AP Physics 1- 5.1 Rotational Kinematics- Study Notes- New Syllabus
AP Physics 1-5.1 Rotational Kinematics – Study Notes
AP Physics 1-5.1 Rotational Kinematics – Study Notes -AP Physics 1 – per latest Syllabus.
Key Concepts:
- Rotational Kinematics
Rotational Kinematics
A rotating system can be described in terms of its angular displacement, angular velocity, and angular acceleration. These are rotational analogues of linear displacement, velocity, and acceleration.
Angular Displacement (\( \theta \))
The angle through which a point or line has been rotated in a specified sense about a specified axis.
Measured in radians (\( \text{rad} \)).
For one complete revolution: \( \theta = 2\pi \, \text{rad} \).
Angular Velocity (\( \omega \))
The rate of change of angular displacement with respect to time.
\( \omega = \dfrac{d\theta}{dt} \)
Units: rad/s.
If constant, the motion is uniform circular motion.
Average Angular Velocity:
Defined as the total angular displacement divided by the total time taken.
\( \omega_{avg} = \dfrac{\Delta \theta}{\Delta t} \)
Angular Acceleration (\( \alpha \))
The rate of change of angular velocity with respect to time.
\( \alpha = \dfrac{d\omega}{dt} \)
Units: rad/s².
Positive \( \alpha \): speed increases, Negative \( \alpha \): speed decreases.
Average Angular Acceleration:
Defined as the change in angular velocity per unit time.
\( \alpha_{avg} = \dfrac{\Delta \omega}{\Delta t} \)
Kinematic Equations of Rotational Motion (constant \( \alpha \))
- \( \omega = \omega_0 + \alpha t \)
- \( \theta = \omega_0 t + \dfrac{1}{2}\alpha t^2 \)
- \( \omega^2 = \omega_0^2 + 2 \alpha \theta \)
Graphical Representation:
- \( \theta \)-t graph: For constant \( \alpha \), parabolic curve (increasing slope).
- \( \omega \)-t graph: Straight line if \( \alpha \) is constant. Slope = \( \alpha \).
- \( \alpha \)-t graph: Horizontal line if \( \alpha \) is constant.
Example:
A wheel starts from rest and accelerates uniformly at \( \alpha = 2 \, \text{rad/s}^2 \).
Find (i) the angular velocity after 5 s, (ii) the angular displacement, and (iii) sketch the motion graphs.
▶️Answer/Explanation
(i) Angular velocity:
\( \omega = \omega_0 + \alpha t = 0 + 2 \cdot 5 = 10 \, \text{rad/s} \)
(ii) Angular displacement:
\( \theta = \omega_0 t + \dfrac{1}{2}\alpha t^2 = 0 + \dfrac{1}{2} \cdot 2 \cdot (5^2) = 25 \, \text{rad} \)
(iii) Graphs:
- \( \theta \)-t: parabola (opening upward)
- \( \omega \)-t: straight line from 0 to 10 rad/s
- \( \alpha \)-t: horizontal line at \( \alpha = 2 \, \text{rad/s}^2 \)
Answer: Angular velocity = \( 10 \, \text{rad/s} \), Angular displacement = \( 25 \, \text{rad} \).
Example:
A fan starts from rest and reaches an angular velocity of \( \omega = 120 \, \text{rad/s} \) in 10 s.
Find (i) the angular acceleration, and (ii) the total angular displacement in this time.
▶️Answer/Explanation
(i) Angular acceleration:
\( \alpha = \dfrac{\Delta \omega}{\Delta t} = \dfrac{120 – 0}{10} = 12 \, \text{rad/s}^2 \)
(ii) Angular displacement:
\( \theta = \dfrac{1}{2} (\omega_0 + \omega) t = \dfrac{1}{2} (0 + 120) \cdot 10 = 600 \, \text{rad} \)
Answer: \( \alpha = 12 \, \text{rad/s}^2 \), \( \theta = 600 \, \text{rad} \)
Example :
A wheel rotates with an initial angular velocity of \( 40 \, \text{rad/s} \) and slows down uniformly at \( \alpha = -4 \, \text{rad/s}^2 \).
Find (i) the time taken to stop, and (ii) the angular displacement before stopping.
▶️Answer/Explanation
(i) Time to stop:
\( t = \dfrac{-\omega_0}{\alpha} = \dfrac{-40}{-4} = 10 \, \text{s} \)
(ii) Angular displacement:
\( \theta = \omega_0 t + \dfrac{1}{2} \alpha t^2 = 40(10) + \dfrac{1}{2}(-4)(100) = 400 – 200 = 200 \, \text{rad} \)
Answer: Time = 10 s, Displacement = \( 200 \, \text{rad} \)
Example:
A wheel makes 30 revolutions in 20 seconds.
Find its (i) average angular velocity, and (ii) angular displacement in radians.
▶️Answer/Explanation
(i) Angular displacement:
One revolution = \( 2\pi \, \text{rad} \)
Total displacement = \( 30 \times 2\pi = 60\pi \, \text{rad} \)
(ii) Average angular velocity:
\( \omega_{avg} = \dfrac{\Delta \theta}{\Delta t} = \dfrac{60\pi}{20} = 3\pi \, \text{rad/s} \approx 9.42 \, \text{rad/s} \)
Answer: \( \theta = 60\pi \, \text{rad} \), \( \omega_{avg} \approx 9.42 \, \text{rad/s} \)