# Torsional Pendulum

Consider a body suspended by a wire. When the body is slightly rotated (twisted) about the vertical axis (i.e., the wire) and released, it starts performing simple harmonic motion (SHM).

The restoring torque is provided by the twisting of the wire. The restoring torque is proportional to the angle of twist i.e., $\tau=c\theta$, where $c$ is the torsional spring constant of the wire.

Let $I$ be moment of inertia of the body about the axis of oscillation. Apply $\tau=I\alpha=I\mathrm{d}^2\theta/\mathrm{d}t^2$ to get, \begin{align} &\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}+\frac{c}{I}\theta=0. \end{align} Compare above equation with general equation of angular SHM \begin{align} &\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}+\omega^2\theta=0, \end{align} to get the angular frequency \begin{align} \omega=\sqrt{\frac{c}{I}} \end{align} and time period \begin{align} T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{I}{c}} \end{align}

The body undergoes torsional SHM for fairly large value of $\theta$ i.e., amplitude need not be small.

## Video based question on torsional SHM

The wire and its length are the same in the three cases. The masses are equal in cases 2 and 3

Question 1. What can you deduce from these experiments

1. The time period increases with an increase in the moment of inertia
2. The time period decreases with an increase in the moment of inertia
3. The time period is independent of the mass
4. The time period is independent of the mass distribution

Question 2. The time periods in three cases are?

## Problem on torsional pendulum

Problem: A torsional pendulum is designed to measure moment of inertia of a body. With a body of known moment of inertia $I_1$, a torsional frequency of oscillation $f_1$ is recorded. The frequency becomes $f_2$ if this body is replaced by another body of unknown moment of inertia $I_2$. The frequency of the pendulum alone, with no added mass, is $f_0$. Find the moment of inertia $I_2$ in terms of the known quantities.

Solution: Let $I_0$ be moment of inertia of the torsional pendulum about its axis of oscillation (without added mass) and $k$ be its torsional spring constant. The frequency of this pendulum is given by \begin{align} f_0=\frac{1}{2\pi}\sqrt{\frac{k}{I_0}}. \end{align} If a body of known moment of inertia $I_1$ is symmetrically placed on the pendulum disc then its moment of inertia becomes $I=I_0+I_1$ and the oscillation frequency changes to \begin{align} f_1=\frac{1}{2\pi}\sqrt{\frac{k}{I_0+I_1}}. \end{align} If a body of unknown moment of inertia $I_2$ is placed on the pendulum disc then its moment of inertia becomes $I=I_0+I_2$ and the oscillation frequency changes to \begin{align} f_2=\frac{1}{2\pi}\sqrt{\frac{k}{I_0+I_2}}. \end{align} Eliminate $k$ and $I_0$ from above equations and solve for $I_2$ to get \begin{align} I_2=I_1\left[\frac{(f_0/f_2)^2-1}{(f_0/f_1)^2-1}\right].\nonumber \end{align}

In physics laboratory, an inertia table is used to find moment of inertia of an unknown body. It works on the principle given in this problem.

## Experiment on torsional oscillations

If a wire is fixed at one end and is twisted at the other end, different parts of the wire get twisted by different angles. Accordingly, if you give torsional vibration to the free end, different parts of the wire will undergo torsional vibration with different amplitudes. In this experiment you will study the damping characteristic, the time period dependence on amplitude and variation of vibration amplitude with distance from the fixed end of a wire under such conditions.

### Procedure

You need a wire fixed at one end and carrying a weight at the other end and with 4 plane mirrors fixed on it, a laser clamped in a stand, a screen with graph paper attached on it, stop watch, graph etc.

You have to find the relation in the vibration amplitudes at 4 different places. This will take time and the vibration amplitudes will decrease during the period. To correct for this you will need the damping characteristic. So first study the damping characteristics.

### Damping

Set your laser so that it falls at the lowest mirror at an angle say 45 deg. Put the screen 1 m to 2 m away from the mirror so as to receive the reflected light normally. Now give a small vibration to the lower end by twisting it by a small angle. The light spot on the screen should also vibrate. Note down the vibration amplitude of the spot as a function of time for about 10 minutes. Draw a graph. From the geometry find the relation between the amplitude of the spot on the screen and the angular amplitude of vibration of the wire at the mirror position.

### Time period versus amplitude

Does time period depend on amplitude? Measure the time for 10 oscillations for different amplitudes, like 1-2 deg, then 10-15 deg, 60-70 deg etc. See if there is a dependence of time period on amplitude. You can decide on the vibration amplitude of the spot on the screen.

### Amplitude versus distance

Vibrate the wire at the end and measure the spot amplitude. Note the time. Let the vibrations continue and set your laser so that it falls on the next mirror at about 45 deg. Measure the spot amplitude now. It will be smaller. Note the time. Amplitude is smaller because of two reasons, you have gone up on the wire and also vibrations are damped. From the time difference between the two measurements, correct for damping and get the amplitude of the 2nd mirror corresponding to that of the 1st mirror. Do the same for 3rd mirror. Get the ratio of the distances from the mirror and the ratio of the corresponding amplitudes. What relation do you find between them?