Verified answer

Consider a simple pendulum system (assume small angular displacement of pendulum bob and simple harmonic motion).

Sketch a diagram to help you. Some knowledge of trigonometry and geometry is assumed.

Let L = length of the pendulum, m = mass, g = acceleration due to gravity, B = angular displacement of the pendulum bob from the equilibrium position (the vertical), x = linear displacement of the pendulum bob from the equilibrium position (the vertical), a = linear acceleration of pendulum bob, w = 2π/T = angular frequency (constant for simple harmonic motion)

Assuming the pendulum executes simple harmonic motion, a = -w^2 x.

Since the pendulum is released from rest, only gravitational force acts on it to cause it to swing.

The pendulum accelerates in the negative direction if its displacement is positive (depends on sign convention; just be consistent). a= -g sin B = -g B (small angle approximation; B in radians)

x = LB (length of arc; B in radians)

-g B = -w^2 LB

g/L = (2π/T)^2

T^2 = (2π)^2 (L/g)

T = 2π√(L/g) [only consider the positive root since T cannot be negative]