Lab Manual: Equivalent Simple Pendulum (Disc Pendulum)

Lab Manual: To Determine the Length of the Equivalent Simple Pendulum (Compound Pendulum - Disc Pendulum)

1. Aim

To determine the length of the equivalent simple pendulum (L) for a given compound pendulum (disc pendulum) and to verify the relationship between the time periods of the two pendulums.

2. Apparatus Used

  • A metallic disc with holes for suspension
  • A knife-edge or rigid support for suspension
  • Stopwatch
  • Meter scale
  • Vernier calipers
  • Thread or thin wire for suspension

3. Diagram

Ring Pendulum Experimental Setup

Figure: Disc pendulum suspended from a point on its circumference.

4. Theory

A compound pendulum is a rigid body capable of oscillating about a horizontal axis passing through it. A disc pendulum is a type of compound pendulum where a disc is suspended from a point on its circumference.

The time period (\( T \)) of a compound pendulum is given by:

\[ T = 2\pi \sqrt{\frac{I}{mgl}} \]

where:

  • \( I \) = Moment of inertia about the axis of suspension
  • \( m \) = Mass of the disc
  • \( g \) = Acceleration due to gravity
  • \( l \) = Distance from the pivot to the center of mass

For a disc:

  • The moment of inertia about the center is \( I_{cm} = \frac{1}{2} mR^2 \)
  • Using the parallel axis theorem, the moment of inertia about the suspension point is:
\[ I = I_{cm} + ml^2 = \frac{1}{2}mR^2 + ml^2 \]

The equivalent simple pendulum is a hypothetical pendulum with the same time period as the compound pendulum. Its length (\( L \)) is given by:

\[ L = \frac{I}{ml} = \frac{\frac{1}{2}R^2 + l^2}{l} \]

5. Formula

  1. Time period of the disc pendulum:
    2. \[ T = 2\pi \sqrt{\frac{\frac{1}{2}R^2 + l^2}{gl}} \]
  2. Length of the equivalent simple pendulum:
    3. \[ L = \frac{\frac{1}{2}R^2 + l^2}{l} \]

6. Procedure

  1. Measure the radius (\( R \)) of the disc using a vernier caliper.
  2. Suspend the disc from a point on its circumference using a knife-edge.
  3. Measure the distance (\( l \)) from the suspension point to the center of the disc.
  4. Displace the disc slightly and release it to allow oscillations.
  5. Measure the time for 20 oscillations using a stopwatch and calculate the time period (\( T \)).
  6. Repeat the experiment for different suspension points (varying \( l \)).
  7. Calculate the equivalent length (\( L \)) using the formula.
  8. Plot a graph of \( T \) vs. \( l \) and analyze the results.

7. Observation Table

S.No. Distance (l) (cm) Time for 20 Oscillations (s) Time Period (T) (s) Equivalent Length (L) (cm)
1 ... ... ... ...
2 ... ... ... ...
3 ... ... ... ...

8. Calculations

  • Using the formula \( L = \frac{\frac{1}{2}R^2 + l^2}{l} \), compute \( L \) for each \( l \).
  • Compare the theoretical and experimental values.

9. Result

The length of the equivalent simple pendulum was found to be L = ____ cm (experimental) and L = ____ cm (theoretical). The results are in good agreement (or discuss deviations).

10. Precautions

  1. Ensure the disc oscillates in a vertical plane without wobbling.
  2. Use a small amplitude (≤ 5°) to ensure SHM conditions.
  3. Measure time for multiple oscillations to reduce error.
  4. Avoid air disturbances during oscillations.
  5. Ensure the knife-edge is rigid and frictionless.

11. Viva Voce Questions

  1. What is a compound pendulum?
  2. How does a disc pendulum differ from a simple pendulum?
  3. What is the significance of the equivalent length of a simple pendulum?
  4. Derive the expression for the time period of a disc pendulum.
  5. Why should the amplitude of oscillation be small?
  6. What is the parallel axis theorem, and how is it used here?
  7. How does the time period change if the disc is suspended from different points?
  8. What are the sources of error in this experiment?

12. Graph / Plot (If Required)

  • Plot T vs. l and observe the variation.
  • The graph should show a minimum time period corresponding to the minimum equivalent length.
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