Mechanics & Materials Laboratory

Flywheel
Apparatus

Determine the moment of inertia of a flywheel experimentally by measuring the angular acceleration produced by a known torque, and compare with the theoretical value.

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1. Objectives

  1. To determine the moment of inertia (\(I\)) of a flywheel experimentally using the energy method.
  2. To compare the experimental moment of inertia with the theoretical value calculated from the flywheel geometry and mass.
  3. To understand the concept of rotational kinetic energy and its conversion from potential energy of a falling mass.
  4. To determine the frictional torque at the flywheel bearings and discuss its effect on the experimental results.
  5. To relate the angular acceleration of the flywheel to the applied torque and verify Newton's second law for rotation.

2. Theory & Equations

2.1 Moment of Inertia

The moment of inertia (also called the second moment of mass) of a rigid body about a given axis is a measure of how strongly the body resists changes in its angular velocity about that axis. For a solid uniform disk of mass \(M\) and radius \(R\) spinning about its central axis, the theoretical moment of inertia is:

\[ I_{\text{theoretical}} = \frac{1}{2} M R^2 \]

2.2 Energy Method — Experimental Determination of \(I\)

A cord is wound around the flywheel axle (radius \(r\)). A mass \(m\) is attached to the free end and allowed to fall through a height \(h\), causing the flywheel to rotate. By the conservation of energy:

\[ mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 + n_1 W_f \]

where:

  • \(m\) = mass of the falling object (kg)
  • \(g\) = gravitational acceleration (9.81 m/s²)
  • \(h\) = height of fall (m)
  • \(v\) = linear velocity of the mass at the instant the cord detaches (m/s)
  • \(\omega\) = angular velocity of the flywheel at that instant (rad/s)
  • \(n_1\) = number of flywheel revolutions during the fall of the mass
  • \(W_f\) = frictional energy lost per revolution (J/rev)

The relationship between linear and angular quantities at the axle:

\[ v = r\omega \quad \text{and} \quad h = 2\pi r \cdot n_1 \]

2.3 Determining Friction Loss (\(W_f\))

After the cord detaches, the flywheel continues to rotate under its own inertia and eventually comes to rest due to friction. If the flywheel completes \(n_2\) additional revolutions before stopping, then all the kinetic energy at detachment is consumed by friction:

\[ \frac{1}{2}I\omega^2 = n_2 W_f \]

Therefore the frictional energy per revolution is:

\[ W_f = \frac{I\omega^2}{2 n_2} \]

2.4 Solving for the Experimental Moment of Inertia

Substituting \(v = r\omega\) and the expression for \(W_f\) back into the energy equation and rearranging:

\[ I = \frac{mgh - \frac{1}{2}m v^2}{\frac{1}{2}\omega^2\left(1 + \dfrac{n_1}{n_2}\right)} \]

Or equivalently, since \(v = r\omega\):

\[ I = \frac{m r^2 \left(\dfrac{2gh}{v^2} - 1\right)}{\left(1 + \dfrac{n_1}{n_2}\right)} \]

2.5 Key supporting relations

Newton's 2nd Law (Rotation)

\(\tau = I\alpha\)

Rotational Kinetic Energy

\(KE_{\text{rot}} = \frac{1}{2}I\omega^2\)

Angular Velocity

\(\omega = \frac{2\pi N}{t}\)

Linear–Angular Link

\(a = r\alpha, \quad v = r\omega\)

Frictional Torque

\(\tau_f = \frac{W_f}{2\pi}\)

Percentage Error

\(\%\text{Error} = \frac{|I_{\text{exp}} - I_{\text{theo}}|}{I_{\text{theo}}} \times 100\%\)

3. Apparatus & Setup

Equipment Required

  • Flywheel apparatus with axle and bearing supports
  • Driving drum (small diameter drum on the axle)
  • Cord (inextensible string wound around the drum)
  • Slotted masses and mass hanger (0.5 kg – 5.0 kg)
  • Metre rule or measuring tape
  • Stopwatch (resolution ± 0.01 s)
  • Vernier calliper for measuring axle/drum diameter
  • Revolution counter or chalk mark for counting turns

Typical Apparatus Data

Flywheel mass, \(M\)≈ 5 – 10 kg
Flywheel radius, \(R\)≈ 0.10 – 0.20 m
Axle/drum radius, \(r\)≈ 0.02 – 0.03 m
Falling mass, \(m\)0.5 – 5.0 kg
Height of fall, \(h\)≈ 0.5 – 1.0 m

Safety Precautions

  • Keep hands, loose clothing and hair away from the spinning flywheel.
  • Ensure the falling mass has a clear path — stand clear below.
  • Do not exceed the recommended maximum load on the hanger.
  • Wear safety goggles during the experiment.

4. Experimental Procedure

  1. Measure apparatus constants — Use the vernier calliper to measure the diameter of the flywheel (\(D = 2R\)) and the diameter of the driving drum (\(d = 2r\)). Record the mass of the flywheel (\(M\)) from the label or by weighing.
  2. Calculate the theoretical moment of inertia — Use \(I_{\text{theo}} = \frac{1}{2}MR^2\).
  3. Set up the falling mass — Wind the cord neatly around the drum. Attach the known mass \(m\) to the cord via the hanger. Measure the height \(h\) from the bottom of the mass to the ground.
  4. Release and time — Release the mass from rest. Start the stopwatch when the mass begins to fall. Record the time \(t_1\) for the mass to reach the ground. Count the number of flywheel revolutions (\(n_1\)) during the fall.
  5. Count free-spin revolutions — After the cord detaches, observe the flywheel spinning freely. Count the number of additional revolutions (\(n_2\)) until the flywheel comes to rest. Record the free-spin time \(t_2\) if required.
  6. Calculate the linear velocity at detachment — Use \(v = \frac{2h}{t_1}\) (assuming uniform acceleration from rest, the average velocity is \(h/t_1\), and the final velocity is twice the average).
  7. Calculate the angular velocity — \(\omega = v / r\).
  8. Repeat for different masses — Repeat steps 3–7 using at least 3 different values of \(m\). Tabulate all results.
  9. Calculate \(I_{\text{exp}}\) and \(W_f\) — Use the energy equations from the theory section for each trial.
  10. Compare and discuss — Calculate the percentage error between \(I_{\text{exp}}\) and \(I_{\text{theo}}\). Discuss sources of error including friction, air resistance, and cord mass.

5. Data Recording Sheet

5.1 Apparatus Constants

ParameterSymbolUnitMeasured Value
Mass of flywheel\(M\)kg
Radius of flywheel\(R\)m
Radius of axle/drum\(r\)m
Height of fall\(h\)m
Theoretical \(I\)\(I_{\text{theo}} = \frac{1}{2}MR^2\)kg·m²

5.2 Experimental Results

Trial \(m\) (kg) \(h\) (m) \(t_1\) (s) \(n_1\) (rev) \(n_2\) (rev) \(v\) (m/s) \(\omega\) (rad/s) \(I_{\text{exp}}\) (kg·m²) \(W_f\) (J/rev)
1
2
3
4

5.3 Sample Calculation Guide

Step 1: Calculate velocity at detachment: \(v = \frac{2h}{t_1}\)

Step 2: Calculate angular velocity: \(\omega = \frac{v}{r}\)

Step 3: Calculate experimental moment of inertia:

\[ I_{\text{exp}} = \frac{m r^2 \left(\frac{2gh}{v^2} - 1\right)}{1 + \frac{n_1}{n_2}} \]

Step 4: Calculate friction energy per revolution: \(W_f = \frac{I_{\text{exp}}\,\omega^2}{2\,n_2}\)

Step 5: Calculate percentage error: \(\%\text{Error} = \frac{|I_{\text{exp}} - I_{\text{theo}}|}{I_{\text{theo}}} \times 100\%\)

5.4 Discussion

5.5 Conclusion

6. Interactive Simulation

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Spin Master — Flywheel Simulation

RPM

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Energy (J)

0.0

Activity Score

0 pts

Spin-Down Timing Game

Practise estimating the flywheel run-down time. Start the timer and press stop when you believe the flywheel has nearly come to rest. The reference stopping time is approximately 8.0 s. The closer your estimate, the more bonus points you earn (up to 20 pts).

Elapsed time

0.0 s

Press "Start" then "Stop" when you think the flywheel is nearly stationary.

7. Concept Check Quiz

Answer all five questions below. Each correct answer is worth 5 points (maximum 25 pts). Your best attempt is recorded.

1. Which quantity describes how strongly a rigid body resists changes in its angular velocity about a given axis?

2. In this experiment, the driving torque on the flywheel is primarily produced by

3. When the driving mass detaches and the flywheel slows to rest, the rotational kinetic energy is mainly converted into

4. For a solid, uniform disk rotating about its central axis, the theoretical moment of inertia is

5. In the energy balance equation \(mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 + n_1 W_f\), the term \(n_1 W_f\) represents

8. Laboratory Report Rubric

Criteria Excellent (9–10) Good (7–8) Satisfactory (5–6) Poor (3–4) Very Poor (0–2)
1. Organisation & Appearance Perfect sequence. All pages numbered. Diagrams clear and labelled. Neat binding. Submitted as single PDF. Good format and tidy. One minor detail missing. Ring-bound or stapled neatly. Rough format. Organisation uneven. Some language errors. Pages not numbered consistently. Sloppy presentation. Inserts loose or torn. Minimal logical structure. Requirements not met or report absent.
2. Objectives & Theory Clearly rephrased in own words. Linked to course learning outcomes. Additional references cited. Objectives identified. Manual paraphrased with reasonable attempt at own wording. Objectives partially stated. Theory section largely copied from manual. Verbatim copy of manual. No additional research. Key objectives missing. Absent.
3. Procedure Complete step-by-step method written clearly. Diagram of setup included and labelled. Procedure present and mostly complete. Minor omission in steps or diagram. Incomplete steps. Diagram missing or unlabelled. Difficult to follow. Procedure largely absent or copied verbatim with no understanding shown. Absent.
4. Results (×2 weighting) Accurate data table. All calculations shown. Figures numbered with captions. Correct units throughout. Sample calculation clear. Correct data with minor errors. Trends identified. Most units correct. Figures provided. Missing data points. Some incorrect calculations. Incomplete tables. Units inconsistent. Major errors in data or calculations. Figures missing or unreliable. Absent or unusable data.
5. Discussion Thorough comparison of experimental and theoretical values. Sources of error analysed. Improvements suggested. Reasonable discussion. Some error analysis. Comparison attempted. Superficial discussion. Error analysis missing or weak. Discussion absent or irrelevant. Absent.
6. Conclusion Clear summary of findings. Objectives achievement stated. Valid suggestions for improvement. Conclusion present. Most objectives addressed. Conclusion incomplete or does not relate to objectives. Conclusion largely absent or irrelevant. Absent.

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