Pendulums
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
A pendulum swings because gravity makes a restoring torque, τ = −mgd sin θ. For small angles, sin θ ≈ θ, so the motion is SHM. A simple pendulum has T = 2π√(L/g); any rigid object swinging on a pivot (a physical pendulum) has T = 2π√(I/mgd).
1. Where the formulas come from
2. What matters
For a simple pendulum, mass cancels: a heavier bob has the same period. Only L and g matter.
On the Moon (g/6), the period is √6 ≈ 2.4 times as long.
“Small angles” means sin θ ≈ θ (within about 1% up to 15°). At large angles the restoring torque grows less than linearly, and the period gets longer.
A physical pendulum acts like a simple pendulum of length I/(md).
3. Example, then try it
Find the period of a 0.25 m simple pendulum. (g = 10 m/s²)
T = 2π√(0.25/10) = 2π(0.158) ≈ 0.99 s, almost exactly 1 s.
A uniform rod 0.60 m long swings on a pivot at one end. Find its period for small swings, and compare it with a simple pendulum 0.60 m long. What length of simple pendulum has the same period as the rod?
Rod: I = ⅓mL², d = L/2, so T = 2π√(⅓mL² ÷ (mg L/2)) = 2π√(2L/3g).
Numbers: T = 2π√(2 × 0.60 ÷ 30) = 2π√0.040 = 2π(0.20) ≈ 1.26 s.
Simple pendulum, 0.60 m: T = 2π√(0.060) ≈ 1.54 s, longer, because all its mass is at the bottom.
Equivalent length: 2L/3 = 0.40 m.
Answer: rod 1.26 s; simple pendulum 1.54 s; equivalent length 0.40 m.
4. Common mistakes
A uniform rod and a simple pendulum have the same length and both swing from one end. Which has the shorter period?
The rod acts like a simple pendulum of length 2L/3, shorter than L, so its period is shorter.
Ready to practice?
5 questions on Pendulums, each with a hint and an explanation for every wrong answer.