Types of movements
rectilinear movement
Rectilinear movement is one in which the mobile[3] describes a trajectory in a straight line.
In uniform rectilinear motion (MRU) the mobile moves along a straight line at constant speed V; acceleration a is zero all the time. This corresponds to the movement of an object thrown into space outside of all interaction, or the movement of an object that slides without friction. With the speed V constant, the position will vary linearly with respect to time, according to the equation:.
where is the initial position of the mobile with respect to the center of coordinates, that is, for .
If the previous equation corresponds to a line that passes through the origin, in a graphical representation of the function, such as the one shown in figure 1.
In this movement the acceleration is constant, so the speed of the mobile varies linearly and the position quadratically with time. The equations that govern this movement are the following:
The final velocity is equal to the initial velocity of the mobile plus the acceleration times the increment of time. if then:.
Final velocity is equal to initial velocity plus acceleration times time.
Starting from the relationship that calculates the speed:
Where , is the final position and its initial velocity, that which has for , we have.
Note that if the acceleration were zero, the previous equations would correspond to those of a uniform rectilinear movement, that is, with constant speed. If the body starts from rest accelerating uniformly, then the .
Two specific cases of MRUA are free fall and vertical throw. Free fall is the movement of an object that falls in the direction of the center of the Earth with an acceleration equivalent to the acceleration of gravity (which in the case of planet Earth at sea level is approximately 9.8 m/s). The vertical throw&action=edit&redlink=1 "Vertical throw (physics) (not yet written)"), on the other hand, corresponds to that of an object thrown in the opposite direction to the center of the earth, gaining height. In this case, the acceleration of gravity causes the object to lose speed, instead of gaining it, until it reaches the state of rest; then, and from there, a free fall movement with zero initial speed begins.
It is a periodic back-and-forth movement, in which a body oscillates to and fro from an equilibrium position in a given direction and at equal time intervals. Mathematically, the path traveled is expressed as a function of time using trigonometric functions, which are periodic. For example, the equation of position with respect to time, in the case of movement in one dimension, is:
either.
which corresponds to a sinusoidal function of frequency, amplitude A and initial phase.
The movements of the pendulum, of a mass attached to a spring or the vibration of atoms in crystalline networks have these characteristics.
The acceleration experienced by the body is proportional to the displacement of the object and in the opposite direction, from the point of balance. Mathematically:.
where is a positive constant and refers to elongation "Elongation (physics)") (displacement of the body from the equilibrium position).
The solution to that differential equation leads to trigonometric functions of the previous form. Logically, a real periodic oscillatory motion slows down in time (mostly due to friction), so the expression of acceleration is more complicated, needing to add new terms related to friction. A good approximation to reality is the study of damped oscillatory movement.
parabolic motion
Parabolic movement can be analyzed as the composition of two different rectilinear movements: one horizontal (according to the x axis) of constant speed and another vertical (according to the y axis) uniformly accelerated, with gravitational acceleration; The composition of both results in a parabolic trajectory.
Clearly, the horizontal component of the velocity remains unchanged, but the vertical component and the angle θ change over the course of the movement.
In figure 4 it is observed that the initial velocity vector forms an initial angle with respect to the x axis; and, as said, for the analysis it is decomposed into the two types of movement mentioned; Under this analysis, the x and y components of the initial velocity will be:
The horizontal displacement is given by the law of uniform motion, therefore its equations will be (if considered):
While the movement along the axis will be uniformly accelerated rectilinear, its equations being:
If we substitute and operate to eliminate time, with the equations that give the positions e , we obtain the equation of the trajectory in the xy plane:.
which has the general form.
y represents a parabola in the y(x) plane. Figure 4 shows this representation, but it has been considered in it (not in the respective animation). This figure also shows that the maximum height in the parabolic trajectory will occur at H, when the vertical component of the velocity is zero (maximum of the parabola); and that the horizontal reach will occur when the body returns to the ground, at (where the parabola cuts the axis).
circular motion
Circular motion in practice is a very common type of motion: It is experienced, for example, by the particles of a disk that rotates on its axis, those of a ferris wheel, those of the hands of a clock, those of the blades of a fan, etc. In the case of a disk rotating around a fixed axis, any of its points describe circular trajectories, making a certain number of revolutions during a certain time interval. For the description of this movement it is convenient to refer to the angles traveled; since the latter are identical for all points on the disk (referring to the same center). The length of the arc traveled by a point on the disc depends on its position and is equal to the product of the angle traveled by its distance from the axis or center of rotation. The angular velocity (ω) is defined as the angular displacement with respect to time, and is represented by a vector perpendicular to the plane of rotation; Its direction is determined by applying the "right-hand rule" or the corkscrew. Angular acceleration (α) turns out to be a variation of angular velocity with respect to time, and is represented by a vector analogous to that of angular velocity, but it may or may not have the same direction (depending on whether it accelerates or retards).
The velocity (v) of a particle is a vector magnitude whose module expresses the length of the arc traveled (space) per unit of time; This module is also called speed or celerity. It is represented by a vector whose direction is tangent to the circular path and coincides with that of the movement.
The acceleration (a) of a particle is a vector quantity that indicates how quickly the velocity changes with respect to time; that is, the change in the velocity vector per unit of time. Acceleration generally has two components: the acceleration tangential to the trajectory and the acceleration normal to it. Tangential acceleration is what causes the variation of the speed module (celerity) with respect to time, while normal acceleration is responsible for the change in direction of speed. The modules of both components of the acceleration depend on the distance at which the particle is with respect to the axis of rotation.
It is characterized by having a constant variable or structural speed so the angular acceleration is zero. The linear velocity of the particle does not vary in module, but it does vary in direction. The tangential acceleration is zero; but there is centripetal acceleration (normal acceleration), which causes the change of direction.
Mathematically, angular velocity is expressed as:.
where is the angular velocity (constant), is the variation of the angle swept by the particle and is the variation of time. The angle traveled in a time interval is:.
In this movement, the angular velocity varies linearly with respect to time, because the mobile is subject to a constant angular acceleration. The equations of motion are analogous to those of a uniformly accelerated rectilinear motion, but using angles instead of distances:
Complex harmonic motion
It is a type of two-dimensional or three-dimensional movement that can be constructed as a combination of simple harmonic movements in different directions. When a structure is subjected to vibrations, the motion of a particular material point can often be modeled by complex harmonic motion if the amplitude of the motion is small.
Complex harmonic motion is interesting because it is usually not periodic motion but quasiperiodic motion that never repeats exactly the same, although it executes almost cycles without repeating exactly. The vector form of a point that executes this movement turns out to be:
where are the maximum amplitudes in the three directions of space, are the oscillation frequencies and the initial phases (the initial conditions allow calculating both the amplitudes and the phases). The frequencies depend on the characteristics of the system (mass, stiffness, etc.).
Uniform circular motion is in fact a case of complex harmonic motion in which the amplitudes in two directions are equal to the radius of the circle, the frequencies in the two directions coincide, and there is a specific phase shift relationship. If the amplitudes are not equal or the phase shift is not exactly as indicated, but the frequencies are equal, it turns out to be the case of an elliptical movement, whose trajectory describes an ellipse.
Rigid solid motion
All the movements described above refer to specific material points, or corpuscles, that is, physical bodies whose dimensions are small with respect to the size of the trajectory, so they can be approached by material points. However, macroscopic physical bodies are not punctual, in many situations the movement of the body as a whole requires a more complex description than assuming that all its points follow a trajectory much greater than the distances between points of the body, so the description of the body as a material point is inadequate and the kinematics of the material point is too simple to adequately describe the kinematics of the body. In these cases, the kinematics of the rigid solid must be used, in which the "path" of the body is given a more complex or richer space than the simple three-dimensional Euclidean space, since it is required to define not only the displacement of the body through said space, but also to specify the changes in orientation of the body in its movement, through rotational movements.