Pendulum

A pendulum , also known as a gravity pendulum (formerly also perpendicular , from the Latin pendere “to hang”) is a body which, rotatably mounted on an axis or a point outside its center of mass , can swing around its own rest position. Its simplest design is the thread pendulum , which consists of a weight suspended from a thread and structurally resembles a plumb bob. In this sense, the spring pendulum and the torsion pendulum are not pendulums .

One property of the gravity pendulum is that its period of oscillation depends only on the length of the thread (more precisely: the distance between the suspension and the center of gravity of the pendulum body), but not on the type, shape or mass of the pendulum body; also almost not of the size of the maximum deflection, provided this is limited to a few angular degrees . This was first established by Galileo Galilei and used to regulate the first precise clocks after in-depth studies by Christiaan Huygens . A seconds pendulum has a length between 99.1 and 99.6 cm , depending on the geographical latitude of the location.

Basics

The pendulum usually consists of a band or a rod, which is weighted down with a mass at the free end. Such a pendulum from its vertical brings one rest position , it swings under the influence of gravity back and is, as long as no damping occurs symmetrically between the apexes as a reversal point of the movement to the lowest possible position of the center of gravity - continue to oscillate - the rest position. When swinging, the potential energy of the mass is converted into kinetic energy and back again. In the rest position the entire energy of the oscillation is available as kinetic energy, at the apex as potential energy. On average over time, the energy is divided equally into kinetic and potential energy according to the virial theorem .

The regularity of the period of oscillation of a pendulum is used in mechanical pendulum clocks . If their pendulums are to be accurate , they must cover the smallest possible and constant amplitudes .

A distinction is made between mathematical pendulums and physical pendulums: The plane mathematical pendulum and the spherical pendulum are idealizing models for the general description of pendulum oscillations. It is assumed that the entire mass of the pendulum is united in one point , which is at a fixed distance from the point of suspension. Such a pendulum is approximately realized by a thread pendulum. The physical pendulum differs from the mathematical pendulum in that it takes into account the shape and size of the pendulum body, which is why the behavior of physical pendulums corresponds more to that of real pendulums. For example, the period of a rod pendulum, in which a pendulum body hangs on a rod with finite mass, is always shorter than the period of a mathematical pendulum of the same length, in which the mass of the suspension can be neglected. The consideration of the movement of the pendulum is simplified for small deflections : Since the restoring force is approximately proportional to the deflection, it is a harmonic oscillator .

The rotation of the earth could be demonstrated with the Foucault pendulum : The Coriolis force acts on the pendulum from outside by changing its oscillation level and deflecting it from oscillation to oscillation in a recurring pattern.

background

Theory: Harmonic Oscillator

With small deflections, the pendulum is a good approximation of a mechanical harmonic oscillator - with large deflections, the oscillations are continuous, but no longer harmonic. The harmonic oscillator is an important model system in physics because it represents a closed, solvable system. It is characterized in that a force proportional to the deflection acts against the deflection direction. With the deflection , the second derivative with respect to time and a proportionality constant, the following applies: ${\ displaystyle \ varphi}$ ${\ displaystyle {\ tfrac {\ mathrm {d} ^ {2}} {\ mathrm {d} t ^ {2}}}}$ ${\ displaystyle \ omega ^ {2}}$ ${\ displaystyle {\ frac {\ mathrm {d} ^ {2} \ varphi} {\ mathrm {d} t ^ {2}}} = - \ omega ^ {2} \ varphi}$ With this constant of proportionality, the harmonic oscillator has a degree of freedom called its angular frequency . The solution to this equation is of a periodic nature, which, depending on the initial physical conditions , can be written as the sum of a sine and cosine function : ${\ displaystyle \ omega}$ ${\ displaystyle \ varphi (t) = A \ sin (\ omega t) + B \ cos (\ omega t)}$ The movement that a harmonic oscillator describes is called harmonic oscillation . Not following the strict definition of the harmonic oscillator, damped harmonic oscillators are sometimes referred to as such. These are modeled in such a way that the amplitude , the maximum deflection, of the oscillation becomes smaller over time.

Mathematical and physical pendulum

The mathematical pendulum is the simplest model of a pendulum: A mass point is suspended from a massless, rigid thread and can accordingly only move in two dimensions on a circular path around the suspension. Its only degree of freedom is the deflection around a position of equilibrium or rest position and the weight acts as a restoring force on the mass point. If the deflection is sufficiently small, the mathematical pendulum can be described as a harmonic oscillator. The angular frequency depends only on the length of the thread and the acceleration due to gravity : ${\ displaystyle l}$ ${\ displaystyle g}$ ${\ displaystyle \ omega = {\ sqrt {\ frac {g} {l}}}}$ The generalization of the mathematical pendulum in three dimensions is called the spherical pendulum . Its coupled system of equations no longer has a simple solution.

In contrast to the mathematical pendulum, the physical pendulum takes into account the expansion of the pendulum body and the mass of the thread. The angular frequency of the physical pendulum also depends on its mass and its moment of inertia , while the distance between the center of gravity and the suspension must be specified: ${\ displaystyle m}$ ${\ displaystyle I}$ ${\ displaystyle l}$ ${\ displaystyle \ omega = {\ sqrt {\ frac {mgl} {I}}}}$ The calculation also only applies to a small deflection.

Coupled pendulums

With two coupled pendulums, two pendulums exert a force on each other that is dependent on both deflections. For example, you connect two identical filament pendulums to each other with a spring in order to observe the natural vibrations and the phenomenon of beating in the demonstration experiment . Bound atoms (e.g. in a molecule or in a solid) can often be approximated by a model of many coupled pendulums. More than two coupled pendulums can show complex oscillation patterns if the fundamental oscillation is superimposed by differently shaped natural oscillations (or oscillation modes ) with higher natural frequencies .

With the double pendulum, a second pendulum is attached to the mass of one pendulum. Among other things, it serves to demonstrate chaotic processes, as movement can be chaotic.

Spring pendulum Play media file
Video: a torsion pendulum

Spring pendulums are not pendulums in the true sense of the word, because, unlike gravity pendulums, they have their own restoring forces that are independent of gravity.

There are u. a. the following variants:

• The linear spring oscillator (also spring pendulum) uses the restoring force of a tensioned helical spring. With horizontal spring oscillators, a mass swings horizontally between two tensioned springs.
• Torsional oscillators perform a circular motion and have a torsional moment of inertia :

Applications

• With the help of Foucault's pendulum , the earth's rotation can be made visible.
• Only about 1.5 cm short pendulums are used in automatic seat belts of cars to detect strong horizontal acceleration within a short distance and to release the lock (next to two centrifugal pawls on the spool).
• Climbers hanging on a safety rope can cause themselves to swing by repeated pushing off in order to reach a position to continue climbing laterally. On the other hand, falling into an upward sloping rope harbors the risk of swinging and hitting the rock.
• In the simplest case, sensors that detect an object's back or tilting, for example to indicate theft, contain a pendulum with an electrical sliding contact in the neutral position.
• Impact pendulums are used to determine the notched impact strength and other strength values ​​of materials or workpieces.
• The wrecking ball is intended to demolish vertical walls and concrete with a horizontal impact. There are also battering rams that hang from 2 pendulum arms, the overhead handles of which are operated by 4 people.
• A bottle swinging on a rope to the ship's side wall is supposed to break apart when the ship is christened.
• A swing is a seat board (or pole) suspended from two ropes or chains for amusement or artistry purposes.
• The boat swing in the amusement park, a rigid pendulum with a gondola, is rocked with physical strength and swing by the people standing in it.
• The artist Carolee Schneemann "draws kinetically" hanging on a rope.
• Dancers on safety ropes with abseiling devices that can be operated with one hand can trigger sideways swinging on a roughly vertical wall by running and pushing off, gradually lowering themselves and thus performing a ballet that can be seen from afar.
• Pendulum plays a role in acrobatics on the trapeze and aerial silk as well as in gymnastics on rings and on the climbing rope .
• The balancing of a rod or a circuit is to keep a rigid inverted pendulum alone by supporting at one point.
• The Riefler pendulum increased the accuracy of pendulum clocks to better than a tenth of a second per day.
• The pendulum figure is a toy.