====== M3. Flywheel ======

===== Introduction =====

===== Theory =====
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


===== Data and method =====

==== Apparatus ====
List the apparatus:
  - Flywheel
  - Stopwatch
  - Weighing scale

==== Procedure ====
Draw the following diagram by hand or on a computer and attach it in the report.
{{ :courses:phy101l:flywheel.png?nolink&400 |}}

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

==== Data ====
Finally provide the data in the following two tables.

Measuring the radius of the flywheel axle:
^ No. of observations ^ Main scale reading ($a$, cm) ^ Vernier scale reading $b$ ^ Vernier scale reading $v=b\times 0.005$ (cm) ^ Diameter $d=a+v$ (cm) ^ Radius $r=d/2$ (cm) ^ Mean radius $r$ (cm) ^
| 1 |
| 2 |
| 3 |

Measuring $n_1$, $n_2$ and $t$:
^ Mass $m$ (gm) ^ $n_1$ ^ Mean $n_1$ ^ n_2 ^ Mean $n_2$ ^ Time $t$ (s) ^ Mean time $t$ (s) ^ Angular speed $\omega$ (rad/s) ^ Average $\omega$ (rad/s) ^ Moment of inertia $I$ (g/cm$^2$) ^ Mean $I$ (g/cm$^2$) ^
| 1000 |  |  |  |  |  |  |  | ::: |  | ::: |
| ::: | | ::: | | ::: | | ::: | ::: | ::: | ::: | ::: |
| 1500 |  |  |  |  |  |  |  | ::: |  | ::: |
| ::: | | ::: | | ::: | | ::: | ::: | ::: | ::: | ::: |
| 2000 |  |  |  |  |  |  |  | ::: |  | ::: |
| ::: | | ::: | | ::: | | ::: | ::: | ::: | ::: | ::: |
| 2500 |  |  |  |  |  |  |  | ::: |  | ::: |
| ::: | | ::: | | ::: | | ::: | ::: | ::: | ::: | ::: |

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