🔄Angular Momentum
Angular Momentum
Angular momentum (L) is the rotational analog of linear momentum:
L = Iω
For a point mass: L = mvr (when v ⊥ r)
Units: kg·m²/s
Conservation of Angular Momentum: If no net external torque acts on a system, its angular momentum is conserved.
L_initial = L_final → I₁ω₁ = I₂ω₂
This explains many real-world phenomena!
Spectacular Demonstrations
Conservation of angular momentum explains:
Figure skater: Pulls arms in (decreases I) → spins faster (ω increases). Spreads arms out → slows down.
Diving: Tucks body (decreases I) → rotates faster. Extends before entering water → slows rotation.
Neutron stars: Giant stars collapse dramatically (R decreases by factor ~10⁵) → spin incredibly fast (up to 700 rotations/second!).
In each case: L = Iω = constant. If I decreases, ω must increase proportionally.
✏️ Worked Example: Spinning Skater
Problem: A skater spins at 2 rev/s with arms out (I = 5 kg·m²). She pulls her arms in, reducing I to 2 kg·m². Find her new angular speed.
📐 Key Equations
Angular Momentum
L = IωL = mvr (point mass, v⊥ r)I_1ω_1 = I_2ω_2 (conservation)Σ τ = (Δ L)/(Δ t)⚠️ Common Mistakes
Misconception: Angular momentum is always conserved in any rotation problem.
✓ Correct thinking: Angular momentum is conserved ONLY when the net external torque on the system is zero.
Why: If friction, air resistance, or an external force exerts a torque, angular momentum changes just like Δp = FΔt in linear momentum.
Misconception: When a skater pulls her arms in, her kinetic energy stays the same because energy is conserved.
✓ Correct thinking: Angular momentum is conserved (no external torque), but rotational kinetic energy actually INCREASES.
Why: KE_rot = L²/(2I). With L constant and I smaller, KE_rot = L²/(2I) increases. The extra energy comes from the work done by the skater's muscles pulling her arms in.
Misconception: Angular momentum and linear momentum are interchangeable concepts.
✓ Correct thinking: They are distinct conserved quantities. A system can have both, and each is conserved independently (given appropriate conditions).
Why: A spinning top at rest has angular momentum but zero linear momentum. They measure fundamentally different aspects of motion.
📝 Practice Problems
Try these problems. Check your answer when ready.
A solid disk (I = 0.5 kg·m²) spins at ω = 10 rad/s. What is its angular momentum?
L = IωA figure skater spins at 3 rev/s with I = 4 kg·m². She pulls her arms in, reducing I to 1.6 kg·m². What is her new angular speed?
A planet orbiting a star moves from aphelion (r = 2×10¹¹ m, v = 2×10⁴ m/s) toward perihelion (r = 8×10¹⁰ m). Find its speed at perihelion.
A 60 kg student sits at the rim of a spinning platform (I_platform = 200 kg·m², R = 2 m, ω_i = 1 rad/s). She walks to the center. Find the new angular speed.
A torque of 5 N·m acts on a flywheel for 3 s. If the flywheel started at rest, what is its final angular momentum?
Δ L = τ Δ tTwo disks are coaxial. Disk A (I_A = 0.8 kg·m²) spins at 20 rad/s and disk B (I_B = 0.4 kg·m²) is at rest. They are suddenly locked together. Find the combined angular speed and how much kinetic energy is lost.
Finished reading through this lesson?