找出鐘擺運動的周期和擺動半徑長度的關係

Let L be the length of the suspension string of the bob
m be the mass of the pendulum bob
g be the acceleration due to gravity

Consider the situation when the string makes a small angle (theta) with the vertical, the restoring force = mg.sin(theta)
hence, restoring torque = mgL.sin(theta)

Since momentum of inertia of the bob about the point of suspension
= m.L^2
using the equation: torque = moment of inertia x angular acceleration
-mgL.sin(theta) = (m.L^2).a, where a is the angular acceleration
hence, a = -g.sin(theta)/L

Assume angle (theta) is small, such that sin(theta) = (theta) approximately
a = - (g/L)(theta)

The motion is thus simple harmonic and the angular frequency square w^2 is given by,
w^2 = (g/L)
Using the realtionship: w = 2.pi/T
where T is the period of oscillation and pi = 3.14159.....
T = 2.pi.square-root[L/g]

This is the required equation.