Perpendicular Axes Theorem Experiment

To Study the Theorem of Perpendicular Axes of Moment of Inertia

1. Aim

To verify the theorem of perpendicular axes for moment of inertia using a regular lamina (e.g., a circular or rectangular plate) and to experimentally confirm that the moment of inertia about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane of the lamina.

2. Apparatus Used

  • A regular lamina (circular or rectangular)
  • A torsion head with suspension wire
  • Stopwatch
  • Vernier calipers
  • Screw gauge
  • Meter scale
  • Weights (if required for calibration)
  • Clamps and stand

3. Diagram

Experimental Setup Diagram

Figure 1: Diagram showing the lamina with X, Y, and Z axes of rotation.

4. Theory

The theorem of perpendicular axes states that for a planar lamina, the moment of inertia about an axis (Z) perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular axes (X and Y) lying in the plane of the lamina and intersecting at the point where the perpendicular axis passes through the lamina.

Mathematically,

\[ I_z = I_x + I_y \]

For a circular lamina:

\[ I_x = I_y = \frac{MR^2}{4} \text{ (about diameter)} \] \[ I_z = \frac{MR^2}{2} \text{ (about central perpendicular axis)} \]

For a rectangular lamina:

\[ I_x = \frac{Mb^2}{12} \] \[ I_y = \frac{Ma^2}{12} \] \[ I_z = \frac{M(a^2 + b^2)}{12} \]

5. Formula

The moment of inertia (\( I \)) of a body oscillating in torsion is given by:

\[ I = \frac{T^2 C}{4 \pi^2} \]

where:

  • \( T \) = Time period of oscillation
  • \( C \) = Torsional constant of the suspension wire

6. Procedure

Step 1: Determine Torsional Constant (C)

  1. Suspend a known object (e.g., a cylinder) of known moment of inertia \( I_0 \).
  2. Measure the time period \( T_0 \) for small oscillations.
  3. Calculate \( C \) using:
    \[ C = \frac{4 \pi^2 I_0}{T_0^2} \]

Step 2: Measure Moment of Inertia about X, Y, and Z Axes

  1. For X-axis: Suspend the lamina so that it oscillates about the X-axis. Measure time period \( T_x \).
  2. For Y-axis: Repeat the process for the Y-axis and measure \( T_y \).
  3. For Z-axis: Suspend the lamina vertically to oscillate about the Z-axis and measure \( T_z \).

Step 3: Calculate Moments of Inertia

Using the formula:

\[ I = \frac{T^2 C}{4 \pi^2} \]

Find \( I_x, I_y, \) and \( I_z \).

Step 4: Verification of Perpendicular Axis Theorem

Check if:

\[ I_z \approx I_x + I_y \]

7. Observation Table

Measurement Time Period (T) Moment of Inertia (I)
X-axis \( T_x = \) ___ \( I_x = \) ___
Y-axis \( T_y = \) ___ \( I_y = \) ___
Z-axis \( T_z = \) ___ \( I_z = \) ___

(Additional columns may be added for multiple trials.)

8. Calculations

  • Calculate \( I_x, I_y, I_z \) using the formula.
  • Verify:
    \[ I_z \approx I_x + I_y \]

9. Result

The experimentally determined values confirm that:

\[ I_z = I_x + I_y \]

within permissible experimental error, thus verifying the theorem of perpendicular axes.

10. Precautions

  1. Ensure the lamina oscillates in a horizontal plane without wobbling.
  2. Use small angular displacements to maintain SHM conditions.
  3. Measure the time period accurately by taking multiple oscillations.
  4. Minimize air resistance by performing the experiment in a still environment.
  5. Ensure the suspension wire is rigid and free from kinks.

11. Viva Voce Questions

  1. State the perpendicular axis theorem.
  2. Does the theorem apply to 3D objects? (No, only for planar laminas.)
  3. What is the physical significance of the moment of inertia?
  4. How does the moment of inertia depend on mass distribution?
  5. Why is a torsion pendulum used in this experiment?
  6. What happens if the lamina is not perfectly horizontal during oscillation?
  7. Derive the expression for the moment of inertia of a circular lamina about its diameter.
  8. How would you minimize errors in time period measurement?
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