Clocks that swing
For a long time, pendulum clocks were among the most accurate clocks in the world. The Dutch scientist Christiaan Huygens invented the pendulum clock in 1656, and the clockmaker Salomon Coster built the first ones to his design. They worked because a pendulum swinging through small arcs takes almost exactly the same time for every swing, big or small. In this lesson you will see why a pendulum is (nearly) a simple harmonic oscillator, derive its period, and use it to measure g at home.
The restoring force on a pendulum
A simple pendulum is a small, heavy bob of mass m on a light string of length L. When the string makes an angle θ with the vertical, two forces act on the bob: the string's tension F_T and gravity mg.
Split gravity into two components:
- mg cos θ along the string (balanced by part of the tension; together they keep the bob on its circular path),
- mg sin θ along the arc, pointing back toward the lowest point.
So the restoring force along the arc is F = −mg sin θ.
This is not quite proportional to the displacement, because of the sine. But for small angles, with θ in radians, sin θ ≈ θ, and the arc length from the bottom is x = Lθ. Then
F ≈ −mg(x/L) = −(mg/L)x
That is Hooke's-law form with an effective spring constant k = mg/L. A pendulum swinging through small angles is a simple harmonic oscillator.
The period of a pendulum
Put k = mg/L into T = 2π√(m/k):
T = 2π√(L/g)
- The mass cancels: a heavy bob and a light bob on the same length of string have the same period.
- The period depends only on the length and on g. Four times the length gives twice the period.
- For small amplitudes the period does not depend on the amplitude. The approximation sin θ ≈ θ is very good for small angles: for amplitudes up to about 15°, T = 2π√(L/g) is accurate to better than 0.5%. Larger swings take slightly longer.
- Because T depends on g, timing a pendulum is a simple way to measure g: g = 4π²L/T².
Energy view: a pendulum is also an energy converter, as in Conservation of Mechanical Energy: Tracks, Pendulums and Loops: Eg at the ends of the swing (h = L(1 − cos θ)) becomes Ek at the bottom.
Hands-on activityMeasure g with a pendulum
You need: a piece of string about 1.2 m long, a heavy small object such as a large metal nut or a few washers, tape, a tape measure, a protractor, and a phone stopwatch.
- Tie the bob to the string and tape the top firmly to the top of a door frame or the edge of a high shelf (a table edge is only high enough for the shorter lengths). Measure L from the pivot to the centre of the bob.
- Pull the bob aside to about 10° and release it from rest (do not push). Time 20 complete swings and divide by 20 to get T.
- Repeat for four or five lengths (for example 1.00, 0.80, 0.60, 0.40 and 0.20 m).
- For each length calculate g = 4π²L/T². Or, better: plot T² against L. The graph should be a straight line through the origin with slope 4π²/g, so g = 4π² ÷ slope.
- Test the claims: does changing the bob's mass change T? Does releasing from 5° instead of 15° change T?
What to look for: your values of g should fall between about 9.6 and 10.0 N/kg. Mass makes no measurable difference; a very large release angle (40° or more) makes T noticeably longer. Common errors: measuring L to the top of the bob, and miscounting swings (count "zero" when you start the timer).