"

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.
Given:
[latex]r=2.00\text{ m}[/latex]
[latex]\omega=1150\text{ rpm}[/latex]
[latex]t=4.00\text{ s}[/latex]
Find:
[latex]\omega = \text{ ?}[/latex]
[latex]\theta = \text{ ?}[/latex]
[latex]v = \text{ ?}[/latex]

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]

  2. Find angular displacement:

    [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. Find linear speed:

    [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]?
Given:
[latex]r=35.0\text{ cm}=0.35\text{ m}[/latex]
[latex]\omega_i=0[/latex]
[latex]\omega_f=2.7\text{ rad/s}[/latex]
[latex]t=12.0\text{ s}[/latex]
Find:
[latex]\alpha=\text{ ?}[/latex]
[latex]v=\text{ ?}[/latex]
[latex]n_{rev}=\text{ ?}[/latex]

Solution:

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

  1. Find angular acceleration:

    [latex]\alpha=\frac{\omega f-\omega i}{t}=\frac{2.7-0}{12.0}=0.225\text{ rad/s}[/latex]

  2. Find the linear velocity:

    [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]

License

Icon for the Creative Commons Attribution-NonCommercial 4.0 International License

College Physics – Fundamentals and Applications Copyright © by Daniela Stanescu, Centennial College is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

Share This Book