Angular Momentum Practice Problems- Step-by-Step Solutions Guide

What You Need to Know Before Starting

Angular momentum is one of those concepts that trips up students because it looks complicated but follows simple rules. The key is understanding that it's just rotational momentum — the equivalent of linear momentum but for spinning objects.

This guide gives you practice problems with actual solutions, not vague explanations. Work through each one. If you get stuck, check the step where you broke down.

The Core Formula You Must Memorize

For a point particle:

L = r × p = mvr sin(θ)

Where:

• L = angular momentum
• r = distance from rotation axis
• p = linear momentum (mv)
• m = mass
• v = velocity
• θ = angle between r and v vectors

For a rotating rigid body:

L = Iω

Where I is the moment of inertia and ω is angular velocity.

Practice Problem 1: Spinning Disk

Problem: A solid disk with mass 4 kg and radius 0.5 m rotates at 300 rpm. What is its angular momentum?

Step 1: Convert rpm to rad/s.

300 rpm × (2π rad/rev) × (1 min/60 s) = 31.4 rad/s

Step 2: Find moment of inertia for a solid disk.

I = ½MR² = ½ × 4 × (0.5)² = 0.5 kg·m²

Step 3: Calculate angular momentum.

L = Iω = 0.5 × 31.4 = 15.7 kg·m²/s

Practice Problem 2: Orbiting Particle

Problem: A 2 kg particle moves in a circle of radius 3 m at 4 m/s. Find its angular momentum.

Step 1: Check the angle. For circular motion, velocity is always perpendicular to the radius, so sin(θ) = 1.

Step 2: Apply the formula.

L = mvr sin(θ) = 2 × 4 × 3 × 1 = 24 kg·m²/s

That's it. No conversion needed since you have direct values.

Practice Problem 3: Conservation of Angular Momentum

Problem: A figure skater spins at 3 rad/s with arms extended. Her moment of inertia is 5 kg·m². She pulls her arms in, reducing her moment of inertia to 2 kg·m². What is her new angular velocity?

Step 1: Apply conservation.

L_initial = L_final

Step 2: Solve for final angular velocity.

I_iω_i = I_fω_f

5 × 3 = 2 × ω_f

ω_f = 15/2 = 7.5 rad/s

The skater spins faster because she reduced her moment of inertia. This is why they pull their arms in.

Practice Problem 4: Torque and Angular Impulse

Problem: A torque of 12 N·m is applied to a wheel for 5 seconds. The wheel starts from rest. What angular momentum does it gain?

Step 1: Use the angular impulse relationship.

ΔL = τ × Δt

Step 2: Plug in values.

ΔL = 12 × 5 = 60 kg·m²/s

Torque applied over time changes angular momentum, just like force applied over time changes linear momentum.

Practice Problem 5: Angular Momentum of a Rod

Problem: A uniform rod (mass 3 kg, length 2 m) rotates about its center at 6 rad/s. Find its angular momentum.

Step 1: Find the moment of inertia for a rod about its center.

I = (1/12)ML² = (1/12) × 3 × (2)² = 1 kg·m²

Step 2: Calculate L.

L = Iω = 1 × 6 = 6 kg·m²/s

Quick Reference: Moment of Inertia Formulas

Where Students Actually Mess Up

1. Forgetting the sin(θ) factor. L = mvr only gives the full value when r and v are perpendicular. If they're not, you must include sin(θ).

2. Using the wrong moment of inertia formula. The axis of rotation matters. A rod spinning end-over-end has a different I than one spinning like a propeller.

3. Mixing up units. Always convert angular velocity to rad/s before multiplying by I. RPMs won't give you correct answers.

4. Ignoring conservation conditions. Angular momentum is only conserved when no external torque acts on the system. If something is applying torque, you can't use L_i = L_f.

How to Approach Any Angular Momentum Problem

Step 1: Identify what type of system you're dealing with — point particle or extended body.

Step 2: Write down what you know: mass, radius, velocity, angular velocity, moment of inertia.

Step 3: Pick the right formula. Point particle? Use L = mvr sin(θ). Rotating body? Use L = Iω.

Step 4: Solve algebraically first if the problem asks for a final expression, then plug in numbers.

Step 5: Check your units. Angular momentum should be kg·m²/s in SI units.

Practice these five steps until they become automatic. Most problems are just variations of the same process.