M3. Flywheel

We will use the following quantities in deriving the final equations.

• $m$ is the mass of the object hung from the flywheel.
• $g$ is the gravitational acceleration near the surface of the earth.
• $r$ is the radius of the flywheel axle.
• $h$ is the height by which the mass has fallen $h=2\pi r n_1$.
• $\omega$ is the maximum angular velocity of the flywheel.
• $n_1$ is the number of revolutions of the flywheel by the time the mass has fallen by $h$.
• $W$ is the work done against friction during one such revolution.

The gravitational potential energy of the mass ($U$) is partly converted to the rotational kinetic energies of the mass ($K_m$) and the flyweheel ($K_w$) and partly lost due to work ($W_f$) done against friction. So

$$ U = K_m + K_w + W, $$

$$ \Rightarrow mgh = \frac{1}{2} mr^2 \omega^2 + \frac{1}{2} I \omega^2 + n_1W $$

After $n_1$ revolutions, the mass is detached and then the flywheel comes to stop after $n_2$ further revolutions. Here the work against friction is used for the change in kinetic energy.

$$ n_2W = \frac{1}{2} I\omega^2 $$

Replacing this in the previous equation we get

$$ mgh - \frac{1}{2} mr^2 \omega^2 = n_2W + n_1W = n_2W(1+\frac{n_1}{n_2}) = \frac{1}{2}I\omega^2(1+\frac{n_1}{n_2}) $$

which finally give the moment of inertia of the flywheel

$$ I= \frac{2mgh-mr^2\omega^2}{\omega^2(1+\frac{n_1}{n_2})}. $$

If $n_2$ revolutions take a time $t$, then the average angular momentum of the flywheel:

$$ \omega_a = \frac{2\pi n_2}{t} = \frac{\omega+0}{2} = \frac{\omega}{2} $$

So the final instantaneous angular momentum

$$ \omega = \frac{4\pi n_2}{t}. $$

And the moment of inertia of the flywheel

List the apparatus:

1. Flywheel
2. Stopwatch
3. Weighing scale

Draw the following diagram by hand or on a computer and attach it in the report.

Then write the procedure of data collection step by step using bullet points and use the diagram while writing the procedure.

Finally provide the data in the following two tables.

Measuring the radius of the flywheel axle:

Measuring $n_1$, $n_2$ and $t$:

Calculate the angular momentum and moment of inertia using the two equations derived in theory.