Question

Show that motion of a simple pendulum is S.H.M.​

Answer 1

1. Consider a simple pendulum of length 'l' with a small bob of mass 'm'.

2. Let the pendulum be displaced from its mean position through a small angle θ.

3. The forces acting on the pendulum are:

(i) Weight (mg) downwards.

(ii) Tension in string (T) towards fixed point.

4. Resolving mg into rectangular components, it is understood that the restoring force developed in the pendulum is mgsinθ.

F = -mgsinθ

For small angles, sinθ ≈ θ  

∴ F = -mgθ ----(1)

Assuming the displacement (y) to be the arc of a circle with radius l at an angle of θ,

y = l x θ

θ = y/l

Substituting in (1)

F = -mg(y/l)

ma = -mg(y/l)

a = -(g/l)y

a ∝ -y

Since a is directly proportional to displacement, the motion of a simple pendulum is simple harmonic.