Q.(a) Show that the oscillations produced in a simple pendulum are simple harmonic.
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Start your 14-day free trial to unlock the full solution →A pendulum bob displaced from its equilibrium (lowest) position experiences a restoring force that, for small swings, is directly proportional to the displacement and always directed back toward equilibrium — which is precisely the defining condition for simple harmonic motion (SHM), F = −kx.
Setup
Consider a simple pendulum: a bob of mass m attached to a light, inextensible string of length L, free to swing about a fixed support. Let θ be the angular displacement of the string from the vertical at some instant, and let x be the corresponding linear (arc) displacement of the bob along its (nearly straight, for small θ) path, so x = Lθ.
Forces on the bob
Two forces act on the bob: gravity mg (vertically down) and the tension T along the string. Resolve gravity into two components:
- Along the string (toward the support): mg cosθ, balanced by the tension T (this component does not affect the swinging motion).
- Perpendicular to the string, i.e. tangential to the bob's arc: mg sinθ, directed back toward the equilibrium (mean) position — this is the restoring force.
So the net tangential (restoring) force is
(the negative sign shows it always opposes the displacement, pulling the bob back toward θ = 0).
Small-angle approximation
For small angular displacements (θ typically less than about 10°, in radians),
so
Since x = Lθ, i.e. θ = x/L,
Identifying SHM
This is exactly of the form
A force directly proportional to displacement and always directed toward the equilibrium position (mean position) is, by definition, the condition for simple harmonic motion. Hence the small-angle oscillations of a simple pendulum are simple harmonic.
Angular frequency and period
Comparing with the standard SHM equation of motion , i.e. , the angular frequency is
and the time period is
Notably, T (and ω) is independent of the mass of the bob and of the amplitude (for small oscillations) — it depends only on the pendulum's length L and the local value of g.
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