Simple PendulumJEE Main

Concept

A simple pendulum is a point mass on a light string. For small swings the restoring torque is proportional to the angular displacement, so the bob performs simple harmonic motion with a period that depends only on the length and gravity — not on the mass or (to first order) the amplitude.

Key formula

Derivation

The tangential restoring force is $-mg\sin\theta$. For small $\theta$, $\sin\theta \approx \theta$, so $ma_t = -mg\theta$ with $a_t = L\ddot\theta$.

This gives $\ddot\theta = -(g/L)\theta$, the SHM equation with $\omega^2 = g/L$.

Hence $T = 2\pi/\omega = 2\pi\sqrt{L/g}$. Energy conservation between the extreme and the lowest point gives the bob's maximum speed $v = \sqrt{2gL(1-\cos\theta_0)}$, valid for any amplitude.

Scenarios to explore

• Simple Pendulum — Small-angle oscillations and the period formula.
• The seconds pendulum — A 1 m pendulum swings with a period close to 2 s — one tick per swing.
• Pendulum on the Moon — With g = 1.62 m/s² the same string takes about 2.5× longer per swing.
• Long pendulum, tiny swing — A 4 m string at 5° — the small-angle formula is almost exact here.

Real-world applications

• Pendulum clocks and metronomes.
• Measuring g experimentally from T and L.
• Seismometers and tuned mass dampers.

JEE exam tips

• $T \propto \sqrt{L}$: to double the period, quadruple the length.
• In a lift accelerating up, replace g with $g + a$ (effective gravity).

Common mistakes

• Thinking a heavier bob swings slower — the period is mass-independent.
• Applying $T = 2\pi\sqrt{L/g}$ at large amplitudes where it is no longer accurate.
• Using $\sin\theta = \theta$ outside the small-angle range.

Exam traps to avoid

• At large amplitude the true period is longer than $2\pi\sqrt{L/g}$.
• A pendulum in free fall (g_eff = 0) does not oscillate at all.