How to measure gravity with a pendulum and a smartphone

The physics behind it

A simple pendulum is a small mass hanging from a string of length L. For small swings, the time of one complete oscillation, the period T, depends only on the length and on gravity:

T = 2π √(L / g)   ⇒   g = 4π² L / T²

The period does not depend on the mass, and for small angles it does not depend on the amplitude either. This is why measuring L and T is enough to find g.

What you need

• a thin, inextensible string about 1 m long;
• a small, heavy weight (a nut, a fishing weight, a bunch of keys);
• a fixed point to hang it from, such as a door frame or a shelf;
• a tape measure;
• a smartphone with a stopwatch or with StarlyLab.

Procedure

1. Make your hypothesis first: do you expect a heavier weight to swing faster? Write it down.
2. Hang the weight and measure L from the hanging point to the centre of the weight, not to its top.
3. Pull the weight aside by a small angle, less than about 10° (for a 1 m string, about 15 cm sideways), and let it go without pushing.
4. Time 10 complete oscillations and divide by 10. Timing many swings reduces the effect of your reaction time.
5. Repeat three times and take the average.
6. Calculate g with the formula and compare it with the expected value: about 9.81 m/s² (in Italy about 9.80 m/s²).

A worked example

With L = 0.800 m we measured 10 oscillations in 17.95 s, so T = 1.795 s:

g = 4π² × 0.800 / 1.795² ≈ 9.80 m/s²

That is less than 0.1% away from the expected value. A single measurement can be lucky, though: the best method uses several lengths.

The better method: a graph

Squaring the formula gives T² = (4π²/g) × L: if you plot T² against L, the points lie on a straight line through the origin, and its slope is 4π²/g.

The best-fit line of these data has a slope of about 4.04 s²/m, so g = 4π² / 4.04 ≈ 9.78 m/s², an error of about 0.3%. The graph also shows at a glance if one measurement is wrong.

Sources of error

• Length: measuring to the top of the weight instead of its centre is the most common mistake.
• Reaction time: starting and stopping the stopwatch adds about 0.1–0.2 s, which is why you time 10 oscillations.
• Large angles: above about 15° the period grows and the formula is no longer accurate.
• Air resistance and an elastic string: use a compact, heavy weight and a string that does not stretch.
• Elliptical swings: release the weight without pushing it sideways.

Do it with StarlyLab

StarlyLab includes a pendulum instrument to measure g and the guided experiment "Measure gravity with a pendulum". The lab notebook asks for your hypothesis first, collects the data in a table, draws the graph with the best-fit line and shows the error from the expected value. At the end you can export a PDF report or the data as CSV. Teachers can find more ready-to-use activities in StarlyLab for teachers.

Frequently asked questions