Simple Pendulum Experiment: Finding g With Formula Steps

Gravity pulls everything down, but what is its exact strength where you stand? It varies slightly from place to place on Earth, and physicists have measured it for centuries with a device of beautiful simplicity: a swinging pendulum. Timing its swings reveals g to impressive precision.

The pendulum experiment needs only string, bob, stopwatch and ruler, yet delivers one of science's fundamental numbers.

Key Concepts

• Working principle: the pendulum's period depends on g through T = 2 pi sqrt(l / g). Rearranged, g = 4 pi^2 l / T^2, so measuring length and period gives gravity.
• Why time many oscillations: a stopwatch has a reaction-time error of about 0.1 to 0.2 s. Timing 20 or 50 swings and dividing spreads that error over many periods, shrinking it dramatically.
• Graphical method: plotting T^2 on the vertical axis against l on the horizontal axis gives a straight line through the origin with slope 4 pi^2 / g, so g = 4 pi^2 / slope. The graph averages out random errors across many lengths.
• Effective length: measured from the suspension point to the centre of the bob, including the bob's radius and any hook length.
• Small angles: keep the swing under about 10 degrees so the SHM formula stays valid.

Formulas

• Gravity from the pendulum: g = 4 pi^2 l / T^2. With l in metres and T in seconds, g comes out in m/s^2.
• Graph method: slope of the T^2 versus l line = 4 pi^2 / g, so g = 4 pi^2 / slope.
• Error control: relative error in g is roughly (delta l / l) + 2 (delta T / T), so the period measurement deserves the most care since its error counts double.
• Assumptions and limits: small angular amplitude, inextensible string, no air drag, rigid support. A wobbly stand or an elliptical swing path both corrupt the result.

Worked Example

A student sets l = 1.00 m and times 20 complete oscillations in 40.1 s. Find g. Then check: if the T^2 versus l graph has slope 4.02 s^2/m, what g does it give?

1. Period T = 40.1 / 20 = 2.005 s.
2. T^2 = 4.020 s^2.
3. g = 4 pi^2 l / T^2 = 39.478 x 1.00 / 4.020 = 9.82 m/s^2.
4. Graph method: g = 4 pi^2 / slope = 39.478 / 4.02 = 9.82 m/s^2.
5. Both methods agree at 9.82 m/s^2, within a whisker of the standard 9.81 m/s^2.

Common Misconceptions

• Timing one swing is not good enough. Human reaction time is a large fraction of a single period. Timing 20 to 50 swings is what makes the experiment precise.
• The bob's mass and swing size do not matter. Neither appears in the formula, so using a heavier bob or swinging slightly wider (within small angles) cannot improve or change the result.
• g is not exactly 9.8 everywhere. It is about 9.78 m/s^2 at the equator and 9.83 m/s^2 at the poles because Earth spins and bulges. A careful pendulum can detect the difference.

Real-World Applications

• Geophysicists map Earth's interior and monitor volcanoes by tracking tiny gravity changes over time.
• Surveyors correct GPS and levelling data for local gravity variations.
• Space agencies must know g precisely along launch paths to compute orbits and trajectories.

Timing a pendulum's swings gives gravity via g = 4 pi^2 l / T^2, sharpened by timing many oscillations and the T^2-versus-l graph: a fundamental constant measured on your bench.