Ring Pendulum Experiment

Lab Manual: To Determine the Length of the Equivalent Simple Pendulum (Ring Pendulum)

1. Aim

To determine the length of the equivalent simple pendulum for a given ring pendulum and to verify the relationship between the time periods of the ring pendulum and its equivalent simple pendulum.

2. Apparatus Used

  • A metal ring with known diameter
  • A rigid support stand
  • A knife-edge for suspension
  • A stopwatch (least count = 0.1 s)
  • A meter scale (least count = 0.1 cm)
  • A vernier caliper (least count = 0.01 cm)

3. Diagram

Ring Pendulum Diagram

A labeled diagram showing the ring suspended from a knife-edge, oscillating about an axis perpendicular to its plane.

4. Theory

A ring pendulum is a rigid body oscillating about a horizontal axis passing through its periphery. The time period of oscillation of a ring pendulum is given by:

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

where:

  • \( I \) = Moment of inertia of the ring about the axis of suspension
  • \( m \) = Mass of the ring
  • \( l \) = Distance from the suspension point to the center of mass (equal to the radius \( R \) of the ring)
  • \( g \) = Acceleration due to gravity

The moment of inertia of the ring about the suspension axis (using the parallel axis theorem) is:

\[ I = mR^2 + mR^2 = 2mR^2 \]

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

\[ L = \frac{I}{ml} = \frac{2mR^2}{mR} = 2R \]

Thus, the length of the equivalent simple pendulum should be equal to the diameter of the ring.

5. Formula

  • Time period of ring pendulum:
    \[ T_{ring} = 2\pi \sqrt{\frac{2R}{g}} \]
  • Length of equivalent simple pendulum:
    \[ L = 2R \]

6. Procedure

  1. Measure the diameter of the ring using a vernier caliper and record it.
  2. Suspend the ring from a rigid support using a knife-edge so that it can oscillate freely in its plane.
  3. Displace the ring slightly and release it to allow small oscillations.
  4. Measure the time for 20 complete oscillations using a stopwatch and calculate the time period \( T \).
  5. Repeat the experiment for different suspension points (if possible) and record observations.
  6. Calculate the equivalent length \( L \) using the formula and compare it with the theoretical value \( 2R \).

7. Observation Table

S.No. Diameter of Ring (2R) (cm) Time for 20 Oscillations (s) Time Period \( T \) (s) \( L = \frac{gT^2}{4\pi^2} \) (cm) Theoretical \( L = 2R \) (cm)
1
2
3

8. Calculations

Sample Calculation:

Given: \( R = 15 \, \text{cm} \), \( T = 1.1 \, \text{s} \), \( g = 980 \, \text{cm/s}^2 \)

\[ L = \frac{gT^2}{4\pi^2} = \frac{980 \times (1.1)^2}{4 \times (3.14)^2} \approx 30.1 \, \text{cm} \]

Theoretical \( L = 2R = 30 \, \text{cm} \)

9. Result

The experimentally determined length of the equivalent simple pendulum is ____ cm, which closely matches the theoretical value of ____ cm (diameter of the ring).

10. Precautions

  • Ensure that the ring oscillates in a vertical plane without any wobbling.
  • Use a stopwatch with a small least count for accurate time measurement.
  • Keep the amplitude of oscillation small (≤ 5°) to ensure simple harmonic motion.
  • Repeat the experiment multiple times to minimize errors.

11. Viva Voce Questions

  1. What is a compound pendulum? How does it differ from a simple pendulum?
  2. Derive the expression for the time period of a ring pendulum.
  3. What is the equivalent simple pendulum?
  4. Why should the amplitude of oscillation be small?
  5. How does the moment of inertia affect the time period of a ring pendulum?
  6. What is the parallel axis theorem, and how is it applied here?
  7. What are the sources of error in this experiment?
Scroll to Top