Q.Show that the motion of a simple pendulum is simple harmonic and hence derive an equation for its time period. What is seconds 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 →A simple pendulum executes SHM for small oscillations because its restoring force is proportional to displacement; this gives T = 2π√(L/g). A seconds pendulum is simply one tuned to have T = 2 s.
Showing the motion is SHM.
Consider a simple pendulum: a point mass m (the bob) suspended by a massless, inextensible string of length L from a fixed support. Let the bob be displaced through a small angle θ from the vertical and released.
Two forces act on the bob: gravity mg (vertically downward) and the tension T along the string. Resolve gravity into two components:
- along the string: mg cos θ (balanced by tension)
- perpendicular to the string (tangential to the arc): mg sin θ, directed back toward the mean position — this is the restoring force.
So the restoring force is
(negative sign because it opposes the displacement).
For small angles, sin θ ≈ θ (in radians). If x is the small arc-length displacement of the bob from the mean position, then x = Lθ, so θ = x/L, and
This is of the form F = −kx with k = mg/L — force directly proportional to displacement and directed opposite to it. This is precisely the defining condition of simple harmonic motion. Hence, for small oscillations, the pendulum's motion is SHM.
Time period. Using Newton's second law, F = ma:
Comparing with the standard SHM equation a = −ω²x:
Since T = 2π/ω:
…
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.