Pendulums
- Describe a simple pendulum as SHM for small angles
- Apply the pendulum period formula, T = 2π√(L/g)
- Identify what does and does not affect a pendulum’s period
A pendulum is SHM for small swings
A simple pendulum — a mass on a string — swings back and forth as gravity provides the restoring force. For small angles (up to about 15°), that restoring force is very nearly proportional to the displacement, so the motion is simple harmonic. Push it too far and the approximation breaks down, but within the small-angle range the swing is beautifully regular.
What sets the period
The period of a simple pendulum is T = 2π√(L/g). Strikingly, it depends only on the length L and the gravitational field g — not on the mass of the bob and not on the amplitude (for small swings). A longer pendulum swings more slowly. This mass-independence is why a grandfather clock is tuned by adjusting the length of its rod, never the weight of its bob.
Find the period of a simple pendulum of length 2.5 m (g = 10 m/s², use 2π ≈ 6.28).
- 1.Compute the ratio inside the root: L/g = 2.5 ÷ 10 = 0.25.
- 2.Take the square root: √0.25 = 0.5.
- 3.Multiply by 2π: T = 6.28 × 0.5.
- 4.Result: T ≈ 3.14 s.
The bob’s mass does not affect the period — it cancels out, just as mass cancels in free fall. A heavy pendulum and a light one of the same length keep identical time.
Two pendulums are identical except that pendulum A has twice the bob mass of pendulum B. How do their periods compare?
A simple pendulum has a length of 0.9 m (g = 10 m/s², use 2π ≈ 6.28). Its period is closest to:
To change a pendulum’s period you must change its length or move it to a different g. Adding mass or (for small swings) changing the amplitude does nothing — a common distractor on the exam.
Answer the 2 checkpoints as you read.
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