Moment of Inertia of a Flywheel by Falling Weight Method: Experiment, Formula & Lab Manual (2026)
Last Updated: May 6, 2026
Determining the moment of inertia of a flywheel by the falling weight method is one of the most common Engineering Physics and B.Sc. mechanics laboratory experiments in 2026 university curricula. The experiment teaches the principle of conservation of energy applied to rotational motion, and gives students hands-on experience with rotational inertia, friction losses, and angular kinematics.
This complete lab manual covers the aim, apparatus, theory, formula derivation, procedure, observation table, calculations, sources of error, precautions, and likely viva-voce questions for the flywheel falling weight experiment.
Aim of the Experiment
To determine the moment of inertia (I) of a flywheel about its axis of rotation using the falling weight method.
Apparatus Required
- Flywheel mounted on a horizontal axle with bearings
- Mass hanger and slotted weights (typically 100 g, 200 g, 500 g, 1 kg)
- Strong inextensible string
- Stop-watch (digital, accuracy 0.01 s preferred)
- Metre ruler / measuring tape
- Vernier caliper (for axle radius measurement)
- Spirit level (to ensure the axle is horizontal)
Theory & Working Principle
A flywheel is a solid disc mounted on the shaft of machines such as turbines, steam engines, diesel engines, and IC engines. Its function is to minimise speed fluctuations by storing rotational kinetic energy when the machine accelerates and releasing it when the machine decelerates. The capacity to store energy depends on the moment of inertia (I), a measure of the body’s resistance to changes in rotational motion.
In this experiment, a mass m is attached to a string wound around the axle of radius r. When the mass is released from height h, it descends and unwinds the string, causing the flywheel to rotate. After the string completely unwinds and the mass detaches, the flywheel continues to rotate freely until it eventually stops due to friction at the bearings.
By applying conservation of energy and accounting for friction losses, we can derive the moment of inertia of the flywheel.
Formula Derivation
Let:
- m = mass attached to string (kg)
- h = height through which mass falls (m)
- r = radius of axle (m)
- v = linear velocity of mass at the moment string detaches
- ω = angular velocity of flywheel at that moment
- n1 = number of rotations of flywheel during descent of mass
- n2 = number of rotations of flywheel after mass detaches, before stopping
- I = moment of inertia of flywheel
- f = energy lost per rotation due to friction
Energy conservation during descent:
m·g·h = ½·m·v² + ½·I·ω² + n1·f
Energy lost to friction (after mass detaches, flywheel rotates n2 times before stopping):
½·I·ω² = n2·f
So: f = ½·I·ω² / n2
Substituting back, with v = ω·r:
I = (m·g·h − ½·m·ω²·r²) / [½·ω²·(1 + n1/n2)]
Or equivalently, when ω is computed from the time taken for the flywheel to come to rest (using uniform deceleration assumption):
I = [2·m·g·h − m·ω²·r²] / [ω²·(1 + n1/n2)]
where ω = 4·π·n2 / t, and t is the time the flywheel takes to come to rest after the string detaches.
Procedure (Step-by-Step)
- Use the spirit level to confirm the flywheel axle is horizontal. Adjust if needed.
- Measure the radius of the axle r using a Vernier caliper. Take the average of three readings.
- Tie one end of the string firmly to the axle and the other end to the mass hanger.
- Place a known mass m (e.g., 200 g + hanger) on the hanger.
- Wind the string around the axle by rotating the flywheel until the mass is at a measured height h above the floor. Count the number of windings, this equals n1, the rotations during descent.
- Release the flywheel and start the stop-watch simultaneously.
- The mass descends, the string unwinds, and at the moment the string fully unwinds and detaches, note that instant.
- The flywheel continues to rotate freely. Count the number of rotations n2 until it comes to rest.
- Note the total time t for the flywheel to come to rest after the string detaches.
- Repeat the experiment 3–5 times with the same mass to get an averaged value.
- Repeat with different masses (300 g, 500 g, 1 kg) to verify consistency of I.
Observation Table
| S.No. | Mass m (kg) | Height h (m) | n1 | n2 | Time t (s) | ω (rad/s) | I (kg·m²) |
|---|---|---|---|---|---|---|---|
| 1 | 0.2 | , | , | , | , | , | , |
| 2 | 0.3 | , | , | , | , | , | , |
| 3 | 0.5 | , | , | , | , | , | , |
Compute mean I from the three trials.
Sample Calculation
For mass m = 0.5 kg, height h = 1.0 m, axle radius r = 0.01 m, n1 = 6, n2 = 12, t = 8 s:
- ω = 4π·n2/t = (4 × 3.14 × 12) / 8 ≈ 18.85 rad/s
- m·g·h = 0.5 × 9.8 × 1.0 = 4.9 J
- ½·m·ω²·r² = 0.5 × 0.5 × 18.85² × 0.01² ≈ 0.0089 J (negligible compared to mgh)
- 1 + n1/n2 = 1 + 6/12 = 1.5
- I ≈ (2 × 4.9) / (18.85² × 1.5) ≈ 9.8 / 533 ≈ 0.0184 kg·m²
Sources of Error
- String slippage on the axle leading to incorrect n1 count
- Bearing friction assumed constant, actual friction varies slightly with rotation speed
- Measurement error in axle radius, small variations get amplified in I = m·r²·… terms
- Reaction time in starting and stopping the stop-watch (parallax in counting rotations)
- Air drag on the flywheel surface (small but non-zero)
- Non-rigid string stretching during descent introduces small velocity error
Precautions
- Use a strong, inextensible string. Cotton thread tends to stretch, prefer thin nylon.
- Wind the string evenly without overlapping turns.
- Release the mass smoothly without giving it any initial push.
- Count rotations carefully, mark a reference point on the flywheel rim with chalk.
- Lubricate bearings before the experiment to ensure consistent friction.
- The axle must be perfectly horizontal, use a spirit level.
- Keep the height h reasonable (typically 0.5–1.5 m). Too small and ω is hard to measure accurately; too large and air drag becomes significant.
- Use the same starting position for repeated trials.
Likely Viva-Voce Questions
Q1. Define moment of inertia.
Moment of inertia is the rotational analogue of mass, it is the measure of an object’s resistance to angular acceleration when subjected to a torque. Mathematically, I = Σ mi·ri² for discrete particles, or I = ∫ r² dm for continuous bodies.
Q2. What are the SI units of moment of inertia?
kg·m² (kilogram metre squared).
Q3. Why does a flywheel store kinetic energy?
Because of its mass distribution far from the axis (high I) and its angular velocity, a flywheel stores energy as ½·I·ω². When the engine load drops, the flywheel slows down and releases this stored energy, smoothing out speed fluctuations.
Q4. Why do we measure n2 in this experiment?
n2 (rotations after mass detaches) lets us calculate the friction loss per rotation. Once we know the friction loss, we can subtract it from the total energy lost during the mass’s descent to isolate the rotational kinetic energy of the flywheel, and therefore I.
Q5. Why not just use I = MR²/2 for a solid disc?
The simple formula assumes a uniform-density solid disc. Real flywheels have spokes, hub thickness, and rim mass concentrations that change the actual moment of inertia. Experimental measurement is needed for a precise value.
Q6. What is the role of the axle radius r?
The axle radius determines the linear-to-angular conversion: v = ω·r. A larger r gives higher torque from the same falling weight, but also a larger ½·m·v² term in the energy balance.
Q7. How does this experiment apply conservation of energy?
Initial PE of falling mass = KE of falling mass + KE of flywheel + Friction losses. By measuring each term, we solve for I, the only unknown.
Q8. What if the bearings are frictionless?
Then n2 would be infinite (flywheel never stops) and the friction term drops to zero, giving I = (m·g·h − ½·m·v²) / (½·ω²) directly. Real bearings always have friction, so n2 is finite.
Real-World Applications of Flywheel Moment of Inertia
- IC engines: Smoothing torque pulses from individual cylinder firings.
- Industrial presses & punching machines: Storing energy between strokes.
- Energy storage systems: Modern flywheel energy storage (FES) units used in grid stabilisation and uninterruptible power supplies.
- Regenerative braking in some hybrid and Formula 1 cars uses flywheel-based KERS.
- Spacecraft attitude control: Reaction wheels (essentially small flywheels) orient satellites without using fuel.
Frequently Asked Questions
What is the formula for moment of inertia of a flywheel by falling weight method?
I = [2·m·g·h − m·ω²·r²] / [ω²·(1 + n1/n2)], where ω = 4π·n2/t is the angular velocity at the moment the string detaches.
Why does the flywheel slow down after the string detaches?
Bearing friction and air drag continuously remove kinetic energy from the rotating flywheel until ω = 0. The number of rotations n2 before it stops is what tells us the friction loss.
What is the unit of moment of inertia?
The SI unit is kilogram metre squared (kg·m²).
Why is the experiment repeated with different masses?
To verify that I is a property of the flywheel only, independent of the descending mass. If repeated trials give similar I values, the experiment is valid.
Can this method work for non-uniform flywheels?
Yes. The falling weight method measures the actual moment of inertia regardless of mass distribution. This is its key advantage over the theoretical I = MR²/2 formula.
What sources of error are most significant?
Bearing friction variations, parallax in counting rotations, and stop-watch reaction-time errors typically dominate. Ensuring smooth bearings, marking a reference point, and using digital timing reduce these.
Related Physics Lab Experiments
- G.M. Counter Experiment
- Resolving Power of Telescope, Microscope and Grating
- Observing Input-Output Waveform of Rectifiers
- Steady-State Sinusoidal AC Current
Conclusion
The falling weight method is one of the most reliable laboratory techniques to determine the moment of inertia of a flywheel because it accounts for real-world friction and gives a result that depends purely on the actual mass distribution of the flywheel. Mastering this experiment helps engineering students grasp the deeper relationship between linear and rotational motion, friction losses, and energy conservation, concepts that recur in dynamics, thermodynamics, and mechanical design throughout the curriculum.
Always perform the experiment with proper safety precautions, maintain a clean lab record with all observations, and verify your calculated I against the manufacturer’s specification or a theoretical estimate based on flywheel geometry.
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