Simple Pendulum and Small-Angle Motion
For a point bob on a light inextensible string of length l and small angular displacement, theta double-dot + (g/l)theta = 0 and T = 2 pi sqrt(l/g).
Why this shows up in the exam
Length-period scaling · Pendulum speed and tension · Comparing clocks at different gravitational fields
Learn the idea
A simple pendulum is approximately harmonic only when sin theta may be replaced by theta. Gravity pulls the bob toward the lowest point; for small angles the tangential pull grows almost linearly with arc displacement.
🧠 Memory hook: Pendulum period uses length over effective gravity.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- omega = sqrt(g/l) — small-angle pendulum angular frequency
- T = 2 pi sqrt(l/g) — small-angle pendulum period
- v² = 2g l(cos theta - cos theta_max) — speed from energy at finite angular displacement
How to approach it
- 1Decide whether small-angle SHM or full energy is needed
- 2Use pivot-to-centre length
- 3Check how local gravity changes before taking a ratio
Common slip-ups that cost marks
- •Applying the small-angle period at arbitrary amplitude
- •Using bob radius instead of pivot-to-centre length
- •Assuming bob mass changes the ideal period
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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