Q.(a) Show that the motion of a simple pendulum is simple harmonic and hence, derive an equation for its time period. What is second's pendulum?
You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
Start your 14-day free trial to unlock the full solution →The restoring torque on a simple pendulum is proportional to (and opposes) the angular displacement for small angles, satisfying the SHM condition; this gives T = 2π√(L/g). A second's pendulum (T = 2 s) has a length of very nearly 1 metre.
(a) SHM of a simple pendulum and its time period:
Consider a simple pendulum: a point mass m suspended by a massless, inextensible string of length L from a fixed support, displaced through a small angle θ from the vertical.
The forces on the bob are gravity (mg, downward) and the tension T along the string. Resolving mg along and perpendicular to the string: the component mg sinθ acts as the restoring force, directed back toward the mean (equilibrium) position, while mg cosθ balances the tension.
Restoring force F = − mg sinθ
For small angular displacements (θ small, in radians), sinθ ≈ θ, and since the arc-length displacement x = Lθ, we have θ ≈ x/L. So:
F = − mg (x/L) = − (mg/L) x
This is of the form F = − k x, with k = mg/L, which is exactly the condition for Simple Harmonic Motion (restoring force directly proportional to displacement, and directed opposite to it). Hence the motion of a simple pendulum, for small oscillations, is simple harmonic.
Comparing with the standard SHM equation F = − mω^2 x:
mω^2 = mg/L ⇒ ω = √(g/L)
Time period: T = 2π/ω = 2π √(L/g)
…
Unlock everything free for 14 days
- Full step-by-step solutions
- Concept-first explanations
- Methods, shortcuts & mistakes
- PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.