# B3-1: The Moment of Inertia of a Cylinder¶

## Apparatus¶

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

## Procedure¶

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.

## Observations¶

For each cylinder:

$$M$$ = ________kg

$$r_a$$ = ________m

$$r$$ = ________m

Tabulate:

## Theory¶

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:

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

Therefore:

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

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

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

## Analysis¶

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 $$2\text{s}$$.