# Torsion pendulum

Torsional oscillator with spiral spring
Rotary transducer (13 seconds, 650 KB)

A torsion pendulum - also known as a rotating pendulum - consists of two masses at the two ends of a rod or a corresponding device. The system can be roughly described as a harmonic oscillator .

Technically, the torsion pendulum is used, among other things, for rotary pendulum clocks , in gravitational scales and in mirror galvanometers .

## construction

The rod is at the distance of the common center of gravity, z. B. in its center, fixed so that the two masses a circular movement around the common center of gravity is possible. There the rod is attached to an axis that can rotate in accordance with the circular movement of the two masses and the rod.

This axle is attached to a spring element in such a way that the rotation puts it in tension and thus absorbs energy. If the entire kinetic energy of the masses is stored in the spring element, the construction comes to a standstill.

The pendulum movement comes about when the spring element returns its energy to the masses in the form of an opposite movement. The masses are accelerated in the opposite direction of rotation until the spring element has relaxed and the entire energy is stored in the movement of the masses, that is, the masses have their highest speed. Due to the inertia of the masses, however, they continue to move and the spring element is tensioned again, the masses are thereby decelerated and give their kinetic energy back to the spring. The cycle begins again with a different direction of movement.

In the case of torsional vibrators with a spiral spring, however, the restoring force is not created by torsion, but by bending in the spiral spring; see for example the balance (clock) .

## Physical description

The restoring torque M is proportional to the deflection of the rotary pendulum and counteracts this:

${\ displaystyle M = -D \ cdot \ varphi}$

${\ displaystyle D}$is the moment of direction , the angle of deflection in radians. ${\ displaystyle \ varphi}$

Another physical relationship exists between the moment of direction and the length of the torsion wire : ${\ displaystyle D}$${\ displaystyle l}$

${\ displaystyle l \ sim {\ frac {1} {D}}}$

The mathematical description of the torsion pendulum hardly differs from the other types of pendulum:

The solution of the differential equation is the same, but it also applies to large deflections, which is not the case with other pendulums. This means that vibration measurements can be carried out much more precisely with a torsion pendulum.

Unlike z. For example, with the (gravity) pendulum swinging back and forth, the linearity of the torque is valid over large angular ranges. In the ideal case, the linearity is given until the elastic limit of the torsion wire is reached. As a result, the differential equation can be solved exactly without the small-angle approximation required for gravitational pendulums , and the oscillation frequency is largely independent of the amplitude. With a suitable design of the pendulum body, the air friction and thus the damping is low. These properties make rotary pendulums suitable as a time standard for clocks - however, the temperature dependence and the long-term stability of the elastic properties of the wire pose a problem.

The following applies to the rotational frequency of an ideal torsional pendulum: ${\ displaystyle f = {\ frac {1} {2 \ pi}} \ cdot {\ sqrt {\ frac {D} {J}}}}$

It is the frequency in Hertz, the Directorate moment and the moment of inertia of the pendulum body. This result would also be obtained from solving the differential equation . ${\ displaystyle f}$${\ displaystyle D}$${\ displaystyle J}$

## Laws of motion as a function of time

Time-path law: ${\ displaystyle \ varphi (t) = {\ hat {\ varphi}} \ cdot \ sin (2 \ pi {f} \ cdot {t})}$

Time-velocity law: ${\ displaystyle \ omega (t) = {\ dot {\ varphi}} (t) = {\ hat {\ varphi}} \ cdot (2 \ pi {f}) \ cdot \ cos (2 \ pi {f} \ cdot {t})}$

Time-acceleration law: ${\ displaystyle \ alpha (t) = {\ ddot {\ varphi}} (t) = - {\ hat {\ varphi}} \ cdot (2 \ pi {f}) ^ {2} \ cdot \ sin (2 \ pi {f} \ cdot {t})}$

## Use in teaching

Since the damping is low due to the design, some experiments can be carried out with rotary pendulums and torsional oscillators: for example, it is possible to install an easily controllable damping system on the oscillator by means of an eddy current brake, which causes almost the only losses. This enables an investigation of damped linear oscillations under easily controllable conditions. Here are some connections that can be proven:

• The undamped natural angular frequency is independent of the damping${\ displaystyle \ omega _ {0}}$
• The damped natural angular frequency with a Lehrsches attenuation${\ displaystyle \ omega _ {d} = \ omega _ {0} \ cdot {\ sqrt {1-d ^ {2}}}}$${\ displaystyle d}$
• The resonance frequency is and the resonance increase is${\ displaystyle \ omega _ {r} = \ omega _ {0} \ cdot {\ sqrt {1-2d ^ {2}}}}$${\ displaystyle {\ frac {1} {2d {\ sqrt {1-d ^ {2}}}}}}$

## use

The torsion pendulum is used in rotary pendulum clocks (mechanical clocks with a rotary pendulum as a time standard). Due to the low damping, such watches can run for a very long time (e.g. 1 year) without being wound.

Cavendish pendulum ( gravitational balance )

In the Cavendish experiment ( gravitational balance ), a rotary pendulum is used to determine the gravitational constant .

The following technical applications have a structure comparable to a torsion pendulum, but the pendulum parameters are secondary and only required for calculating and dimensioning the damping:

• Moving coil instruments and galvanometer drives ( mirror galvanometers , galvanometer drives for laser scanners) often have a tension band bearing for the coil. The restoring force is applied by the torsion of the ribbon. The moment of inertia of the coil / the pointer / the mirror and the band form an oscillatory system similar to a rotary pendulum, which in this case has to be dampened as well as possible.