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Rotational Motion: Torque, Angular Momentum, and Energy in A-Level Science
What Is Rotational Motion?
Rotational motion occurs when an object spins around an axis.
Key Concepts in Rotational Motion
Torque (\( \tau \))
Torque is the rotational equivalent of force:
\[
\tau = rF \sin\theta
\]
Where:
- \( \tau \): Torque (\( \text{N·m} \))
- \( r \): Lever arm distance (\( \text{m} \))
- \( F \): Force (\( \text{N} \))
- \( \theta \): Angle between force and lever arm
Angular Momentum (\( L \))
Angular momentum describes rotational motion:
\[
L = I\omega
\]
Where:
- \( I \): Moment of inertia (\( \text{kg·m}^2 \))
- \( \omega \): Angular velocity (\( \text{rad/s} \))
Rotational Kinetic Energy
The energy of a rotating object:
\[
KE_{\text{rot}} = \frac{1}{2}I\omega^2
\]
Real-Life Applications of Rotational Motion
Engineering
- Calculating torque in machines and tools
Sports Science
- Analyzing angular momentum in gymnastics and diving
Astronomy
- Understanding the rotation of planets and stars
Example Problem
A wheel with a moment of inertia of \( 2.5 \, \text{kg·m}^2 \) rotates at \( 10 \, \text{rad/s} \). Calculate its rotational kinetic energy.
- Formula:
\[
KE_{\text{rot}} = \frac{1}{2}I\omega^2
\]
- Substitute Values:
\[
KE_{\text{rot}} = \frac{1}{2} \cdot 2.5 \cdot 10^2 = 125 \, \text{J}
\]
Common Mistakes in Rotational Motion Problems
- Confusing torque with force
- Mixing up angular velocity and linear velocity
- Forgetting to square angular velocity in energy calculations
Practice Questions
- A \( 2 \, \text{m} \) lever arm applies a force of \( 50 \, \text{N} \) at \( 90^\circ \). Calculate the torque.
- Calculate the angular momentum of a rotating object with \( I = 5 \, \text{kg·m}^2 \) and \( \omega = 3 \, \text{rad/s} \).
- Describe one real-world application of rotational kinetic energy.