Pendulum Time Period
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Direct answer
Galileo timed a swinging cathedral lamp against his pulse and found the beat unchanged by how far it swung — for small amplitudes a simple pendulum keeps time by T = 2π√(L/g), where L is the suspension-point-to-bob-centre length and g the local gravitational acceleration. Neither the bob's mass nor (within the small-angle approximation sinθ ≈ θ) the amplitude appears: only length and gravity set the clock. A seconds pendulum (T = 2 s) needs L ≈ 0.99 m at g = 9.8 m/s². Because T responds to g, pendulums measure gravity — g = 4π²L/T² is the laboratory staple — and because length drifts with temperature or altitude, pendulum clocks gain and lose time in ways NEET loves to quantify.
What you must remember
- The formula: T = 2π√(L/g); quadrupling the length doubles the period; frequency = 1/T.
- What is absent: bob mass and small amplitude — isochronism holds only while sinθ ≈ θ; at large amplitudes the period grows slightly.
- Seconds pendulum: T = 2 s (one second per swing each way); L = gT²/4π² ≈ 0.99 m at g = 9.8 m/s² — exactly 1 m if g = π² ≈ 9.87 m/s².
- Measuring g: g = 4π²L/T²; the graph of T² against L is a straight line of slope 4π²/g, which launders out measurement error — NCERT's experiment logic.
- Effective length: L includes the string plus the bob's radius, measured to the centre of the bob.
- Clock-error formula: ΔT/T = ½(ΔL/L − Δg/g); a summer lengthening (ΔL = LαΔθ) slows the clock by ½αΔθ fractionally.
- Lift and location variants: ascending a hill (g smaller) lengthens T, clock loses; in a lift accelerating upward, g_eff = g + a shortens T; in free fall g_eff = 0 and the pendulum stops swinging entirely.
- Damping aside: light damping shrinks amplitude but barely shifts the period — damped oscillations keep fair time.
A clock that loses seven minutes a day
A pendulum clock keeps correct time at g = 9.8 m/s² with L = 1 m, so T = 2π√(1/9.8) ≈ 2.006 s. Carry it to a hill station where g = 9.7 m/s²: T grows as 1/√g, so the fractional change is ΔT/T = ½ × (0.1/9.8) ≈ 0.0051. Each oscillation now takes about 0.0102 s too long, and the clock performs 86400/2 ≈ 43200 oscillations a day, losing 43200 × 0.0102 ≈ 441 s — over seven minutes daily. Run the same arithmetic for summer heat: with steel's α = 1.2 × 10^-5 /K and a 30°C rise, ΔT/T = ½αΔθ = 1.8 × 10^-4, costing about 15 s per day. Small fractional changes, visible daily losses — this is the examination heartbeat of the chapter.
Where NEET sets the trap
The mass distractor never retires: "a heavier bob swings slower" is false, because the restoring acceleration g sinθ carries no mass. Amplitude questions exploit the small-angle boundary — the period is amplitude-independent only for small swings, an assertion-reason staple. Clock numericals reverse the reasoning: given that a clock loses 5 minutes a day, find the fractional change in length or g; students who compute ΔT/T from the wrong base (per-oscillation versus per-day) land off by orders of magnitude. The free-fall lift is the philosophical favourite: with g_eff = 0 there is no restoring force at all, so T is not infinite-but-regular — the pendulum simply does not oscillate. Effective length (to the bob's centre) quietly adds or subtracts centimetres in laboratory numericals.
Frequently asked questions
Does the mass of the bob affect a pendulum's period?
No; T = 2π√(L/g) contains no mass — a 10 g and a 100 g bob on equal strings keep identical time.
What is a seconds pendulum and how long is it?
One with period 2 s; its length is gT²/4π² ≈ 0.99 m where g = 9.8 m/s² — close to a metre by design of the standard value.
How does a pendulum clock behave on the Moon?
With g_moon = g/6 the period grows by √6 ≈ 2.45 times, so the clock runs grossly slow — every nominal second takes 2.45 real seconds.
Why does a pendulum clock lose time in summer?
Thermal expansion lengthens the pendulum, raising T by the fraction ½αΔθ; for steel with a 30°C rise that is roughly 15 seconds lost per day.
What happens to a pendulum inside a freely falling lift?
With g_eff = 0 there is no restoring torque; the pendulum does not oscillate at all — period effectively infinite.