AP Physics – Angular Momentum and Conservation

AP Physics – Angular Momentum and Conservation

AP Physics: Angular Momentum & Conservation – Exam-Ready Study Guide

What This Is

Angular momentum is the rotational equivalent of linear momentum—it measures how hard it is to stop a spinning object. The conservation of angular momentum states that if no external torque acts on a system, its total angular momentum stays constant. This principle explains why figure skaters spin faster when they pull their arms in, why planets orbit the Sun in stable paths, and even how cats land on their feet (by twisting their bodies mid-air). On the AP exam, you’ll see angular momentum in collisions, orbits, and rotational motion problems, often paired with energy conservation or torque.

Key Terms & Concepts

• Angular momentum (L): A vector quantity describing an object’s rotational motion. For a point mass: L = r × p = mvr? (where r = radius, p = linear momentum, v? = velocity perpendicular to r). For a rotating rigid body: L = I? (where I = moment of inertia, ? = angular velocity).
• Conservation of angular momentum: If net external torque (?) = 0, then L_initial = L_final. This applies to isolated systems (e.g., ice skaters, collapsing stars).
• Torque (?): The rotational equivalent of force. ? = r × F = rF? (where F? = force perpendicular to r). Torque changes angular momentum: ? = ?L/?t.
• Moment of inertia (I): A measure of an object’s resistance to rotational motion. Depends on mass distribution (e.g., I = mr² for a point mass, I = ½mr² for a solid disk).
• Precession: The slow rotation of a spinning object’s axis (e.g., a wobbling top or Earth’s 26,000-year axial precession).
• Right-hand rule (for angular momentum): Point fingers in the direction of r, curl toward v (or ?), and your thumb shows L’s direction.
• Parallel-axis theorem: I = I_cm + md², where I_cm = moment of inertia about the center of mass, d = distance to the new axis.
• Angular impulse: ?L = t (change in angular momentum equals torque × time).
• Kepler’s 2nd Law (equal areas in equal times): A consequence of angular momentum conservation—planets sweep out equal areas in equal times as they orbit the Sun.
• Gyroscopic stability: Spinning objects resist changes to their axis (e.g., bicycle wheels, frisbees).

Step-by-Step: Solving Angular Momentum Problems

1. Identify the system and external torques
2. Is the system isolated (no external torque)? If yes, L is conserved.

3. If torque is present, use ? = ?L/?t.

4. Draw a diagram

5. Label r, v, ?, and axes of rotation.

6. For collisions, show before/after states.

7. Write initial and final angular momentum

8. For point masses: L = mvr? (use r? = perpendicular distance from axis).

9. For rigid bodies: L = I? (may need to calculate I first).

10. Apply conservation (if-= 0)

11. L_initial = L_final-solve for unknowns (e.g., ?, r, v).

12. Example: Skater pulls arms in-I decreases-? increases.

13. Check units and directions

14. Angular momentum is a vector—watch for signs (clockwise vs. counterclockwise).

15. Use the right-hand rule to confirm directions.

16. Combine with other concepts (if needed)

17. Energy conservation: K_rot = ½I?².
18. Linear momentum: p = mv (for collisions).

Common Mistakes

• Mistake: Forgetting that r in L = mvr is the perpendicular distance from the axis, not the radius of a circle. Correction: Use r? (e.g., for a planet orbiting the Sun, r? = distance from Sun to planet’s path).

• Mistake: Assuming angular momentum is conserved when external torque exists (e.g., friction, gravity). Correction: Only conserve L if net-= 0. Example: A spinning top slows due to friction (?-0).

• Mistake: Mixing up I for different shapes (e.g., using I = mr² for a disk instead of I = ½mr²). Correction: Memorize common moments of inertia (see Last-Minute Cram Sheet).

• Mistake: Ignoring vector directions (e.g., adding L for clockwise and counterclockwise spins without signs). Correction: Assign positive/negative based on direction (e.g., counterclockwise = +).

• Mistake: Confusing angular momentum (L) with linear momentum (p). Correction: L depends on r (distance from axis), while p does not.

AP Exam Insights

• FRQs often combine angular momentum with:
• Collisions (e.g., a bullet hits a rod, causing it to rotate).
• Orbital motion (e.g., Kepler’s 2nd Law, satellite orbits).
• Energy conservation (e.g., a falling object unwinds a string, converting U to K_rot).
• Multiple-choice traps:
• Non-isolated systems: Questions may imply L is conserved when it’s not (e.g., a spinning wheel slowing due to friction).
• Units: Watch for kg·m²/s (SI unit for L) vs. J·s (equivalent but less common).
• Vector directions: Questions may ask for the direction of L (use the right-hand rule!).
• Tricky distinction: Torque vs. angular momentum
• Torque (?) changes L (like force changes p).
• L is conserved if ? = 0 (like p is conserved if F = 0).

Quick Check Questions

1. A figure skater with arms extended spins at 2 rad/s. When she pulls her arms in, her moment of inertia decreases by a factor of 3. What is her new angular velocity?
2. (A) 0.67 rad/s
3. (B) 2 rad/s
4. (C) 6 rad/s

5. (D) 18 rad/s Answer: (C) 6 rad/s. L_initial = L_final-I? = I?- = (I?/I?) = 3 × 2 = 6 rad/s.

6. A 0.5 kg ball on a 1 m string moves in a horizontal circle at 4 m/s. What is its angular momentum about the center?

7. (A) 0.5 kg·m²/s
8. (B) 1 kg·m²/s
9. (C) 2 kg·m²/s

10. (D) 4 kg·m²/s Answer: (C) 2 kg·m²/s. L = mvr = 0.5 × 4 × 1 = 2 kg·m²/s.

11. A student sits on a frictionless stool holding a spinning bicycle wheel. When they flip the wheel upside down, what happens to the student?

12. (A) They spin in the same direction as the wheel.
13. (B) They spin in the opposite direction.
14. (C) They remain stationary.
15. (D) They move linearly. Answer: (B) They spin in the opposite direction. Total L must stay zero (initially at rest), so flipping the wheel reverses its L, causing the student to spin oppositely.

Last-Minute Cram Sheet

1. Conservation of L: L_initial = L_final if ? = 0.
2. L = I? (rigid body) or L = mvr? (point mass).
3. Torque changes L: ? = ?L/?t.
4. Right-hand rule: Fingers = r, curl = v or ?, thumb = L.
5. Common I values:
6. Point mass: I = mr²
7. Rod (center): I = ?mL²
8. Disk: I = ½mr²
9. Sphere: I = ?mr²
10. Parallel-axis theorem: I = I_cm + md².
11. Kepler’s 2nd Law: Planets sweep equal areas in equal times (L conserved).
12. r in L = mvr is perpendicular distance, not radius!
13. External torque (e.g., friction) breaks conservation of L.
14. Units: L = kg·m²/s, ? = N·m, I = kg·m².