Wave motion or simple sinusoidal motion, is the movement that can be analyzed as the composition of two rectilinear motions, a horizontal uniform rectilinear motion and a simple vertical harmonic rectilinear motion. This movement occurs in a continuous medium in which a disturbance propagates from one particle to neighboring particles unless there is a net flow of mass, even when there is energy transport in the medium. It corresponds to the ideal trajectory of a body that moves in a medium that offers no resistance to progress and that is subject to a uniform gravitational field.
Rotary movements
Three-dimensional rotations are of special practical interest because they correspond to the geometry of the physical space in which we live (naturally as long as medium-scale regions are considered, since for large distances the geometry is not strictly Euclidean). In three dimensions it is convenient to distinguish between plane or rectangular rotations, which are those in which the rotated vector and the one that determines the axis of rotation form a right angle, and conical rotations, in which the angle between these vectors is not right. Plane rotations have a simpler mathematical treatment, as they can be reduced to the two-dimensional case described above, while conic rotations are much more complex and are generally treated as a combination of plane rotations (especially the Euler angles and the Euler-Rodrigues parameters).
When studying the motion of a rigid solid, it is convenient to decompose it into a translational motion plus a rotational motion:
To describe the translation we only need to calculate the resulting forces and apply Newton's laws as if they were material points.
On the other hand, the description of rotation is more complex, since we need some magnitude that accounts for how the mass is distributed around a certain point or axis of rotation (for example an axis that passes through the center of mass). This magnitude is the inertia tensor that characterizes the rotational inertia of the solid.
That rigid solid inertia tensor is defined as a second-order symmetric tensor such that the quadratic form constructed from the tensor and the angular velocity ω gives the kinetic energy of rotation, that is:.
Not only can kinetic energy be expressed simply in terms of the inertia tensor, if we rewrite expression (3) for angular momentum by introducing into it the definition of the inertia tensor, we have that this tensor is the linear application that relates angular velocity and angular momentum:
According to the mechanics of the rigid solid, in addition to rotation around its axis of symmetry, a gyroscope generally presents two main movements: precession and nutation.
In a gyroscope we must take into account that the change in the angular momentum of the wheel must occur in the direction of the moment of force acting on the wheel.
Wave and vibrational movement
Wave movement is the movement performed by an object whose trajectory describes an undulation.
A type of frequent wave motion, sound that involves propagation in the form of longitudinal elastic waves (whether audible or not), usually through a fluid (or other elastic medium) that is generating the vibratory motion of a body.
A complex harmonic motion is a linear superposition motion of simple harmonic motions. Although a simple harmonic motion is always periodic, a complex harmonic motion is not necessarily periodic, although it can be analyzed by Fourier harmonic analysis. A complex harmonic motion is periodic only if it is the combination of simple harmonic motions whose frequencies are all rational multiples of a base frequency. In classical mechanics, the trajectory of a two-dimensional complex harmonic motion is a Lissajous curve.
Movement record
Technology today offers us many ways to record the movement made by a body. Thus, to measure speed there is traffic radar whose operation is based on the Doppler effect. The tachymeter is an indicator of a vehicle's speed based on the frequency of rotation of the wheels. Walkers have pedometers that detect the characteristic vibrations of the step and, assuming a characteristic average distance for each step, allow the distance traveled to be calculated. The video, together with the computer analysis of the images, also allows the position and speed of the vehicles to be determined.
molecular motion
Molecular dynamics (MD) is a simulation technique in which atoms and molecules are allowed to interact for a period, allowing a visualization of the movement of the particles. It was originally conceived within theoretical physics, although today it is mainly used in biophysics and materials science. Its field of application ranges from catalytic surfaces to biological systems such as proteins. Although X-ray crystallography experiments allow us to take “static photographs” and the NMR technique gives us clues to molecular motion, no experiment is capable of accessing all the time scales involved. It is tempting, although not entirely correct, to describe DM as a “virtual microscope” with high spatial and temporal resolution.
In general, molecular systems are complex and consist of a large number of particles, so it would be impossible to find their properties analytically. To avoid this problem, the DM uses numerical methods. DM represents an intermediate point between experiments and theory. It can be understood as an experiment on the computer.
We know that matter is made up of particles in motion and interaction at least since Boltzmann's time in the 20th century. But many still imagine molecules as static models in a museum. Richard Feynman said in 1963 that "everything living things do can be understood through the leaps and contortions of atoms."
One of the most important contributions of molecular dynamics is to raise awareness that DNA and proteins are machines in motion. It is used to explore the relationship between structure, movement and function.
Molecular dynamics is a multidisciplinary field. Its laws and theories come from Mathematics, Physics and Chemistry. It uses algorithms from Computer Science and Information Theory. It allows materials and molecules to be understood not as rigid entities, but as animated bodies. It has also been called “numerical mechanical statistics” or “Laplace's view of Newtonian mechanics,” in the sense of predicting the future by animating the forces of nature.
To use this technique correctly, it is important to understand the approximations used and avoid falling into the conceptual error that we are simulating the real and exact behavior of a molecular system. The integration of the equations of motion are poorly conditioned, which generates cumulative numerical errors, which can be minimized by appropriately selecting the algorithms, but not completely eliminated. On the other hand, the interactions between particles are modeled with an approximate force field "Force field (physics)"), which may or may not be suitable depending on the problem we want to solve. In any case, molecular dynamics allows us to explore their representative behavior in phase space.
In DM, we must balance the computational cost and the reliability of the results. In classical DM, Newton's Equations are used, the computational cost of which is much lower than that of quantum mechanics. This is why many properties that may be of interest, such as bond formation or breaking, cannot be studied using this method since it does not contemplate excited states or reactivity.
There are hybrid methods called QM/MM(Quantum Mechanics/Molecular Mechanics) in which a reactive center is treated in a quantum way while the environment around it is treated in a classical way. The challenge in this type of methods results in precisely defining the interaction between the two ways of describing the system...
The result of a molecular dynamics simulation is the positions and velocities of each atom of the molecule, for each instant in discretized time. This is called trajectory.
Brownian motion is the random motion observed in particles found in a fluid medium (liquid or gas), as a result of collisions with the molecules of said fluid. This phenomenon is named in honor of the Scottish biologist and botanist Robert Brown. In 1827, while looking through a microscope at particles trapped in cavities within a grain of pollen in water, he noted that the particles moved through the liquid; but he was not able to determine the mechanisms that caused this movement. Atoms and molecules had been theorized as components of matter, and Albert Einstein published a paper in 1905 explaining in detail how the movement Brown had observed was the result of pollen being moved by individual water molecules. The direction of the atomic bombardment force is constantly changing, and at different times, the particle is hit more on one side than the other, leading to the random nature of the motion.
Brownian motion is among the simplest stochastic processes, and is akin to two other simpler and more complex stochastic processes: the random walk and Donsker's theorem).
• - Physics.
• - Quantity of movement.
• - Kinematics of the rigid solid.
• - Mechanics.
• - Thrust force.
• - Relative speed.
• - Centripetal acceleration.
• - A small part of this article corresponds to the information acquired by the encyclopedic book Nature Studies, Yaditzha Irausquin (2008)..
• - Physics – Physical Science Study Committee (1966). ISBN 978-0-669-97451-5.
[8] ↑ Halzen, Francis; D.Martin, Alan (1984). Universidad de Wisconsin, ed. Quarks and Lepons: An Introducory Course in Modern Particle Physics. Universidad de Durham (1ª edición). Canadá: Wiley. p. 396. ISBN QC793.5.Q2522H34 |isbn= incorrecto (ayuda).
[9] ↑ Hofmann: Historia de la matemática ISBN 968-18-6286-4.
[10] ↑ I. Bronshtein, K Semendiaev Manual para ingenieros y estudiantes Editorial Mir Moscú (1973).
A
Q
velocity
The potential energy reaches its maximum at the ends of the trajectory and has a null value (zero) at the point x = 0, that is, the equilibrium point.
The kinetic energy will change throughout the oscillations as does the speed:
The kinetic energy is zero at -A or +A (v=0) and the maximum value is reached at the equilibrium point (maximum speed Aω).
In the case of fluids, Bernoulli's principle states that the total energy of a fluid at any time consists of three components:.
One way to represent the equation of Bernoulli's principle is expressed as the sum of the kinetic energy, the flow energy, and the gravitational potential energy per unit mass:
In a streamline each type of energy can increase or decrease by virtue of the decrease or increase of the other two. Although Bernoulli's principle can be seen as another form of the law of conservation of energy, it is actually derived from the conservation of momentum.
Classical kinematics deals with the study of the movement of bodies in general, and, in particular, the simplified case of the movement of a material point. For systems of many particles, such as fluids, the laws of motion are studied in fluid mechanics.
The movement traced by a particle is measured by an observer with respect to a reference system. From a mathematical point of view, Kinematics expresses how the position coordinates of the particle (or particles) vary as a function of time. The function that describes the trajectory traveled by the body (or particle) depends on the speed (the speed with which a mobile changes position) and the acceleration (variation of speed with respect to time).
The motion of a particle (or rigid body) can be described according to the values of velocity and acceleration, which are vector quantities "Vector (physics)").
• - If the acceleration is zero, it gives rise to a uniform rectilinear movement and the speed remains constant over time.
• - If the acceleration is constant with the same direction as the speed, it gives rise to uniformly accelerated rectilinear movement and the speed will vary over time.
• - If the acceleration is constant with a direction perpendicular to the speed, it gives rise to uniform circular motion, where the magnitude of the speed is constant, changing its direction with time.
• - When the acceleration is constant and is in the same plane as the speed and the trajectory, we have the case of parabolic motion, where the velocity component in the direction of acceleration behaves as a uniformly accelerated rectilinear motion, and the perpendicular component behaves as a uniform rectilinear motion, generating a parabolic path by composing both.
• - When the acceleration is constant but is not in the same plane as the speed and trajectory, the Coriolis effect is observed.
• - In simple harmonic motion there is a periodic back-and-forth motion, like that of a pendulum, in which a body oscillates back and forth from the equilibrium position in a given direction and in equal time intervals. Acceleration and velocity are functions, in this case, sinusoidal functions of time.
When considering the translational movement of an extended body, in the case of being rigid, knowing how one of the particles moves, it is deduced how the others move. Thus it is enough to describe the motion of a point particle such as the center of mass of the body to specify the motion of the entire body. In the description of the rotation movement, we must consider the axis of rotation with respect to which the body rotates and the distribution of particles with respect to the axis of rotation. The study of the rotational movement of a rigid solid is usually included in the topic of rigid solid mechanics because it is more complicated. An interesting movement is that of a top, which when spinning can have a precession and nutation movement.
When a body has several movements simultaneously, such as one of translation and another of rotation, each one can be studied separately in the reference system that is appropriate for each one, and then the movements can be superimposed.
In Newtonian mechanics the movement of a particle in three-dimensional space is represented by a vector function:.
The image set is called trajectory and is obtained by integrating the previous differential equation with the appropriate boundary conditions. Since the differential equation can be complicated, integrals of motion are sometimes sought to allow finding the trajectory more easily. For a system of n free particles that exert actions at instantaneous distances, the previous idea generalizes:
If there are ligatures "Ligation (physics)") in the motion it may be simpler and cheaper to move to a generalized coordinate system and work with an abstract formulation typical of analytical mechanics.
A law of motion is a quantitative relationship between variables necessary to describe the motion of bodies. Historically, classical mechanics emerged after Isaac Newton's formulation of three quantitative "laws" that described the movement of a material particle.
Newton's Laws, also known as Newton's Laws of Motion, are three principles from which most of the problems posed by dynamics are explained, particularly those related to the movement of bodies. They revolutionized the basic concepts of physics and the movement of bodies in the universe.
While they constitute the foundations not only of classical dynamics but also of classical physics in general. Although they include certain definitions and in some sense can be seen as axioms, Newton claimed that they were based on quantitative observations and experiments; they certainly cannot be derived from other, more basic relations. The demonstration of its validity lies in its predictions and the validity of those predictions was verified in each and every case for more than two centuries.
The studies that he carried out can be defined with the following three laws that he postulated:
• - The first law of motion refutes the Aristotelian idea that a body can only stay in motion if a force is applied to it. Newton states that:
• - Newton's second law of motion states that:.
Note: it must be taken into account that 1 N= kg m/s2.
• - Newton's Third Law states that:.
Furthermore, each of these forces separately obeys the second law. Together with the previous laws, this allows us to state the principles of conservation of linear momentum and angular momentum.
A
Q
velocity
The potential energy reaches its maximum at the ends of the trajectory and has a null value (zero) at the point x = 0, that is, the equilibrium point.
The kinetic energy will change throughout the oscillations as does the speed:
The kinetic energy is zero at -A or +A (v=0) and the maximum value is reached at the equilibrium point (maximum speed Aω).
In the case of fluids, Bernoulli's principle states that the total energy of a fluid at any time consists of three components:.
One way to represent the equation of Bernoulli's principle is expressed as the sum of the kinetic energy, the flow energy, and the gravitational potential energy per unit mass:
In a streamline each type of energy can increase or decrease by virtue of the decrease or increase of the other two. Although Bernoulli's principle can be seen as another form of the law of conservation of energy, it is actually derived from the conservation of momentum.
Classical kinematics deals with the study of the movement of bodies in general, and, in particular, the simplified case of the movement of a material point. For systems of many particles, such as fluids, the laws of motion are studied in fluid mechanics.
The movement traced by a particle is measured by an observer with respect to a reference system. From a mathematical point of view, Kinematics expresses how the position coordinates of the particle (or particles) vary as a function of time. The function that describes the trajectory traveled by the body (or particle) depends on the speed (the speed with which a mobile changes position) and the acceleration (variation of speed with respect to time).
The motion of a particle (or rigid body) can be described according to the values of velocity and acceleration, which are vector quantities "Vector (physics)").
• - If the acceleration is zero, it gives rise to a uniform rectilinear movement and the speed remains constant over time.
• - If the acceleration is constant with the same direction as the speed, it gives rise to uniformly accelerated rectilinear movement and the speed will vary over time.
• - If the acceleration is constant with a direction perpendicular to the speed, it gives rise to uniform circular motion, where the magnitude of the speed is constant, changing its direction with time.
• - When the acceleration is constant and is in the same plane as the speed and the trajectory, we have the case of parabolic motion, where the velocity component in the direction of acceleration behaves as a uniformly accelerated rectilinear motion, and the perpendicular component behaves as a uniform rectilinear motion, generating a parabolic path by composing both.
• - When the acceleration is constant but is not in the same plane as the speed and trajectory, the Coriolis effect is observed.
• - In simple harmonic motion there is a periodic back-and-forth motion, like that of a pendulum, in which a body oscillates back and forth from the equilibrium position in a given direction and in equal time intervals. Acceleration and velocity are functions, in this case, sinusoidal functions of time.
When considering the translational movement of an extended body, in the case of being rigid, knowing how one of the particles moves, it is deduced how the others move. Thus it is enough to describe the motion of a point particle such as the center of mass of the body to specify the motion of the entire body. In the description of the rotation movement, we must consider the axis of rotation with respect to which the body rotates and the distribution of particles with respect to the axis of rotation. The study of the rotational movement of a rigid solid is usually included in the topic of rigid solid mechanics because it is more complicated. An interesting movement is that of a top, which when spinning can have a precession and nutation movement.
When a body has several movements simultaneously, such as one of translation and another of rotation, each one can be studied separately in the reference system that is appropriate for each one, and then the movements can be superimposed.
In Newtonian mechanics the movement of a particle in three-dimensional space is represented by a vector function:.
The image set is called trajectory and is obtained by integrating the previous differential equation with the appropriate boundary conditions. Since the differential equation can be complicated, integrals of motion are sometimes sought to allow finding the trajectory more easily. For a system of n free particles that exert actions at instantaneous distances, the previous idea generalizes:
If there are ligatures "Ligation (physics)") in the motion it may be simpler and cheaper to move to a generalized coordinate system and work with an abstract formulation typical of analytical mechanics.
A law of motion is a quantitative relationship between variables necessary to describe the motion of bodies. Historically, classical mechanics emerged after Isaac Newton's formulation of three quantitative "laws" that described the movement of a material particle.
Newton's Laws, also known as Newton's Laws of Motion, are three principles from which most of the problems posed by dynamics are explained, particularly those related to the movement of bodies. They revolutionized the basic concepts of physics and the movement of bodies in the universe.
While they constitute the foundations not only of classical dynamics but also of classical physics in general. Although they include certain definitions and in some sense can be seen as axioms, Newton claimed that they were based on quantitative observations and experiments; they certainly cannot be derived from other, more basic relations. The demonstration of its validity lies in its predictions and the validity of those predictions was verified in each and every case for more than two centuries.
The studies that he carried out can be defined with the following three laws that he postulated:
• - The first law of motion refutes the Aristotelian idea that a body can only stay in motion if a force is applied to it. Newton states that:
• - Newton's second law of motion states that:.
Note: it must be taken into account that 1 N= kg m/s2.
• - Newton's Third Law states that:.
Furthermore, each of these forces separately obeys the second law. Together with the previous laws, this allows us to state the principles of conservation of linear momentum and angular momentum.