Movement (physics)

As movement in the physical sense is the change in the location of a mass point or a physical body with time.

The two disciplines of physics , posing as kinesiology deal with the movement are:

the kinematics of the teaching of the description of movement
the momentum (in the technical mechanics : the kinetics ) as the doctrine of the causes of movement

Movement of mass points

A mass point is the theoretical idealization of a physical body. It is assumed that the entire mass of the body is united in a single point, and that rotations of the body around its own axis are therefore irrelevant for the description of the movement.

The entirety of all places at which such a mass point is in the course of a movement is called a trajectory or trajectory . Trajectories are always continuously (ie, in the mathematical sense constantly ) and the movement provided at any point of the trajectory comes to a standstill, even smooth (ie, in the mathematical sense differentiable ). If the location is known at all times , the function is called the path-time law of motion. ${\ displaystyle t}$ ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle {\ vec {r}} (t)}$

Relativity of motion

For a clear description of the location, the speed, etc., a reference system is required that defines both the origin of the coordinates and the state of rest. The reference system can be chosen arbitrarily; however, the description of the movement depends on this choice. As a rule, it is assumed that the observer himself is at rest. Since different observers describe the same movement differently, a suitable formulation often includes the term “relative movement”. For example, a person in the passenger seat of a moving car moves from the perspective of a pedestrian at the edge of the road (read: “relative” to the pedestrian), while from the driver's point of view he is resting.

Speed and acceleration

The speed is the ratio of the length of a small, at least approximately straight section of the trajectory to the time it takes for the mass point to cover this section of the path. The smaller the distance, the more precisely a certain instantaneous speed can be assigned to a place and time . The speed has a direction that corresponds to the direction of movement at the respective point in time, and an amount that is often referred to colloquially as the speed . The speed is a vector that is tangential to the trajectory at the relevant point.

As you progress on the trajectory, both the amount and the direction of the speed can change. Colloquially, the first is referred to as accelerating or braking, the second often as turning or “making a bow”. In physics and technology, the term acceleration is used for everything together . Acceleration is a vector that is defined as the ratio of the change in the speed vector to the amount of time that change occurs. A tangential acceleration only changes the amount, a normal acceleration only the direction of the speed. In the general case, the vector sum of the tangential acceleration and normal acceleration gives the entire acceleration vector.

From a mathematical point of view, the path-time law of a point-like object, i.e. the position vector , is a continuous function of time. If it is also differentiable , the first derivative forms the velocity vector, the second derivative the acceleration vector. ${\ displaystyle {\ vec {r}} (t)}$

Dynamics of the mass point

In Newtonian mechanics, movements are influenced by forces . Newton summarized the effect of the forces in Newton's three laws :

Law of inertia: If no external forces act on a mass point or - which is equivalent - if it is in equilibrium of forces , its state of motion does not change. This means that neither the direction of movement nor the amount of speed change. The mass point therefore moves uniformly and in a straight line.
Basic equation of mechanics: When a resultant force acts on a mass point, the latter experiences an acceleration which is greater, the stronger the force and the smaller its mass . This is expressed by the equation . ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {a}}}$ ${\ displaystyle m}$ ${\ displaystyle {\ vec {F}} = m {\ vec {a}}}$
Interaction principle (" actio = reactio "): When a body exerts a force on a second body, it also experiences a force from it. The two forces have the same magnitude but opposite directions.

The equation of motion results from Newton's second law as a differential equation , the solution of which is the path-time law . In classical mechanics , the equations of motion are ordinary differential equations of the second order in time. By defining location and speed at a specific point in time as the initial conditions , the further development of time is clearly determined. In other words: if you know all the forces acting on it, you can - based on the initial conditions - predict the movement of the object or calculate it back. In classical mechanics, the movements of mass points are strictly deterministic . One speaks of a chaotic movement, however, if the equation of motion is such that the smallest changes in the initial conditions result in large changes in the resulting movement. Then a prediction of the future development of the system is de facto impossible. An example of this is the three-body problem .

In addition to Newton's equations of motion, there are other formulations of dynamics, see D'Alembert's principle , Lagrange formalism and Hamiltonian mechanics .

pulse

Since the behavior of a mass point depends very much on its inertia , which is given by its mass , it often makes sense to describe the movement not in terms of speed but in terms of momentum . For this applies: ${\ displaystyle m}$ ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {p}} = m {\ vec {v}}}$

As long as there are no forces acting, the momentum does not change. ( Conservation of momentum )

Impulse can be transmitted from one body to another. A force acts between the two bodies.

The attacking force determines the rate at which a body's momentum changes over time.

If one looks at several mass points that influence each other but do not experience any external forces, the total momentum does not change. The center of gravity of the system then moves uniformly and in a straight line. ( Focal point )

Kinetic energy

Every mass point that moves has a certain kinetic energy, also known as "kinetic energy". In the non-relativistic mechanics, the kinetic energy is calculated according to the equation . ${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} mv ^ {2} = {\ frac {p ^ {2}} {2m}}}$

Rigid body movement

The movement of a rigid body can be broken down into the movement of the center of gravity ( translation ) and rotational movements of the body around axes that go through the center of gravity. For the former, the same applies as described for mass points. The equations of motion for the rotation are called Euler's equations . Stable rotary movements only occur around those axes with respect to which the body's moment of inertia is minimal or maximal.

Movement of liquids and gases

The movement of deformable bodies (especially liquids and gases ) can no longer be described by a few trajectories.

Depending on the type of movement, a distinction is made between the following cases:

steady flow : the flow pattern is constant over time.
Laminar flow : The fluid can be broken down into individual flow threads that do not mix.
turbulent flow : the flow is neither stationary nor laminar. Turbulence occurs in all size scales.

The Reynolds number helps to characterize a flow .

The equations of motion of liquids and gases are the Navier-Stokes equations . They are derived from the basic equation of mechanics .

Special forms of movement of individual objects

Straight, uniform movement

One speaks of rectilinear uniform motion when the trajectory is a straight line section and the speed is the same at every point on the trajectory. A rectilinear, uniform movement exists precisely when the acceleration is zero everywhere.

Evenly accelerated movement

With a uniformly accelerated movement , the acceleration has the same amount and the same direction at every point on the trajectory. The trajectory of a uniformly accelerated movement is either a straight line section or a parabola .

Circular motion

In the case of a circular movement , the trajectory is circular. The velocity vector forms a right angle with the radius at any point in time and therefore points in a tangential direction. If the amount of speed in a circular motion is the same everywhere, then it is a uniform circular motion . With it, the tangential acceleration is zero and the normal acceleration is always directed towards the center of the circle.

Periodic movement

In the case of a periodic movement, the observation object returns to the starting point after a certain time, the period , and has the same direction and the same speed. Periodic movements have closed trajectories. The circular motion is a special case of a periodic motion.

Harmonic oscillation

Another example of a periodic movement is harmonic oscillation , in which the change in location over time follows a sine function. A classic example of a harmoniously oscillating object is a spring pendulum . In general, every object that is slightly deflected from its equilibrium oscillates harmoniously . Using Fourier analysis , every periodic movement can be represented as the sum of harmonic oscillations, the frequencies of which are integral multiples of the fundamental frequency , the reciprocal of the period.

Ergodic movement

In an ergodic movement, the trajectory evenly fills a section of space.

Statistical consideration of numerous objects

The movements of a large number of similar objects, e.g. B. the molecules of a gas are described statistically. The totality of all possible motion states of all objects that are compatible with the measured state variables (e.g. energy, volume and number of particles) is called an ensemble . One then postulates that all possible states of motion are equally likely and from this derives statements about the probability distributions of the physical quantities. The Maxwell-Boltzmann distribution shows, for example, the (probability) distribution of the magnitude of the particle velocities in an ideal gas .

Movement on a microscopic scale

The idea of point-like particles moving with well-defined speeds on a trajectory is in truth a model that is only sustainable from a certain size of the scale. The model of the trajectory fails, for example, with the movement of electrons in an atom , of conduction electrons in a metal, of protons and neutrons in an atomic nucleus or of photons .

In order to analyze the situations mentioned, one has to switch to the more exact representation, quantum mechanics , in which physical objects are described by a wave function . The wave function can be used to derive the probability that an object is at a certain location or has a certain speed. The Heisenberg uncertainty principle limits the accuracy of a simultaneous measurement of the position and velocity; in addition, every measurement affects the wave function and changes the future time evolution of the probabilities.

Individual evidence

↑ Richard Courand, Herbert Robbins: What is Mathematics? 5th edition. limited preview in Google Book search.

[1] Richard Courand, Herbert Robbins: What is Mathematics? 5th edition. limited preview in Google Book search.