Moment of Inertia of a Flywheel

Determination of Moment of Inertia of a Flywheel

(Rotation Before and After String Detachment)

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1. Aim

To determine the moment of inertia (I) of a flywheel by studying its rotational motion before and after the string is detached.

2. Apparatus Used

  • Flywheel mounted on an axle
  • String with a mass hanger
  • Slotted masses (weights)
  • Stopwatch
  • Vernier caliper / Screw gauge (to measure radius)
  • Meter scale
  • Spirit level (to ensure horizontal alignment)

3. Diagram

flywheel_diagram

Flywheel Setup:

  • Flywheel (F) with axle
  • String (S) wound around the axle
  • Mass (m) attached to the free end of the string

4. Theory

When a mass m is attached to the string wound around the flywheel's axle and released, it accelerates downward due to gravity, causing the flywheel to rotate.

The kinetic energy of the falling mass is converted into:

  1. Rotational kinetic energy of the flywheel
  2. Potential energy loss of the mass
  3. Work done against friction

After the string detaches, the flywheel continues rotating but slows down due to frictional torque (τ).

The moment of inertia (I) can be calculated using the energy conservation principle and rotational dynamics equations.

5. Formula

The moment of inertia (I) is given by:

\[ I = \frac{m \cdot r \cdot (g - a)}{a} - \frac{\tau \cdot t_1}{\omega_0} \]

Where:

  • m = mass attached (kg)
  • r = radius of the axle (m)
  • g = acceleration due to gravity (9.81 m/s²)
  • a = linear acceleration of the mass (m/s²)
  • τ = frictional torque (N·m)
  • t₁ = time taken for the flywheel to stop after string detachment (s)
  • ω₀ = initial angular velocity just after detachment (rad/s)

(Derivation steps can be included if required.)

6. Procedure

A. Before String Detachment

  1. Measure the radius (r) of the axle using a Vernier caliper.
  2. Wind the string around the axle and attach a known mass (m) to its free end.
  3. Release the mass and allow it to fall, rotating the flywheel.
  4. Use a stopwatch to measure the time (t) taken for the mass to fall a known height (h).
  5. Calculate linear acceleration (a) using:
    \[ h = \frac{1}{2} a t^2 \implies a = \frac{2h}{t^2} \]

B. After String Detachment

  1. Once the string detaches, measure the time (t₁) until the flywheel comes to rest.
  2. Determine the initial angular velocity (ω₀) just after detachment:
    \[ \omega_0 = \frac{v}{r} = \frac{a t}{r} \]
  3. Repeat the experiment for different masses and record observations.

7. Observation Table

Mass (m) (kg) Height (h) (m) Time of Fall (t) (s) Acceleration (a) (m/s²) Time to Stop (t₁) (s) ω₀ (rad/s)
0.05 1.0 2.5 0.32 10.2 4.0
0.10 1.0 1.8 0.62 8.5 6.9
... ... ... ... ... ...

8. Calculations

Using the formula:

\[ I = \frac{m \cdot r \cdot (g - a)}{a} - \frac{\tau \cdot t_1}{\omega_0} \]

For m = 0.05 kg, a = 0.32 m/s², t₁ = 10.2 s, ω₀ = 4.0 rad/s:

\[ I = \frac{0.05 \times 0.02 \times (9.81 - 0.32)}{0.32} - \frac{\tau \times 10.2}{4.0} \]

(Assuming τ is negligible or calculated separately.)

Repeat for different masses and average the results.

9. Result

The moment of inertia (I) of the flywheel is found to be ______ kg·m².

10. Precautions

  1. Ensure the flywheel is horizontal using a spirit level.
  2. Minimize friction in the bearings.
  3. Use a smooth, thin string to avoid slipping.
  4. Measure height (h) accurately.
  5. Take multiple readings to reduce errors.

11. Viva Voce Questions

  1. What is the physical significance of the moment of inertia?
  2. How does frictional torque affect the experiment?
  3. Why do we measure time after string detachment?
  4. What happens if the axle radius is increased?
  5. How would air resistance impact the results?

Graph / Plot (If Required)

  • Plot ω₀ vs. t₁ to study deceleration due to friction.
  • Plot I vs. m to check consistency.

Conclusion:
This experiment successfully determines the moment of inertia of a flywheel using rotational dynamics principles. Proper measurements and calculations ensure accurate results.

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