The simple pendulum

A simple pendulum is a small heavy bob on a light string of length L. (A swinging solid body, like a clock's pendulum rod, is a physical pendulum.)

Pull the bob aside by an angle θ and look at the forces on it:

• The tension pulls along the string, and the part mg cos θ of the weight balances it along the string.
• The part mg sin θ, across the string, pulls the bob back towards the mean: the restoring force is –mg sin θ.
• Taking torques about the support gives the same thing: Iα = –mgL sin θ, with I = mL².

This is not yet –kx. But for small angles, sin θ ≈ θ (θ in radians; good up to about 20°), and the arc the bob has moved is x = Lθ. So F ≈ –(mg/L)x: SHM, with k/m = g/L.

So T = 2π√(L/g), for small swings.

• It depends on L and g only: not on the mass of the bob, and not on the amplitude (as long as it is small). That is why pendulums were used to keep time.
• Example: a pendulum 2.45 m long has T = 2π√(2.45/9.8) = 2π × 0.5 ≈ 3.14 s.
• Since T ∝ √L, making the string four times longer only doubles the period.

Where the effective gravity changes, use that value: T ∝ 1/√g.

• In a lift accelerating upwards at 2.2 m s⁻², the bob feels g + a = 12 m s⁻², and the 2.45 m pendulum swings faster: T = 2π√(2.45/12) ≈ 2.8 s.
• In a lift falling freely, the effective g is zero and the pendulum does not swing at all.
• On the moon, where g is about a sixth of the earth's, the same pendulum swings about √6 ≈ 2.4 times more slowly.