Pendulum: Length vs Period

Period of a simple pendulum is T = 2π√(L/g) — proportional to √L.

Doubling length increases period by √2 (~1.41×).

T² vs L is LINEAR with slope 4π²/g. Lab method to measure g.

Length L is measured from pivot to center of mass of bob.

For compound pendulum (extended body): T = 2π·√(I/Mgd), where I = moment of inertia about pivot, d = distance from pivot to COM.

Pendulum clock: typical L ≈ 1 m for T = 2 s. Grandfather clocks tune by adjusting L slightly.

Used in seismometers, accelerometers, time-standards before atomic clocks.

Temperature effects: warm air expands the pendulum rod ⇒ longer L ⇒ slower clock.