Kinematics

The kinematics is the branch of physics that describes the movement of solid objects without considering the causes that originate (the forces) and mainly limited to the study of history in function of time. For this, it uses speeds and accelerations, which describe how the position changes as a function of time. The speed is determined as the quotient between the displacement and the time used, while the accelerationis the quotient between the change of speed and the time used.

Basic elements of kinematics
The basic elements of kinematics are space, time and a mobile.

In classical mechanics, the existence of an absolute space is admitted, that is, a space prior to all material objects and independent of the existence of these. This space is the stage where all physical phenomena occur, and it is assumed that all the laws of physics are rigorously fulfilled in all regions of physics. Physical space is represented in classical mechanics by means of a Euclidean space.

Analogously, classical mechanics admits the existence of an absolute time that takes place in the same way in all regions of the Universe and that is independent of the existence of material objects and the occurrence of physical phenomena.

The simplest mobile that can be considered is the material or particle point; when in the kinematics this particular mobile case is studied, it is called particle kinematics, and when the mobile under study is a rigid body it can be considered a system of particles and extensive analogue concepts; in this case it is called kinematics of the rigid solid or the rigid body.

Foundation of classical kinematics
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, but does not study why bodies move but merely describes their trajectories and how to reorient themselves in their Advance. For many particle systems, for example 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 the mathematical point of view, the kinematic expresses how varying the coordinates of position of the particle (s) versus time. The mathematical 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 the speed with respect to time).

The movement of a particle (or rigid body) can be described according to the velocity and acceleration values, which are vector magnitudes:

If the acceleration is zero, it gives rise to a uniform rectilinear motion and the speed remains constant over time.
If the acceleration is constant with the same direction as the velocity, it gives rise to the uniformly accelerated rectilinear motion and the velocity will vary over time.
If the acceleration is constant with direction perpendicular to the velocity, it gives rise to the uniform circular motion, where the modulus of the velocity is constant, changing its direction with time.
When the acceleration is constant and is in the same plane as the speed and trajectory, parabolic motion takes place, where the component of the velocity in the direction of acceleration behaves as a uniformly accelerated rectilinear motion, and the perpendicular component It behaves as a uniform rectilinear motion, and a parabolic trajectory is generated when composing both.
When the acceleration is constant but not in the same plane as the speed and trajectory, the Coriolis effect is observed.
In the simple harmonic movement there is a periodic oscillating movement, like that of the pendulum, in which a body oscillates from one side to the other from the equilibrium position in a certain direction and at equal intervals of time. Acceleration and speed are functions, in this case, sinusoidal of time.

When considering the translation movement of an extensive body, in the case of being a rigid body, knowing how one of the particles moves, it is deduced how the others move. More concretely:

In a two-dimensional plane motion if the 2-point movement of the solid is known, the movement of the entire solid is determined
In a general three-dimensional movement, movement is determined if the 4-point movement of the solid is known.

Thus, considering a point of the body, for example the center of mass of the body or any other, the movement of the whole body can be expressed as:

where:

, is the position of a point on the body at time t.
, is the position of the reference point (for example the center of gravity) at time t.
, is a rotation matrix that accounts for the rotation of the body around itself at time t, to calculate this matrix is enough to know the position of other 3 points in addition to the reference point (or 1 point more if the movement is flat).
In the description of the rotation movement given by 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 rotation movement of a rigid solid is usually included in the theme of rigid solid mechanics, since it is more complicated (the main direction of associated with the eigenvalue 1, gives the axis of rotation at each instant t).

An interesting movement is that of a spinning top, that when turning can have a precession and nutation movement. When a body has several movements simultaneously, such as one of translation and another of rotation, you can study each one separately in the reference system that is appropriate for each one, and then superimpose the movements.

Coordinate systems
In the study of movement, the most useful coordinate systems are seeing the limits of the path to be traveled or analyzing the geometric effect of the acceleration that affects the movement. Thus, to describe the movement of a heel forced to move along a circular ring, the most useful coordinate would be the angle traced on the ring. In the same way, to describe the movement of a particle subjected to the action of a central force, the polar coordinates would be the most useful.

In the vast majority of cases, the kinematic study is done on a system of Cartesian coordinates, using one, two or three dimensions, according to the trajectory followed by the body.

Movement record
Technology today offers us many ways to record the movement effected by a body. Thus, to measure the speed of vehicles, the traffic radar is available whose operation is based on the Doppler effect. The tachometer is an indicator of the speed of a vehicle based on the frequency of rotation of the wheels. The walkers have pedometers that detect the characteristic vibrations of the passage and, assuming a characteristic average distance for each step, they allow to calculate the distance traveled. The video, together with the computer analysis of the images, also allows to determine the position and speed of the vehicles.

Types of movements

Rectilinear movement
It is the one in which the mobile describes a straight line trajectory.

Uniform rectilinear motion
In this movement the mobile moves along a straight line at a constant V velocity; the acceleration a is zero all the time. This corresponds to the movement of an object thrown into space outside of any interaction, or to the movement of an object that glides without friction. Since the velocity V is 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 to say for .

the previous equation corresponds to a line that passes through the origin, in a graphical representation of the function .

Rectilinear motion uniformly accelerated or varied
In this movement the acceleration is constant, so the mobile speed varies linearly and the position quadratically with time. The equations that govern this movement are the following:

the final speed is equal to the initial speed of the mobile plus the acceleration due to the increase in time. so so:

the final velocity is equal to the initial velocity plus the acceleration for time.

Starting from the relation that calculates the speed:

Where , initial speed, the one you have for , we have.

Note that if the acceleration were zero, the previous equations would correspond to those of a uniform rectilinear motion, that is, with speed constant. If the body part of the rest accelerating uniformly, then the .

Two specific cases of MRUA are free fall and vertical shooting. The free fall is the movement of an object that falls towards the center of the Earth with an acceleration equivalent to the acceleration of gravity (which in the case of the planet Earth at sea level is approximately 9.8 m / s 2). The vertical shot, 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 winning it, until reaching the state of rest; then, and from there, a free fall movement with zero initial velocity begins.

Simple harmonic movement
It is a periodic back and forth motion, in which a body oscillates on either side of a position of equilibrium in a certain direction and at equal intervals of time. 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, for the case of movement in a dimension is:

or

which corresponds to a frequency sinusoidal function ,, amplitude A and initial phase .

The movements of the pendulum, of a mass attached to a spring or the vibration of the atoms in the crystal lattices are of these characteristics.

The acceleration experienced by the body is proportional to the displacement of the object and the opposite direction, from the point of equilibrium. Mathematically:

where it is a positive constant and ,refers to the elongation (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 oscillatory periodic movement slows down in time (mostly 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 movement
The 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 y axis) uniformly accelerated, with the 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 in the course of the movement.
The initial velocity vector forms an initial angle with respect to the x axis ; and, as said, for the analysis it is broken down into the two types of movement mentioned; Under this analysis, the components according to x and y of the initial velocity will be:

The horizontal displacement is given by the law of uniform motion, therefore its equations will be (if one considers
):

As long as the movement according to the axis will be rectilinear uniformly accelerated, being its equations:

If you replace and operate to eliminate time, with the equations that give the positions and , the equation of the trajectory in the xy plane is obtained:

which has the general form

and represents a parabola in the plane y (x). This representation is shown, but in it it has been considered (not so in the respective animation). In this figure it is also observed that the maximum height in the parabolic trajectory will occur in H, when the vertical component of the velocity is null (maximum of the parabola); and that the horizontal reach will occur when the body returns to the ground, in (where the parabola cuts the axis ).

Circular movement
The circular movement in practice is a very common type of movement: They experience it, for example, 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 pallets of a fan, etc. In the case of a disk rotating around a fixed axis, any of its points describes circular trajectories, performing a certain number of turns during a certain time interval. For the description of this movement it is convenient to refer anglestours; since the latter are identical for all points of the disc (referred to the same center). The length of the arc traveled by a point of the disk depends on its position and is equal to the product of the angle traveled by its distance to 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. The angular acceleration (α) turns out to be variation of angular velocity with respect to time, and it is represented by a vector analogous to that of the 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 modulus expresses the length of the arc traveled (space) per unit of time time; said 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 magnitude that indicates the speed with which the velocity changes with respect to time; that is, the change of the velocity vector per unit of time. Acceleration generally has two components: acceleration tangential to the trajectory and acceleration normal to it. The tangential acceleration is what causes the variation of the velocity modulus (celerity) with respect to time, while the normal acceleration is responsible for the change in velocity direction. The modules of both acceleration components depend on the distance the particle is from the axis of rotation.

Uniform circular motion
It is characterized by having a variable speed or structural constant so that the angular acceleration is zero. The linear velocity of the particle does not vary in modulus, but in direction. The tangential acceleration is zero; but there is centripetal acceleration (normal acceleration), which is the cause of the change of direction.

Mathematically, the angular velocity is expressed as:

where is the angular velocity (constant), is the variation of the angle swept by the particle and It is the variation of time. The angle traveled in a time interval is:

Uniformly accelerated circular motion
In this movement, the angular velocity varies linearly with respect to time, since the mobile is subjected to a constant angular acceleration. The equations of motion are analogous to those of the uniformly accelerated rectilinear, but using angles instead of distances:

being , the constant angular acceleration.

Complex harmonic movement
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 movement of a particular material point can often be modeled by a complex harmonic movement if the amplitude of the movement is small.

The complex harmonic movement is interesting because it is usually not a periodic movement but a quasiperiodic movement that never exactly repeats itself, although it executes almost cycles without exactly repeating itself. The vector form of a point that executes this movement turns out to be:

where they are the maximum amplitudes in the three directions of space, are the oscillation frequencies and the initial phases (the initial conditions allow to calculate both the amplitudes and the phases). The frequencies depend on the characteristics of the system (mass, rigidity, etc.).

The 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 relation of specific gaps . If the amplitudes are not equal or the lag is not exactly the given, the trajectory of this movement, but the frequencies are equal, the result is an elliptical movement .

Rigid solid movement
All the movements described above refer to concrete material points, or corpuscles, that is to say physical bodies whose small dimensions with respect to the size of the trajectory so they can be approximated 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 that 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 those cases the kinematics of the rigid solid must be used, in which the “trajectory” of the body is given a more complex or richer space than the simple three-dimensional Euclidean space, since it is necessary to define not only the displacement of the body through said space, but to specify the changes of orientation of the body in its movement, by means of rotation movements.

Mathematical formulation with differential calculus

The velocity is the temporal derivative of the position vector and the acceleration is the temporal derivative of the velocity:

or its integral expressions:

where they are the initial conditions.

Movement on Earth
When observing the movement on Earth of bodies such as masses of air in meteorology or projectiles, there are deviations caused by the so-called Coriolis Effect. They are used to prove that the Earth is rotating on its axis. From the cinematic point of view it is interesting to explain what happens when considering the trajectory observed from a reference system that is in rotation, the Earth.

Suppose a cannon at the equator launches a projectile northward along a meridian. An observer located to the north on the meridian observes that the projectile falls to the east of the predicted thing, deviating to the right of the trajectory. Similarly, if the projectile had fired along the meridian to the south, the projectile would also have deviated to the east, in this case to the left of the trajectory followed. The explanation of this “deviation”, caused by the Coriolis Effect, is due to the rotation of the Earth. The projectile has a velocity with three components: the two that affect the parabolic shot, towards the north (or the south) and upwards, respectively, plus a third component perpendicular to the previous ones due to the projectile, before leaving the canyon, has a speed equal to the speed of rotation of the Earth at the equator. This last component of speed is the cause of the deviation observed because although the angular speed of rotation of the Earth is constant over its entire surface, it is not the linear speed of rotation, which is maximum at the equator and null in the center of the poles. Thus, the projectile proceeds towards the north (or the south), moves faster towards the east than the surface of the Earth, so the deviation mentioned is observed. Logically,

Another interesting case of movement on Earth is that of the Foucault pendulum. The plane of oscillation of the pendulum does not remain fixed, but we observe it spinning, turning clockwise in the northern hemisphere and counterclockwise in the southern hemisphere. If the pendulum oscillates at the equator, the plane of oscillation does not change. On the other hand, at the poles, the rotation of the oscillation plane takes a day. For intermediate latitudes it takes higher values, depending on the latitude. The explanation for such a turn is based on the same principles previously made for the artillery projectile.

Relativistic kinematics
In relativity, what is absolute is the speed of light in a vacuum, not space or time. Every observer in an inertial reference system, no matter its relative speed, will measure the same speed for light as another observer in another system. This is not possible from the classical point of view. The transformations of movement between two reference systems must take into account this fact, from which the Lorentz transformations arose. They show that spatial dimensions and time are related, so in relativity it is normal to talk about spacetime and a four-dimensional space.

There is much experimental evidence of relativistic effects. For example, the time measured in a laboratory for the disintegration of a particle that has been generated with a velocity close to that of light is higher than the decay measured when the particle is generated at rest with respect to the laboratory. This is explained by the relativistic temporal dilation that occurs in the first case.

The kinematics is a special case of differential geometry of curves, in which all the curves are parameterized in the same way: over time. For the relativistic case, the coordinate time is a relative measure for each observer, therefore the use of some type of invariant measure is required as the relativistic interval or equivalently for particles with mass the own time. The relationship between the coordinate time of an observer and the proper time is given by the Lorentz factor.

Source from Wikipedia