Determination of Moment of Inertia using Torsion Pendulum

1. Aim

To determine the moment of inertia of a disc using a torsion pendulum by:

• Measuring the time period of oscillations without the disc (only suspension wire)
• Measuring the time period of oscillations with the disc attached
• Comparing experimental and theoretical values of the moment of inertia

2. Apparatus Used

• Torsion pendulum setup (thin metallic wire, rigid support, and disc)
• Stopwatch
• Vernier calipers / Screw gauge (for measuring disc and wire dimensions)
• Meter scale
• Balance (to measure mass of the disc)

3. Diagram

Figure 1: Torsion pendulum setup with and without disc

4. Theory

A torsion pendulum consists of a wire fixed at one end and a disc attached at the other. When twisted, the restoring torque causes oscillations.

The time period (T) depends on:

• Moment of inertia (I) of the oscillating system
• Torsional rigidity (C) of the wire

The moment of inertia of the disc is found by:

1. Measuring the time period without the disc (T₀) → gives the inertia of the wire
2. Measuring the time period with the disc (T₁) → gives the combined inertia of wire + disc
3. Using the difference to find the disc's moment of inertia

5. Formula

(a) Time Period of Torsional Oscillations

(b) Torsional Constant (C) of the Wire

(c) Moment of Inertia of the Disc (Experimental)

(d) Theoretical Moment of Inertia of Disc

6. Procedure

Part A: Without the Disc (Only Wire Inertia)

1. Set up the torsion pendulum with only the wire (no disc)
2. Gently twist the wire and measure the time for 20 oscillations (repeat 3 times)
3. Calculate the average time period ($T_0$)

Part B: With the Disc (Wire + Disc Inertia)

1. Attach the disc to the wire
2. Measure the time for 20 oscillations (repeat 3 times)
3. Calculate the average time period ($T_1$)

Part C: Measurements

1. Measure:
   • Length of wire ($l$)
   • Diameter of wire ($d$) → radius ($r = d/2$)
   • Mass of disc ($M$)
   • Radius of disc ($R$)

7. Observation Table

Additional Measurements:

• Mass of disc ($M$) = _____ kg
• Radius of disc ($R$) = _____ m
• Length of wire ($l$) = _____ m
• Diameter of wire ($d$) = _____ m → Radius ($r$) = _____ m
• Modulus of rigidity ($\eta$) = _____ N/m² (if given)

8. Calculations

1. Torsional Constant (C):

2. Moment of Inertia of Disc (Experimental):

3. Theoretical Moment of Inertia of Disc:

4. Percentage Error:

9. Result

• Moment of inertia of disc (experimental): $I_{disc} =$ _____ kg·m²
• Moment of inertia of disc (theoretical): $I_{theory} =$ _____ kg·m²
• Percentage error: _____ %

10. Precautions

1. Ensure the wire is clamped tightly to avoid slipping
2. Oscillations should be purely torsional (no side swings)
3. Measure time for multiple oscillations to reduce errors
4. Use a thin and uniform wire for accurate results
5. Avoid air drafts that may disturb oscillations
6. Ensure the disc is perfectly horizontal during oscillations
7. Take measurements at room temperature to avoid thermal effects on the wire

11. Viva Voce Questions

Graph/Plot (If Required)

• Plot $T^2$ vs. $I$ for different masses (if multiple discs are used)
• The slope should be $\frac{4\pi^2}{C}$