Section 5.1: Rotational Kinematics
This section introduces rotational motion, angular displacement, velocity, and acceleration. We draw parallels to linear motion for easier understanding.
Rotational Kinematics Formulas:
- Angular displacement: \( \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2 \)
- Angular velocity: \( \omega = \omega_0 + \alpha t \)
- Relation to linear motion: \( s = r\theta, v = r\omega, a = r\alpha \)
Example 1
A wheel starts from rest and accelerates at \( 2 \, \text{rad/s²} \). Find its angular velocity after 5 s.
\( \omega = \omega_0 + \alpha t = 0 + 2 \times 5 = 10 \, \text{rad/s} \)
Example 2
A disk rotates with \( \omega_0 = 5 \, \text{rad/s} \) and \( \alpha = 1 \, \text{rad/s²} \) for 4 s. Find angular displacement.
\( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 = 5*4 + 0.5*1*16 = 20 + 8 = 28 \, \text{rad} \)
Practice Problems
- A wheel accelerates from rest at 3 rad/s² for 6 s. Find angular velocity.
- A turntable rotates at 10 rad/s and slows with α = -2 rad/s². How long until it stops?
- A disk rotates with α = 0.5 rad/s² for 10 s. Find angular displacement.
- Relate linear distance traveled at rim radius 0.5 m to angular displacement 4 rad.
- A rotating wheel with ω₀ = 3 rad/s accelerates at 1 rad/s² for 7 s. Find final ω and θ.