Torsional Pendulum EX-5521A Page 1 of 6

Torsional Pendulum
EQUIPMENT

INCLUDED:
1 Torsion Pendulum Accessory ME-6694
1 Large Rod Stand ME-8735
1 60 cm Long Steel Rod ME-8977
1 Rotational Inertia Accessory ME-3420
1 Rotary Motion Sensor PS-2120A
1 High Resolution Force Sensor PS-2189
NOT INCLUDED, BUT REQUIRED:
1 Pliers for bending wire
1 Mass Balance SE-8757B
1 Calipers SE-8710
1 850 Universal Interface UI-5000
1 PASCO Capstone UI-5400

INTRODUCTION

The torsional pendulum consists of a torsion wire attached to a Rotary Motion Sensor with an
object (a disk, a ring, or a rod with point masses) mounted on top of it. The period of oscillation
is measured from a plot of the angular displacement versus time. To calculate the theoretical
period, the rotational inertia is determined by measuring the dimensions of the object and the
torsional spring constant is determined from the slope of a plot of force versus angular
displacement.

The dependence of the period on the torsional constant and the rotational inertia is explored by
using different diameter wires and different shaped objects.

Written by Ann Hanks

Torsional Pendulum EX-5521A Page 2 of 6

THEORY

Consider a wire securely fixed on both ends. If the wire is twisted, it will exert a restoring torque
when trying to return to its original untwisted position. For small twists, the restoring torque is
proportional to the angular displacement of the wire.

𝜏𝜏 = 𝜅𝜅𝜅𝜅 𝜏𝜏 = 𝜅𝜅𝜅𝜅 (1)

The proportionality constant, κ, depends on the

properties of the wire and is called the torsional spring
constant.
θ Equilibrium
Position When the object attached to the wire is twisted and
released, the object executes simple harmonic motion
with a period, T, given by
𝐼𝐼
𝑇𝑇 = 2𝜋𝜋�𝜅𝜅 (2)
where I is the rotational inertia of the object about the
Wire axis of rotation.

Theoretically, the rotational inertia, I, of a ring is given

by
1
𝐼𝐼 = 2 𝑀𝑀(𝑅𝑅12 + 𝑅𝑅22 ) (3)

where M is the mass of the ring, R1 is the inner radius of the ring, and R2 is the outer radius of
the ring. The rotational inertia of a disk is given by
1
𝐼𝐼 = 2 𝑀𝑀𝑅𝑅 2 (4)

where M is the mass of the disk and R is the radius of the disk.

The rotational inertia of a point mass rotating in a circle of radius r is given by

𝐼𝐼 = 𝑀𝑀𝑅𝑅 2 (5)

Written by Ann Hanks

Torsional Pendulum EX-5521A Page 3 of 6

SET UP

1. Start with the 0.032" diameter wire. Use pliers to bend each end of the wire into an "L"
shape.

2. Fit the bent ends of the wire under the screws and washers of the upper and lower clamps, as
illustrated in Figures 1 and 2. Make sure the screws are firmly tightened.

Figure 1: Upper Clamp Figure 2: Lower Clamp

Figure 3: Setup

3. Adjust the Rotary Motion Sensor on the support rod such that the guide on the upper clamp is
aligned with the slot on the shaft of the Rotary Motion Sensor. See Figure 3.

4. Adjust the height of the set up so that the upper clamp is approximately half way up the shaft
of the Rotary Motion Sensor (see Figure 3). NOTE: When switching to a new diameter
wire, try to keep the length of the wires, from clamp to clamp, relatively constant.

5. Plug the Rotary Motion Sensor and Force Sensor into the interface.

Written by Ann Hanks

Torsional Pendulum EX-5521A Page 4 of 6

PROCEDURE

A. Determining the Torsional Spring Constant

1. Measure the radius of the medium pulley of the Rotary Motion Sensor in meters. Enter
this radius (not diameter!) into the Capstone calculator window where it asks for the
experimental constants. The torque is calculated using τ=rF, where F is the force
measured using the Force Sensor.

2. In PASCO Capstone, create a graph of torque vs. angle.

3. Attach about 20 cm of string to the Rotary Motion Sensor by tying it around the small
pulley. Then thread the string through the notch in the medium pulley and wrap the
string around the medium pulley 3 times. Attach the Force Sensor to the end of the
string.

4. Set the sample rate for both sensors on 20 Hz.

5. Hold the force sensor parallel to the table at the height of the large pulley and prepare to
pull it straight out as shown in Figure 4.

Figure 4: Measuring the Torque

6. Let the string go slack and press the tare button on the Force Sensor. Click the RECORD
button in PASCO Capstone and pull the Force Sensor horizontally until the pulley turns
about one revolution. Click on STOP.

7. Use the Fit Tool to determine the slope of the graph of Torque vs. Angle. This slope is
equal to the torsional spring constant for the wire (see Equation 1). Record the spring
constant and the error in the spring constant.

Written by Ann Hanks

Torsional Pendulum EX-5521A Page 5 of 6

B. Determining the Rotational Inertia

1. Measure the mass and radius of the disk.

2. Calculate the rotational inertia of the disk using Equation (4).

C. Calculating the Theoretical Period of Oscillation

Using the rotational inertia of the disk and the torsional spring constant for the wire,
calculate the theoretical period using Equation (2). Use the error in the spring constant to
estimate the error in the theoretical period.

D. Measuring the Period of Oscillation

1. Remove the Force Sensor. The string can still be attached in this part of the
experiment as long as it does not impede the oscillation.

2. Change the sample rate to 200 Hz. Create a graph of the angle vs. time. Twist the disk
1/4 of a turn.

3. Start recording and release the disk.

4. After several oscillations have been completed, click on STOP.

5. Use the Coordinates Tool to find the period of oscillation. Measure the time of
several periods and then divide by the number of periods.

6. Compare the measured and calculated values of the period using a percent difference.

measured − calculated
%difference = x100
calculated

E. Repeating the Experiment

1. Repeat Steps B through D with the ring added to the top of the disk.

2. Remove the disk and the ring. Repeat Steps B through D using the rod with a point
mass on each end of the rod. Invert the 3-step pulley on the Rotary Motion Sensor
before attaching the rod (see Figure 5). For this part of the lab, the rotational inertia
of the rod is ignored because it is small compared to the point masses. However, you
can take the rotational inertia of the rod into account. For a thin rod of length L and
1
mass m, the rotational inertia is 𝐼𝐼 = 12 𝑚𝑚𝐿𝐿2 .

Written by Ann Hanks

Torsional Pendulum EX-5521A Page 6 of 6

Figure 5: Point Masses

3. Replace the wire with a wire of different diameter but same length. Return to using the
disk. Repeat Steps A, C, and D.

QUESTIONS

1. Which of the wires was harder to twist? What does κ tell you about how much a wire
resists bending and twisting?

2. Which of the wires oscillated faster (smaller Period)?

3. How does the period relate to which wire is harder to twist? Explain.

4. Using the same wire, which object had the least rotational inertia?

5. Using the same wire, which object oscillated faster?

6. How does the period depend on the rotational inertia of the object?

7. How much error is caused by ignoring the rod in the point mass part of the experiment?

8. Was there any other source of rotational inertia that was ignored in this experiment?

9. How could you use a torsional pendulum to determine the rotational inertia of any object
that could be mounted on the Rotary Motion Sensor?

Written by Ann Hanks