Bifilar Suspension Experiment

Determination of Moment of Inertia Using Bifilar Suspension Method

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1. Aim

To determine the moment of inertia of a given rectangular body (bar) about an axis passing through its center using the bifilar suspension method.

2. Apparatus Used

  • A rectangular bar (metal or wooden)
  • Two parallel strings/threads of equal length
  • A rigid support (e.g., retort stand)
  • Measuring scale (meter scale)
  • Stopwatch
  • Vernier calipers (for measuring small distances)
  • Weighing balance (to measure mass of the bar)

3. Diagram

Figure 1: Bifilar suspension setup showing the bar suspended by two parallel strings
Figure 1: Bifilar suspension setup showing the bar suspended by two parallel strings

4. Theory

The bifilar suspension method involves suspending a body (bar) horizontally using two parallel strings of equal length. When the bar is twisted slightly in the horizontal plane and released, it undergoes torsional oscillations.

The moment of inertia (I) of the bar about its central axis can be determined by measuring the time period of oscillation. The restoring torque is provided by the tension in the strings, and the system behaves like a torsional pendulum.

5. Formula

The moment of inertia (I) of the bar is given by:

\[ I = \frac{m \cdot g \cdot d^2 \cdot T^2}{16 \pi^2 L} \]

Where:

  • m = Mass of the bar
  • g = Acceleration due to gravity (9.81 m/s²)
  • d = Distance between the two strings
  • T = Time period of oscillation
  • L = Length of the suspension strings

6. Procedure

Step 1: Setup the Bifilar Suspension

  1. Suspend the bar horizontally using two parallel strings of equal length (L) attached to a rigid support.
  2. Ensure that the strings are vertical and symmetrically placed at equal distances (d/2) from the center of the bar.

Step 2: Measurement of Parameters

  1. Measure the length of the strings (L) using a meter scale.
  2. Measure the distance (d) between the two strings using a vernier caliper.
  3. Measure the mass (m) of the bar using a weighing balance.

Step 3: Determination of Time Period

  1. Gently twist the bar in the horizontal plane and release it to allow small oscillations.
  2. Measure the time (t) for 20 complete oscillations using a stopwatch.
  3. Repeat the experiment 3-5 times and take the average time period (T = t/20).

Step 4: Calculation

  1. Substitute the values of m, g, d, T, L into the formula to calculate the moment of inertia (I).

7. Observation Table

S.No Length of strings (L) (m) Distance between strings (d) (m) Mass of bar (m) (kg) Time for 20 oscillations (t) (s) Time period (T = t/20) (s)
1
2
3
Mean

8. Calculations

Using the mean values:

\[ I = \frac{m \cdot g \cdot d^2 \cdot T^2}{16 \pi^2 L} \]

Substitute the values and compute I.

9. Result

The moment of inertia of the given bar about its central axis is found to be ______ kg·m².

10. Precautions

  • Ensure that the strings are perfectly vertical and of equal length.
  • The oscillations should be small (less than 10°) to maintain SHM conditions.
  • Avoid any external disturbances during the experiment.
  • Measure the time period accurately by taking multiple readings.
  • Ensure that the bar remains horizontal during oscillations.

11. Viva Voce Questions

  1. What is the principle behind the bifilar suspension method?
  2. Why should the oscillations be small?
  3. How does the length of the strings affect the time period?
  4. What would happen if the strings were not parallel?
  5. How can you minimize errors in this experiment?
  6. What is the physical significance of the moment of inertia?
  7. How does mass distribution affect the moment of inertia?
  8. Can this method be used for irregular bodies? Why or why not?
  9. What is the role of gravity in this experiment?
  10. How would air resistance affect the results?

(Optional: If required, a graph of T vs. d² can be plotted to verify the relationship.)

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