11.2. Solved Examples: Circular Motion

Example 1.

A disc with a radius of 2.00 m rotates at 1150 rpm.

1. Express its angular speed in rad/s.
2. Find its angular displacement in 4.00 s.
3. Find the linear speed (in m/s) 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][rpm][latex] into radians per second [latex][rad/s][latex]. 1 minute = 60 seconds; 1 revolution = 2[latex]\small \pi[/latex]
   [latex]1150\frac{rev}{min}\times\frac{1min}{60s}\times\frac{2\pi}{1rev}=120.4\text rad/s[/latex]
2. [latex]\omega=\frac{\theta}{t}[/latex];
   Rearranging the formula, we get:
   [latex]\theta=\omega t=120.4\times4.0=481.6\text{rad}[/latex]
3. [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 35.0 cm long string. The rock uniformly accelerates from rest to 2.7 [latex]\text{rad/s}[/latex] in 12.0 [latex]s[/latex].

1. Find its angular acceleration.
2. Find the linear velocity of the rock after 12.0 s.
3. How many revolutions did the rock make during the 12.0 s?

Solution:

This is an accelerated circular motion, since the angular velocity changes from zero to 2.77 [latex]\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 rad) divided by the length of a circle (in rad):
   [latex]n_rev=\frac{\theta}{2\pi}=\frac{\frac{\omega_i+\omega_f}{2}\times t}{2\pi}=2.6rev[/latex]