The Simple Pendulum

A pendulum is not strictly SHM. But for small angles — less than about 15° — it is a near-perfect approximation. This is one of the most elegant results in physics.

The motion

Bob swings in an arc. The restoring force is the component of gravity tangential to the arc:

$F = -mg\sin\theta \approx -mg\theta \quad \text{(for small } \theta\text{)}$

This linear approximation makes it SHM. The angular frequency is:

$\omega = \sqrt{\frac{g}{L}}, \quad T = 2\pi\sqrt{\frac{L}{g}}$

Pendulum length 150 angle 15
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Watch the θ(t) graph — a perfect cosine. KE and PE exchange continuously just as in the spring-mass system.

Period depends on length, not mass

The mass cancels completely in the derivation. A heavy bob and a light bob on the same string swing identically.

Short pendulum ($L$ small, faster):

Pendulum length 80 angle 15
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Long pendulum ($L$ large, slower):

Pendulum length 200 angle 15
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Both start at the same 15° angle. The longer pendulum takes more time per cycle. T displayed in the readout — longer string, longer period.

$T \propto \sqrt{L}$

Doubling the length increases the period by $\sqrt{2} \approx 1.41$.

Period is independent of amplitude (for small angles)

Pull the bob to 10° or 20° — the period barely changes. This is the isochronous property again, now in a pendulum:

Pendulum length 150 angle 8
Snapshot at extreme
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Pendulum length 150 angle 25
Snapshot at extreme
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Same length, different angles. Period is approximately the same for both — as long as the angle stays small.

For large angles, the $\sin\theta \approx \theta$ approximation breaks down and the period increases. At 90° the error is about 18%.

At the extremes — maximum PE, zero KE

Bob momentarily at rest at maximum angle. All energy is gravitational PE:

Pendulum length 150 angle 20
Snapshot at extreme
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At the centre — maximum KE, zero PE

Bob passes through lowest point at maximum speed. All energy is kinetic:

Pendulum length 150 angle 20
Snapshot at centre
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Effect of gravity

On the Moon ($g = 1.6$ m/s²), the same pendulum swings much slower:

Pendulum length 150 angle 15 g 9.8
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Pendulum length 150 angle 15 g 1.6
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Period on Moon is $\sqrt{9.8/1.6} \approx 2.47$ times longer than on Earth.

Key results

$\omega = \sqrt{\frac{g}{L}}, \quad T = 2\pi\sqrt{\frac{L}{g}}, \quad f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$

• $T$ depends on $L$ and $g$ only — not mass, not amplitude (for small angles)
• $T \propto \sqrt{L}$ — double length → period $\times\sqrt{2}$
• $T \propto 1/\sqrt{g}$ — weaker gravity → longer period