Section 5.1: Rotational Motion

Just as objects can undergo linear motion, they can also rotate around an axis. Rotational motion describes the movement of objects that spin, roll, or turn about a fixed axis.

In rotational motion, we replace linear quantities with their rotational counterparts:

  • Angular displacement (θ): analogous to distance (measured in radians)
  • Angular velocity (ω): rate of change of angular displacement (rad/s)
  • Angular acceleration (α): rate of change of angular velocity (rad/s²)
\[ \theta = \frac{s}{r} \] where: \( \theta \) = angular displacement (radians)
\( s \) = arc length (m)
\( r \) = radius (m)

The connection between linear and angular quantities is given by:

\[ v = r \omega \quad \quad a = r \alpha \] where: \( v \) = linear velocity
\( a \) = linear acceleration
\( \omega \) = angular velocity
\( \alpha \) = angular acceleration

Example: Spinning Wheel

A wheel of radius 0.5 m rotates with an angular velocity of 4 rad/s. Find the linear speed of a point on its rim.

Using \( v = r \omega \): \( v = 0.5 \times 4 = 2 \, \text{m/s} \) The point on the rim moves at 2 m/s.

Example: Angular Acceleration

A ceiling fan starts from rest and reaches an angular velocity of 20 rad/s in 5 seconds. Find its angular acceleration.

Angular acceleration: \( \alpha = \frac{\Delta \omega}{\Delta t} = \frac{20 - 0}{5} = 4 \, \text{rad/s}^2 \) The angular acceleration is 4 rad/s².

Practice Problems

  1. A wheel makes 10 revolutions in 2 seconds. What is its angular velocity in rad/s?
  2. A CD has a radius of 0.06 m and spins at 30 rad/s. Find the linear speed of a point on its edge.
  3. A grinder accelerates from 0 to 100 rad/s in 20 seconds. Find its angular acceleration.
  4. A bicycle wheel rotates at 2 revolutions per second. What is its angular velocity in rad/s?
  5. If a pulley of radius 0.2 m has an angular acceleration of 5 rad/s², find the linear acceleration of a point on its rim.
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