Solving Angular Momentum Problems- Techniques and Examples

What Angular Momentum Actually Is

Angular momentum is the rotational equivalent of linear momentum. If an object is spinning, it has angular momentum. Simple as that.

The formula is L = Iω, where:

• L = angular momentum (measured in kg·m²/s)
• I = moment of inertia (depends on mass distribution)
• ω = angular velocity (radians per second)

There's also the cross-product form: L = r × p, where r is the position vector and p is linear momentum. Use this when you're dealing with point particles.

The Conservation Law You Need to Know

When no external torque acts on a system, angular momentum is conserved. This is the key to solving most problems.

The equation is straightforward:

L(initial) = L(final)

Or expanded:

I₁ω₁ = I₂ω₂

That's it. Whatever the object loses in rotational speed, it gains in moment of inertia, and vice versa.

Moments of Inertia You'll Actually Use

You need to know these common moments of inertia. Memorize them.

The Step-by-Step Method

Step 1: Identify What's Conserved

Ask yourself: Is there any external torque? Friction counts. Air resistance counts. If your system is isolated, angular momentum is conserved.

Step 2: Define Your Initial and Final States

Write down what you know about the system before and after the event. This could be:

• A figure skater pulling in their arms
• A person on a rotating platform
• Colliding objects

Step 3: Set Up the Conservation Equation

Write I₁ω₁ = I₂ω₂ and plug in what you know. Solve for the unknown.

Step 4: Check Your Units

Angular velocity must be in radians per second. If you have RPM, convert it: ω = RPM × 2π / 60.

Example 1: The Figure Skater Problem

A figure skater spins at 4 rad/s with arms extended. Her moment of inertia with arms out is 3 kg·m². She pulls her arms in, reducing her moment of inertia to 1.5 kg·m². What is her new angular velocity?

Solution:

Using conservation:

I₁ω₁ = I₂ω₂

3 × 4 = 1.5 × ω₂

ω₂ = 12 / 1.5 = 8 rad/s

She spins twice as fast. This is why skaters pull in their arms—they reduce their moment of inertia and spin faster.

Example 2: The Rotating Platform Problem

A person stands on a frictionless turntable holding a spinning wheel. The wheel has angular velocity of 10 rad/s and moment of inertia of 0.5 kg·m². The person and turntable have a combined moment of inertia of 2 kg·m². What happens when the person flips the wheel upside down (reversing its direction)?

Solution:

Initial angular momentum:

L(total) = L(wheel) + L(person+table) = (0.5 × 10) + 0 = 5 kg·m²/s

After flipping, the wheel spins at -10 rad/s (negative because direction reversed). Let ω(p) be the person's angular velocity:

5 = (0.5 × -10) + (2 × ωₚ)

5 = -5 + 2ωₚ

2ωₚ = 10

ωₚ = 5 rad/s

The person starts spinning in the opposite direction. The total angular momentum is conserved.

Example 3: Particle Moving in a Circle

A particle of mass 2 kg moves at 3 m/s in a circle of radius 0.5 m. What is its angular momentum about the center?

Solution:

Using L = r × p:

L = r × mv = (0.5)(2)(3) = 3 kg·m²/s

The direction is perpendicular to the plane of motion (use right-hand rule to determine "in" or "out").

Common Mistakes That Will Cost You Points

• Forgetting to convert units — RPM to rad/s is a common trap
• Using wrong moment of inertia — make sure you're using the correct axis
• Sign errors — clockwise and counterclockwise have opposite signs
• Confusing torque with angular momentum — torque changes angular momentum, not the momentum itself
• Ignoring external torques — if there's friction, conservation doesn't apply

Torque and Angular Momentum Connection

When you need to find torque and angular momentum together, remember:

τ = dL/dt

This is the rotational equivalent of F = dp/dt. If torque is zero, dL/dt = 0, meaning L is constant.

For constant torque:

τ = Iα

where α is angular acceleration. This works exactly like F = ma for linear motion.

Practical Tips for Exams

When you see an angular momentum problem:

• Draw a diagram immediately
• Label all known quantities
• Write the conservation equation before plugging in numbers
• Identify the axis of rotation
• Check if the problem asks for magnitude only or direction too

When Conservation Doesn't Apply

Angular momentum is not conserved when:

• External torques act on the system
• There's significant friction at the axis
• Objects deform and change mass distribution without external forces (rare, but possible)
• The problem explicitly states that external forces are involved

In these cases, use τ = dL/dt or work-energy methods instead.

The Bottom Line

Angular momentum problems follow a predictable pattern. Identify your initial and final states, apply I₁ω₁ = I₂ω₂, and solve for the unknown. The math is straightforward—the hard part is setting up the problem correctly.

Know your moments of inertia. Know when conservation applies. Check your units. That's all there is to it.