What sets a pendulum’s period?
What sets a pendulum’s period?
- Set length to 1 m and record the period.
- Increase length to 2 m; predict the period ratio first.
- Change release angle from 4° to 12° and compare.
What to measure
L in m; g in m/s²; θ in radians; T in seconds.
The idea underneath
For small angles, the tangential restoring acceleration is proportional to angular displacement. The result is harmonic motion with a period set by length and gravity, not bob mass. The model restricts release angles to 14° so the small-angle approximation remains meaningful. Increasing length slows the rhythm; increasing gravity speeds it up. Track the bob’s changing height and speed: gravitational potential energy becomes kinetic energy and back. The three-dimensional suspension is a visual aid; the bob is constrained to a plane.
θ(t)=θ₀ cos(√(g/L)t) · T=2π√(L/g)L in m; g in m/s²; θ in radians; T in seconds.
Build the argument
- Draw the forces and choose coordinates before writing a differential equation.
- Check the result against energy conservation and dimensional consistency.
- Change one parameter at a time and compare a ratio rather than relying on visual impressions.
Worked example 1
At L=1 m and g=9.81 m/s², T=2.006 s.
Worked example 2
At L=2 m the period is 2.837 s, √2 times longer.
Applications
Oscillators appear in clocks, vibration isolation, sensors and mechanical structures. Conserved quantities make complicated motion easier to predict.
Common misconception
A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.
A further investigation
Find the length needed for a 2 s period.
Check your understanding
Doubling length changes the period by…
- √2
- 2
- 1/2
Explanation
T scales with √L. A 2 s period needs L≈0.994 m at standard gravity.
Model limits
Small-angle analytic model; massless rigid suspension, no air drag or pivot friction.