Flywheel moment of inertia and rotational dynamics lab report

1. Title and aim

Uniformly accelerated angular motion and determination of flywheel moment of inertia. Investigate the fundamental dynamic relation between applied net torque τ, moment of inertia I, and angular acceleration α for a precision heavy flywheel mounted on low-friction bearings, verifying the rotational analogue of Newton's second law α = τ / I across varying applied torques, and extracting the flywheel's moment of inertia from the inverse slope of the angular acceleration versus torque graph.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Flywheel moment of inertia of 0.0500 kg m², differing by 0.000% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Heavy Steel Flywheel and Axle Drum: A massive, dynamically balanced 4.000 kg steel disk mounted on a precision horizontal axle with a concentric 20.0 mm radius cord drum.
• Low-Friction Precision Ball Bearings and Rigid Wall Mount: Rigid cast-iron bearing housings supporting the flywheel axle with deep-groove miniature ball bearings to minimize rotational friction.
• High-Resolution Optical Rotary Encoder and Digital Tachometer: An optical encoder disc with 360 pulses per revolution, coupled to a digital microprocessor counter measuring angular velocity and acceleration.
• Calibrated Slotted Weight Set, Tow Cord, and Impact Cushion: Precision brass weights suspended by an inextensible monofilament cord wrapped around the drum, landing on a resilient energy-absorbing foam pad.
• Flywheel disk mass M = 4.000 kg, outer radius R = 0.1581 m, axle drum radius r = 0.020 m (20.0 mm).
• Digital optical encoder measuring angular acceleration α to 0.001 rad s⁻².
• Reference moment of inertia I = 0.0500 kg m²; local gravity g = 9.80665 m s⁻².

4. Theory and graph choice

The rotational dynamics of rigid bodies forms an essential pillar of classical mechanics and mechanical engineering. In direct analogy to linear translation, where an unbalanced net force produces linear acceleration inversely proportional to inertial mass (F = m · a), an unbalanced external torque τ acting on a rigid body free to rotate about a fixed axis produces an angular acceleration α inversely proportional to the body's moment of inertia: τ = I · α. The moment of inertia I represents the rotational inertia or kinetic resistance of a physical body to changes in its rotational state of motion. Unlike linear mass m, which depends solely on the total quantity of matter, the moment of inertia depends both on the total mass M and critically on how that mass is geometrically distributed relative to the axis of rotation: I = ∫ r² dm. For a solid uniform cylindrical disk of mass M and radius R rotating about its central symmetry axis, integration yields I = 1/2 M R²; for a thick annular wheel or rim with inner radius R₁ and outer radius R₂, the moment of inertia is given by I = 1/2 M (R₁² + R₂²). Experimentally determining the moment of inertia of a heavy laboratory flywheel involves applying a known torque τ via a light flexible cord wrapped around an axle drum of radius r, tensioned by a falling mass m under gravity. As the mass descends through height h, the cord exerts a tangential tensile force T, generating an accelerating torque τ = T · r. The resulting uniformly accelerated angular motion is recorded by a high-resolution optical rotary encoder or spark timing chronometer to measure the angular acceleration α = d²θ/dt². By systematically varying the applied torque across a series of calibrated values while keeping the flywheel geometry fixed, students map the linear relationship α = (1 / I) · τ. An ordinary least-squares regression on the α versus τ dataset yields a straight line whose slope S equals the reciprocal of the moment of inertia: S = 1 / I. Inverting the slope provides an accurate, unbiased experimental determination of the flywheel's moment of inertia I = 1 / S, which can then be compared with the theoretical value calculated from direct geometric and mass metrology.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units kg⁻¹ m⁻².

5. Method

1. Inspect the flywheel mounting bracket with a spirit level to verify horizontal alignment of the axle. Measure the diameter of the axle drum at three orthogonal orientations using Vernier calipers, and calculate the mean drum radius r = 0.0200 m (20.0 mm). Give the unweighted wheel a gentle spin to confirm smooth, silent rotation.
2. Record the manufacturer's certified mass of the flywheel disk M = 4.000 kg, or weigh the disk on a heavy-duty platform balance. Measure the outer radius R = 0.1581 m using an engineering steel rule. Compute the theoretical geometric moment of inertia I_geom = 1/2 M R² = 0.0500 kg m².
3. Attach a loop at the end of the monofilament cord to the small driving peg on the axle drum. Wind the cord evenly in a single layer along the drum without overlapping turns. Suspend the initial hanging mass corresponding to an applied torque of τ = 0.030 N m (hanging mass m ≈ 153.0 g).
4. Hold the flywheel at rest and zero the digital rotary encoder display. Release the wheel smoothly, allowing the weight to descend. The falling weight accelerates the wheel until the cord unhooks automatically from the peg at the floor. Record the constant angular acceleration α displayed on the digital tachometer to 0.001 rad s⁻².
5. Repeat the winding, release, and timing procedure for applied torques of 0.060 N m, 0.090 N m, 0.120 N m, and 0.150 N m by adjusting the slotted mass suspended from the cord. Perform three successive trials at each torque setting to confirm repeatability.
6. Plot angular acceleration α on the vertical axis against applied torque τ on the horizontal axis. Perform an ordinary least-squares linear fit to evaluate the slope S = 1 / I. Compute the experimental moment of inertia as I = 1 / S, evaluate statistical uncertainty, and compare with the geometric value.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 1.9722e-31; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 2.7026e-15. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.6891e-16.

Propagating only the relevant fit coefficient gives a fit-only standard error of 6.7566e-18 kg m². This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Optical encoder angular quantization (±0.001 rad s⁻²), drum radius caliper measurement uncertainty (±0.05 mm), cord thickness contributing an effective radius r + d_cord/2, bearing friction torque variations during rotation, and the small unmodelled linear inertia of the falling mass (m r² ~0.00008 kg m²) contribute uncertainty. The OLS inverse-slope standard error reflects random timing scatter across torque stations, but does not capture shared static weight calibration errors or non-uniform cord winding layers.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Wind the driving cord in a single uniform layer along the axle drum without overlapping coils to maintain a constant effective torque radius r.
• Ensure the cord loop detaches cleanly from the axle peg when the falling weight reaches the floor to prevent reverse winding and parasitic braking.
• Align the hanging weight trajectory vertically below the drum tangent to avoid lateral sideloading or bending moments on the axle bearings.
• Place a soft, energy-absorbing landing cushion directly beneath the falling mass to catch the slotted weights without mechanical shock.

This virtual experiment assumes a perfectly rigid, symmetric flywheel with zero bearing friction, zero aerodynamic windage, and an inextensible cord of negligible mass. It neglects the minor equivalent inertia m r² of the falling weight and assuming an exact single-layer winding. Generated readings reflect ideal Newtonian rotational dynamics; physical experiments will exhibit small random variations from bearing friction fluctuations. Theoretical angular acceleration: α = τ / I. Linearized relation: α = (1 / I) · τ. The ordinary least-squares regression extracts the slope S = 1 / I, from which moment of inertia is recovered as I = 1 / S. Reference flywheel moment of inertia I = 0.0500 kg m².

12. Conclusion

The worked example illustrates how to obtain Flywheel moment of inertia from measurements, graph variables and a fitted coefficient. The estimate is 0.0500 kg m². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.