Short on time? This page is the recap for the calculus-based course. For the derivation, diagrams and worked examples, use the full study guide.
Recap
- A physical pendulum is a rigid body swinging about a fixed axis. d is the distance from the axis to the centre of mass.
- Gravity acts at the centre of mass. Its lever arm is d sin θ, so the restoring torque is −mgd sin θ.
- For small angles in radians, sin θ ≈ θ. Then τ_net = Iα gives d²θ/dt² = −(mgd/I)θ: angular SHM, with solution θ = θ_max cos(ωt + φ).
- A simple pendulum is the special case of a point mass on a light string.
- A torsion pendulum has a restoring torque from a twisted wire, proportional to the angle.
Key relationships
| Relationship | Units | Notes |
|---|---|---|
| τ = −mgd sin θ ≈ −mgdθ | N·m | θ in rad; small angles |
| ω = √(mgd/I), T = 2π√(I/(mgd)) | rad/s, s | I about the pivot: I = I_cm + md² |
| T = 2π√(l/g) | s | simple pendulum: I = ml², d = l |
| l_eq = I/(md) | m | simple pendulum with the same period |
| τ = −κθ; T = 2π√(I/κ) | N·m; s | κ in N·m/rad; any angle the wire obeys |
| dθ/dt max = ωθ_max | rad/s | at the bottom of the swing |
Assumptions behind the numbers
- Fixed, frictionless pivot; no air resistance; rigid body.
- Small amplitude for gravity pendulums (below about 15°, the period error is under half a percent). Larger swings have slightly longer periods.
- The simple pendulum bob is a point mass; the string is light and does not stretch.
- g = 9.8 m/s².
Mistakes to avoid
- I_cm instead of I about the pivot. Add md².
- d to the end of the body. d goes to the centre of mass.
- Simple-pendulum formula for a plank or hoop. Use 2π√(I/mgd).
- Degrees in sin θ ≈ θ. Radians only.
- ω taken as the angular velocity. ω is fixed; dθ/dt varies.
- g in a torsion pendulum. The wire supplies the torque.
- “Mass never matters.” Scaling all the mass changes nothing; adding mass at one place usually changes T.
Quick self-check
- A uniform rod 1.5 m long swings from one end. What is its period? (I = mL²/3, d = L/2, so T = 2π√(2L/(3g)) = 2π√(1.0/9.8) ≈ 2.0 s)
- A torsion pendulum has κ = 0.020 N·m/rad and I = 0.0050 kg·m². What is T? (2π√0.25 = π ≈ 3.1 s)
- A pendulum has T = 1.5 s and amplitude 0.12 rad. What is its greatest angular speed? (ω = 2π/1.5 = 4.19 rad/s; ωθ_max ≈ 0.50 rad/s)
Next: practice questions.