Angular Momentum
- Define angular momentum as L = Iω for a rigid body and L = r × p for a particle
- Relate the net torque to the rate of change of angular momentum, τ = dL/dt
- Compute the angular momentum in simple rotating situations
Angular momentum is rotational momentum
The rotational analog of linear momentum is angular momentum. For a rigid body spinning about a fixed axis it is L = Iω. For a single particle it is L = r × p, whose magnitude is rmv sin θ — the momentum times the perpendicular distance from the axis. A particle moving in a straight line still has angular momentum about a point it does not head toward.
Torque changes angular momentum
The rotational form of Newton's second law is τ_net = dL/dt — a net torque is exactly what changes angular momentum, just as a net force changes linear momentum (F = dp/dt). This is the deepest statement of rotational dynamics, and it immediately implies that if the net external torque is zero, angular momentum cannot change.
A 0.5 kg particle moves at 4 m/s along a straight line whose perpendicular distance from a fixed point O is 2 m. Find its angular momentum about O.
- 1.For a particle, L = rmv sin θ, where r sin θ is the perpendicular (lever-arm) distance from O to the line of motion.
- 2.Here that perpendicular distance is given directly as 2 m.
- 3.Multiply: L = (perpendicular distance)(mv) = 2 × (0.5 × 4).
- 4.So L = 2 × 2 = 4 kg·m²/s.
A disk with a moment of inertia of 3 kg·m² spins at 4 rad/s. What is its angular momentum?
For a rotating system, the net external torque is equal to:
Keep L = Iω and τ = dL/dt distinct: one is the angular momentum, the other is the rule for how torque changes it. The single most useful consequence is that zero net external torque forces angular momentum to stay constant.
Answer the 2 checkpoints as you read.
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