The Swinging Clock

Why it matters

Galileo timed a swinging cathedral lamp against his own pulse and noticed something scandalous: the swing's WIDTH didn't change its timing. Neither, it turns out, does its weight.

The idea in plain words

The clockmaker's equation. Tap the terms.

T = 2π√(l/g)

What length gives the seconds pendulum (T = 2 s)?

l ≈ 0.99 m — just under a metre, as your slider found

• second: T — time period — one COMPLETE to-and-fro oscillation
• metre: l — length from support to the bob's CENTRE — the only knob you have
• m/s²: g — 9.8 m/s² — pendulum clocks run slow up a mountain (g drops)
• the exam graph: T² ∝ l — quadruple the length to double the period — the graph of T² vs l is a straight line
• l = gT²/4π² = 9.8 × 4 ÷ 39.5

Predict first

Swap a pendulum's 100 g bob for a 200 g bob (same length). The time period…

T = 2π√(l/g). Mass isn't in the formula — gravity pulls harder on the heavier bob but ALSO has more to move. The two effects cancel perfectly.

• stays exactly the same — correct
• doubles
• halves

What you do

Build the seconds pendulum (T = 2 s), then sweep the mass slider across 100 g and watch T not care.

Check yourself

One oscillation means…

A→B→A is one oscillation; A→B is only half. Frequency = oscillations per second.

To DOUBLE a pendulum's period, its length must become…

T ∝ √l: doubling T needs l × 4. The square root hides in every pendulum answer.

Why time 20 oscillations and divide, rather than time one?

A 0.2 s human error spread over 20 swings is 0.01 s per swing — precision by patience.

• a full to-AND-fro trip (back to the start, same direction) — correct
• one swing across
• any movement of the bob
• four times — correct
• double
• half
• your reaction-time error gets divided by 20 too — correct
• the pendulum needs a warm-up
• one swing is too fast to see