Core answer: gravitational potential energy PE=mgh converts to kinetic energy via ½mv²=mgh when falling; pendulum period T=2π√(L/g) depends only on length and gravity — not bob mass or (small) angle; circular motion needs centripetal force F=mv²/r, so doubling speed quadruples the required force.
Three formulas, one table
| Scenario | Formula | Key variables |
|---|---|---|
| Gravitational PE | PE=mgh | mass, height, g |
| Pendulum period | T=2π√(L/g) | length L (mass-free) |
| Centripetal force | F=mv²/r | speed squared, inverse radius |
Falling conversion: ½mv²=mgh → v=√(2gh) — same result as free fall.
PE to kinetic conversion
Energy chain: rest at height (pure PE) → falling (PE→KE) → bottom (pure KE). Friction and drag convert some mechanical energy to heat — which is why swings decay and perpetual motion fails.
| Height | Speed if fully converted |
|---|---|
| 1 m | 4.43 m/s |
| 10 m | 14 m/s |
| 50 m (coaster) | 31.3 m/s ≈ 113 km/h |
Two pendulum surprises
- Mass is irrelevant: a 1kg and 10kg bob share the same period at equal length — Galileo noticed this watching cathedral lamps.
- Small angles only: T=2π√(L/g) is accurate below ~5°; at 60° the real period runs ~7% longer.
A 1m pendulum ticks at T≈2.0s — the seconds pendulum inside antique clocks.
Where centripetal force comes from
It is not a new force but the net inward resultant: friction for a cornering car, track normal plus gravity for a loop, gravity for a satellite. Since v is squared, cornering at 80 vs 40 km/h needs 4× the friction — beyond wet-road grip, you slide.
Example: roller coaster loop speed
Loop radius 8m. At the top, gravity alone must supply the centripetal force: mg=mv²/r → v=√(gr)=√(9.8×8)=8.85 m/s minimum. Real designs carry 1.5× margin plus launch height reserve.
Example: swing and pendulum clock
A 2.5m swing: T=2π√(2.5/9.8)≈3.17s. Pushing in sync with this natural period (resonance) pumps energy in; random pushes cancel out. Clock running fast? Lower the bob (longer L, longer T) to slow it — not the reverse.
Common pitfalls
- “Heavier bobs swing slower”: mass never appears in the formula — the same wrong intuition as “heavy objects fall faster”.
- “Centripetal is its own force”: free-body diagrams contain gravity/normal/friction; centripetal is their net effect.
- “Speed alone clears a loop”: radius matters too — smaller r needs less v since F=mv²/r.
- “PE belongs to the object”: strictly it belongs to the object-Earth system; “the object’s PE” is convenient shorthand.