Section 5.3: Rotational Dynamics
Rotational dynamics describes how forces cause rotational motion. Just as Newton’s second law relates force to linear acceleration, rotational dynamics relates torque to angular acceleration using the rotational analogue of mass, called moment of inertia.
\( \tau \) = net torque (N·m)
\( I \) = moment of inertia (kg·m\(^2\))
\( \alpha \) = angular acceleration (rad/s\(^2\))
The moment of inertia depends on how mass is distributed relative to the axis of rotation. A greater distance of mass from the axis results in a larger moment of inertia and thus more resistance to changes in rotational motion.
Example: Solid Disc Rotation
A solid disc of mass 10 kg and radius 0.5 m is acted on by a torque of 20 N·m. Find the angular acceleration of the disc.
For a solid disc about its central axis: \( I = \tfrac{1}{2}MR^2 = \tfrac{1}{2}(10)(0.5^2) = 1.25 \, \text{kg·m}^2 \). Using \( \tau = I\alpha \): \( \alpha = \tfrac{\tau}{I} = \tfrac{20}{1.25} = 16 \, \text{rad/s}^2 \). The angular acceleration is 16 rad/s².
Example: Rod Pivoted at One End
A uniform rod of mass 5 kg and length 2 m is pivoted at one end. A force of 30 N is applied perpendicular to the free end. Find the angular acceleration of the rod.
Torque: \( \tau = F \cdot r = 30 \times 2 = 60 \, \text{N·m} \). Moment of inertia of a rod about one end: \( I = \tfrac{1}{3}ML^2 = \tfrac{1}{3}(5)(2^2) = \tfrac{20}{3} \approx 6.67 \, \text{kg·m}^2 \). Angular acceleration: \( \alpha = \tfrac{\tau}{I} = \tfrac{60}{6.67} \approx 9 \, \text{rad/s}^2 \). The angular acceleration is about 9 rad/s².
Practice Problems
- A wheel of radius 0.2 m and moment of inertia 0.5 kg·m² is subjected to a torque of 4 N·m. Find its angular acceleration.
- A disc of mass 8 kg and radius 0.4 m is rotated by a torque of 10 N·m. Determine \( \alpha \).
- A child applies a force of 25 N tangentially to the rim of a 0.5 m radius wheel. If the wheel’s mass is 12 kg, find the angular acceleration (disc model).
- A rod of length 1.5 m and mass 4 kg is pivoted at its center. A 20 N force is applied at one end perpendicular to it. Find the angular acceleration.
- Explain why mass distributed farther from the axis increases the difficulty of rotating an object.