B3-1: The Moment of Inertia of a Cylinder



2 mounted rails; 2 different cylinders with axles; metre rule; micrometer screw gauge; stack of books; stop watch; piece of chalk; triple beam balance; graph paper


  1. Set up the apparatus as shown above with \(PR\) less than \(4\text{cm}\). Measure \(s=AB\) and measure \(PQ\). Record \(s\) and \(PQ\).
  2. Measure and record \(PR\). Calculate \(\sin\theta = \frac{PR}{PQ}\).
  3. Place a cylinder at \(A\). Record the time, \(t\), for the cylinder, starting from rest, to roll from \(A\) to \(B\).
  4. Determine the linear acceleration, \(a\), of the cylinder using your readings of \(s\) and \(t\).
  5. Increase \(PR\) and repeat steps 2, 3, and 4. Increase \(PR\) three more times, repeating steps 2, 3, and 4 to obtain five sets of readings.
  6. Measure the axle diameter and find the axle radius, \(r_a\). Find the cylinder radius, \(r\). Measure the mass, \(M\), of the cylinder and axle.
  7. Repeat steps 2 to 6 for the second cylinder.


For each cylinder:

\(M\) = ________kg

\(r_a\) = ________m

\(r\) = ________m




The cylinder loses potential energy (\(PE\)) and gains kinetic energy (\(KE\)) as it moves from A to B. Conservation of energy requires:

\[\begin{split}\big(\text{PE lost}\big) &= \big(\text{KE gained}\big) \\ Mgh &= Mgs\ (\sin\theta) \\\end{split}\]

Ignoring friction this becomes the \(KE\) of the cylinder where the total \(KE\) is:

\[KE = \big(\text{linear KE }\big)+\big(\text{rotational KE}\big)\]


\[\begin{split}Mgs\ (\sin\theta) &= \frac{1}{2}Mv^2+\frac{1}{2}I\omega ^2 \\\end{split}\]

Substitute \(v^2 = 2as\) and \(\omega = \frac{v}{r_a}\) :

\[a = \left(\frac{Mgr_a^2}{Mr_a^2 + I}\right) \sin\theta \tag{check this yourself!}\]


  1. Plot a graph of \(a\) against \(\sin\theta\) for each cylinder on the same sheet of graph paper. Find the gradient of each line.
  2. Given that \(a=\left(\frac{Mgr_a^2}{Mr_a^2 + I}\right)\sin\theta\), find \(I\) for each cylinder.
  3. From theory \(I = \frac{1}{2}Mr^2\) where \(r = \text{cylinder radius}\). Calculate \(I\) using this to check your value from 2 above. Give the % error for your value from 2.
  4. If \(I = Mk\), find the radius of gyration, \(k\), for each cylinder.
  5. Calculate the torque necessary to steadily accelerate each cylinder from rest to an angular velocity of \(30\text{rad/s}\) in \(2\text{s}\).