11.2. Solved Examples: Circular Motion

Example 1.

A disc with a radius of [latex]2.00 \text{ m}[/latex] rotates at [latex]1150 \text{ rpm}[/latex].

1. Express its angular speed in [latex]\text{rad/s}[/latex].
2. Find its angular displacement in [latex]4.00 \text{ s}[/latex].
3. Find the linear speed (in [latex]\text{m/s}[/latex]) of a point on the rim of the disc.

Solution:

1. This is a uniform circular motion. The angular velocity in rad/s can be found by converting the revolutions per minute [latex]\text{[rpm]}[/latex] into radians per second [latex]\text{[rad/s]}[/latex].

   1 minute = 60 seconds; 1 revolution = 2[latex]\pi[/latex]

   [latex]1150\frac{rev}{min}\times\frac{1min}{60s}\times\frac{2\pi}{1rev}=120.4\text{ rad/s}[/latex]

   [latex]\omega=\frac{\theta}{t}[/latex];

   Rearranging the formula, we get:

   [latex]\theta=\omega t=120.4\times4.0=481.6\text{ rad}[/latex]

   [latex]v=\omega r=120.4\times2.00=240.8\text{ m/s}[/latex]

Example 2.

A child swings a rock at the end of a [latex]35.0 \text{ cm}[/latex] long string. The rock uniformly accelerates from rest to [latex]2.7 \text{ rad/s}[/latex] in [latex]12.0\text{ s}[/latex].

1. Find its angular acceleration.
2. Find the linear velocity of the rock after [latex]12.0 \text{ s}[/latex].
3. How many revolutions did the rock make during the [latex]12.0 \text{ s}[/latex]?

Solution:

This is an accelerated circular motion, since the angular velocity changes from zero to [latex]2.77\text{ rad/s}[/latex].

1. [latex]\alpha=\frac{\omega f-\omega i}{t}=\frac{2.7-0}{12.0}=0.225\text{ rad/s}[/latex]
2. [latex]v=\omega r=2.7\times0.35=0.945\text{ m/s}[/latex]
3. The number of revolutions can be calculated as the ratio of total displacement in the given time (in [latex]\text{rad}[/latex]) divided by the length of a circle (in [latex]\text{rad}[/latex]):

   [latex]n_{rev}=\frac{\theta}{2\pi}=\frac{\tfrac{\omega_i+\omega_f}{2}\times t}{2\pi}=2.6rev[/latex]