Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which does not use r × p or integrate time-dependent torques). For the derivations, graphs and worked examples, use the full study guide.
Recap
- A rigid body on a fixed axis has L = Iω, with I about that axis. L takes the sign of ω.
- A point object has L = r × p about a chosen point O. Its size is rmv sin φ = mvd, where d is the perpendicular distance from O to the line of motion. In the plane, L_z = x p_y − y p_x.
- An object moving in a straight line has L about any point not on its path. About a point on its path, L = 0.
- Angular impulse = ∫τ dt. It points the same way as the torque.
- From τ_net = Iα with constant I: τ_net = dL/dt. For a point object, d(r × p)/dt = r × F_net gives the same result.
Key relationships
| Relationship | When to use it | Graph meaning |
|---|---|---|
| L = Iω | Rigid body, fixed axis | — |
| L = r × p; L = mvd | Point object, chosen point | d is the perpendicular distance to the path |
| angular impulse = ∫τ dt | Any torque | Signed area under τ–t graph |
| angular impulse = τΔt | Constant torque only | Rectangle |
| τ_net = dL/dt | Always (fixed axis or point) | Slope of L–t graph |
| ∫τ_net dt = L₂ − L₁ | Rotational impulse–momentum theorem | Area under τ_net–t equals ΔL |
Units: kg·m²/s = N·m·s.
Assumptions behind the numbers
- One fixed axis or reference point, used for every quantity in the problem.
- The body is rigid (constant I) when you use τ_net = Iα to derive the theorem.
- An inertial reference frame; axles frictionless unless a friction torque is given.
Mistakes to avoid
- Not stating the axis or point. L has no meaning without it.
- “Straight-line motion has no angular momentum.”
- Constant-torque formula τΔt for a torque that changes with time.
- Area under τ–θ (work) mistaken for area under τ–t (angular impulse).
- Losing the sign of L₀ when the body reverses.
- Reading L = 0 from τ = 0. Zero torque means zero slope.
- Using only one torque in ∫τ dt = ΔL. The theorem needs the net torque.
Quick self-check
- A flywheel with I = 0.050 kg·m² turns at 40 rad/s. What is L? (2.0 kg·m²/s)
- A 0.30 kg ball moves at 5.0 m/s along a line 0.60 m from point O. Its L about O? (0.30 × 5.0 × 0.60 = 0.90 kg·m²/s)
- A net torque τ = 2.0t (N·m) acts from t = 0 to 3.0 s on a wheel. Change in L? (∫₀³ 2.0t dt = 9.0 N·m·s)
Next: practice questions.