## Torsion Pendulum

Life swings like a pendulum backward and forward between pain and boredom.
Arthur Schopenhauer

### 1 Introduction

Oscillations show up throughout physics. From simple spring systems in mechanics to atomic bonds in quantum physics to bridges blowing the wind, physical systems often act like oscillators when they are displaced from stable equilibria.

In this experiment you will observe the behavior of a simple sort of oscillator: the torsion pendulum. In general a torsion pendulum is an object that has oscillations which are due to rotations about some axis through the object. This apparatus allows for exploring both damped oscillations and forced oscillations.

### 2 Theory

Note that angular frequency (ω in rad/s) and frequency (f in Hz.) are not the same. Most the equations below concern ω, in many cases it is easier to measure f.

In the damped case, the torque balance for the torsion pendulum yields the differential equation:

 (1)

where J is the moment of inertia of the pendulum, b is the damping coefficient, c is the restoring torque constant, and θ is the angle of rotation [

In the underdamped case (β < ω0):

 (2)

with the oscillation frequency ω1 = , initial amplitude θ0, and phase γ.

In the critically damped case (β = ω0):

 (3)

In the overdamped case (β > ω0):

 (4)

where ω2 = .

For the forced oscillation case, an external torque is added to Equation 1:

 (5)

where ω is the driving frequency and τ0 is the driving torque [

### 3 Equipment

In this experiment you will use the torsion pendulum, the power supply for the driving motor, a low voltage power supply for the eddy current damper, two digital multimeters, and a stop watch.

Figure 1 shows the torsion pendulum and associated electronics. The motor which is used to force the pendulum (which will only be used in the second half of the experiment) is shown on the left of the diagram. The eddy current damping device is shown on the bottom of the diagram.