DETERMINATION OF MOMENT OF INERTIA OF A COMPOUND PENDULUM BY METHOD OF COINCIDENCES

1. AIM

To determine the moment of inertia of a compound pendulum using the method of coincidences, and to verify the parallel axis theorem.

2. APPARATUS USED

• Compound pendulum (a rigid body with holes at different distances from the center of mass)
• Knife edge or pivot for suspending the pendulum
• Laboratory second's pendulum with adjustable length
• Stop watch
• Meter scale
• Vernier caliper
• Screw gauge
• Weight balance
• Spirit level

3. DIAGRAM

Fig. Experimental Setup for Determination of Moment of Inertia of a Compound Pendulum by Method of Coincidences

4. THEORY

A compound pendulum is any rigid body that can oscillate freely under the influence of gravity about a horizontal axis passing through the body. The time period of oscillation of a compound pendulum depends on:

• The mass of the pendulum
• The location of the center of mass
• The moment of inertia about the axis of suspension

When a compound pendulum oscillates with small amplitude, its time period is given by:

Where:

• \(T\) is the time period of oscillation
• \(I\) is the moment of inertia about the axis of suspension
• \(m\) is the mass of the pendulum
• \(h\) is the distance between the center of mass and the axis of suspension
• \(g\) is the acceleration due to gravity

According to the parallel axis theorem, the moment of inertia of a body about any axis is equal to the sum of the moment of inertia about a parallel axis passing through its center of mass and the product of its mass and the square of the distance between the two axes:

Where \(I_{cm}\) is the moment of inertia about the center of mass.

The method of coincidences uses a second's pendulum (a simple pendulum with a time period of exactly 2 seconds) alongside the compound pendulum. When both pendulums start swinging together, they gradually go out of phase and then come back into phase after a certain number of oscillations. The observation of these coincidences helps determine the precise time period of the compound pendulum.

5. FORMULA

The moment of inertia of the compound pendulum about the axis of suspension is given by:

Where:

• \(I\) is the moment of inertia about the axis of suspension (in kg·m²)
• \(m\) is the mass of the pendulum (in kg)
• \(g\) is the acceleration due to gravity (in m/s²)
• \(h\) is the distance between the center of mass and the axis of suspension (in m)
• \(T\) is the time period of oscillation (in s)

For the method of coincidences, the time period of the compound pendulum is calculated as:

Where:

• \(T\) is the time period of the compound pendulum
• \(n\) is the number of oscillations of the compound pendulum between two successive coincidences
• The sign \(\pm\) is + if the compound pendulum is faster than the second's pendulum and - if it is slower

The moment of inertia about the center of mass (\(I_{cm}\)) can be calculated using:

6. PROCEDURE

1. Set up the compound pendulum on a stable support with the knife edge or pivot.
2. Balance the setup using the spirit level to ensure the axis of oscillation is horizontal.
3. Measure the mass (\(m\)) of the compound pendulum using the weight balance.
4. Locate the center of mass of the pendulum by balancing it horizontally.
5. Choose a hole for suspension and measure the distance (\(h\)) from the center of mass to this point using a meter scale.
6. Set up the second's pendulum nearby, ensuring it has a time period of exactly 2 seconds.
7. Displace both pendulums slightly (approximately 5° from vertical) and release them simultaneously.
8. Count the number of oscillations (\(n\)) of the compound pendulum between two successive coincidences.
9. Record the time (\(t\)) for these \(n\) oscillations using a stopwatch.
10. Calculate the time period (\(T\)) of the compound pendulum using the formula \(T = \frac{2n}{n\pm1}\).
11. Alternatively, calculate \(T = \frac{t}{n}\) from direct timing measurements.
12. Repeat steps 7-11 at least three times and take the average value of \(T\).
13. Change the position of suspension by selecting a different hole and repeat steps 5-12.
14. Perform the experiment for at least 5 different suspension points.
15. Calculate the moment of inertia for each suspension point using the formula \(I = mgh\left(\frac{T}{2\pi}\right)^2\).
16. Plot a graph of \(I\) versus \(h^2\).
17. Calculate the moment of inertia about the center of mass (\(I_{cm}\)) using the slope and intercept of the graph.

7. OBSERVATION TABLE

Table 1: Physical Parameters of the Compound Pendulum

Table 2: Measurements for Different Suspension Points

8. CALCULATIONS

1. Calculate the time period (\(T\)) for each suspension point using:

   \[ T = \frac{2n}{n\pm1} \]

   or

   \[ T = \frac{t}{n} \]

2. Calculate the moment of inertia (\(I\)) for each suspension point using:

   \[ I = mgh\left(\frac{T}{2\pi}\right)^2 \]

3. Plot a graph of \(I\) versus \(h^2\) (This should be a straight line according to the parallel axis theorem: \(I = I_{cm} + mh^2\))

4. Determine the slope and intercept of the graph:

   • The slope should be equal to \(m\) (mass of the pendulum)
   • The intercept should be equal to \(I_{cm}\) (moment of inertia about the center of mass)

5. Calculate \(I_{cm}\) using the graph and also using the equation:

   \[ I_{cm} = I - mh^2 \]

   for each suspension point and find the average value.

9. RESULT

1. The moment of inertia of the compound pendulum about different axes of suspension:

   • \(I_1\) = ___ kg·m² (at \(h_1\) = ___ m)
   • \(I_2\) = ___ kg·m² (at \(h_2\) = ___ m)
   • \(I_3\) = ___ kg·m² (at \(h_3\) = ___ m)
   • \(I_4\) = ___ kg·m² (at \(h_4\) = ___ m)
   • \(I_5\) = ___ kg·m² (at \(h_5\) = ___ m)

2. The moment of inertia of the compound pendulum about its center of mass:

   • From the graph: \(I_{cm}\) = ___ kg·m² (intercept of the \(I\) vs \(h^2\) graph)
   • From calculations: \(I_{cm}\) (average) = ___ kg·m²

3. Verification of the parallel axis theorem:

   • Theoretical slope = \(m\) = ___ kg
   • Experimental slope from the graph = ___ kg
   • Percentage error = ___ %

10. PRECAUTIONS

1. The amplitude of oscillation should be kept small (less than 5°) to ensure simple harmonic motion.
2. The knife edge should be sharp, and the suspension should be friction-free.
3. The second's pendulum should be accurately set to have a time period of exactly 2 seconds.
4. Both pendulums should be started simultaneously for accurate coincidence observations.
5. Ensure that the axis of oscillation is perfectly horizontal using a spirit level.
6. The center of mass should be determined accurately.
7. Measurements of distances and times should be taken with precision.
8. The counting of oscillations should be done carefully, without missing any oscillations.
9. The suspension points should be selected such that they are at different distances from the center of mass.
10. The laboratory environment should be free from air currents that could disturb the pendulum motion.
11. The time period should be measured for a large number of oscillations to minimize errors.
12. Multiple readings should be taken for each suspension point to increase accuracy.

11. VIVA VOICE QUESTIONS

1. What is a compound pendulum? How does it differ from a simple pendulum?
2. Explain the principle of the method of coincidences.
3. State and explain the parallel axis theorem.
4. Why is the relationship between \(I\) and \(h^2\) expected to be linear?
5. Why must we keep the amplitude of oscillation small?
6. How would the time period change if the compound pendulum is suspended from a point closer to its center of mass?
7. What happens to the time period when the pendulum is suspended from its center of mass?
8. How would you find the radius of gyration of the compound pendulum?
9. What is meant by the center of oscillation of a compound pendulum?
10. What is a second's pendulum? Why is it used in this experiment?
11. How does the moment of inertia vary with the position of the axis of suspension?
12. If the compound pendulum is taken to the moon, how would its time period change?
13. What are the possible sources of error in this experiment?
14. Explain the significance of the intercept in the \(I\) vs \(h^2\) graph.
15. What is the physical significance of the radius of gyration?

12. GRAPH/PLOT

Plot a graph with:

• \(h^2\) (in m²) on the x-axis
• Moment of inertia \(I\) (in kg·m²) on the y-axis

The graph should be a straight line according to the equation \(I = I_{cm} + mh^2\). The slope of this line represents the mass (\(m\)) of the pendulum, and the y-intercept gives the moment of inertia about the center of mass (\(I_{cm}\)).