How to Find g Using a Simple Pendulum (Method + Graph)

Here's how to find g using a simple pendulum: you can measure acceleration due to gravity with a piece of string, a small weight and a stopwatch. Time how long the pendulum takes to swing at a few different lengths, plot T² against L, and the gradient of your straight line gives you g: g = 4π² ÷ gradient. With care, a school lab gets within a couple of percent of 9.81 m/s².

Where the formula comes from

For small swings, the time for one full oscillation (there and back) is

T = 2π √(L / g)

where L is the length from the pivot to the centre of the bob. Two things are missing from that formula, and it's worth noticing them. The mass isn't in it: a heavier bob is pulled harder but is also harder to get moving, and the two effects cancel exactly. The size of the swing isn't in it either, as long as it stays small (under about 15°).

Square both sides and you get T² = (4π²/g) × L. That's the equation of a straight line through the origin, with gradient 4π²/g. That's why we plot T² rather than T: a T squared vs L graph is a straight line, and you can read a gradient off a straight line accurately.

The simple pendulum experiment, step by step

1. Clamp the string between two small blocks of wood so the pivot is a fixed point, not a loop that slides.
2. Measure L from the pivot to the middle of the bob. Start with about 1.0 m.
3. Pull the bob a little to one side (less than 15°) and let it go. Don't push it.
4. Wait for a couple of swings so it settles, then time 10 full oscillations. Start the stopwatch as the bob passes through the middle, where it's moving fastest.
5. Divide by 10 to get T. Repeat the timing and take an average.
6. Shorten the string by 10 to 20 cm and repeat, until you have five or six lengths.

Why 10 swings, and why the middle?

Your reaction time on a stopwatch is roughly 0.1 to 0.2 seconds at each end, and it doesn't shrink just because the thing you're timing is short. Time one swing of about 2 s and that error is 5 to 10%. Time 10 swings (about 20 s) and the same error is spread over all ten: 0.5 to 1%.

Timing from the middle helps because the bob races through it and your eye can pick the moment precisely. At the ends it slows down and hangs for an instant, so it's hard to say exactly when it turned.

A worked example

L (m)	Time for 10 swings (s)	T (s)	T² (s²)
0.40	12.7	1.27	1.61
0.60	15.5	1.55	2.40
0.80	17.9	1.79	3.20
1.00	20.1	2.01	4.04
1.20	22.0	2.20	4.84

Plot T² (up) against L (across). The points sit close to a straight line through the origin. Its gradient is about 4.04 ÷ 1.00 = 4.03 s² m⁻¹, so

g = 4π² ÷ 4.03 = 39.48 ÷ 4.03 ≈ 9.80 m/s².

Use the gradient of your best line, not one row of the table. The line averages out the small errors in each reading.

What drags your answer away from 9.81

• Measuring L to the wrong place. Measuring to the top or bottom of the bob instead of its centre makes every L wrong by the same amount. Your graph line then misses the origin. That's a clue: if the line doesn't pass through (0, 0), check your lengths.
• Swinging too wide. Big swings take slightly longer than the formula says, so your g comes out low. At 60° the period is about 7% longer.
• A stretchy or loose pivot. If the string slides or the clamp flexes, the real length changes during the swing.
• An elliptical swing. If the bob traces an oval instead of a flat arc, it's no longer a simple pendulum. Release it gently in one plane.
• Counting wrong. "One" is the first time it comes back to where you started, not the moment you let go. Starting the count at "one" instead of "zero" is a classic way to be out by 10%.

Air resistance

A real pendulum slowly swings less and less far, because air resistance turns its energy into heat. Luckily this hardly changes the period, so it doesn't spoil the experiment. Just take your timings in the first minute, while the swing is still a good size.

Frequently asked questions

Does the mass of the bob affect the period?

No. A heavier bob is pulled down harder by gravity, but it also needs a bigger force to get it moving, and the two effects cancel exactly. That's why mass doesn't appear in T = 2π√(L/g). In practice a heavy, small bob still works best, because air resistance affects it less.

What is the period of a 1 metre pendulum?

About 2.0 seconds on Earth. Put L = 1.00 m and g = 9.81 m/s² into T = 2π√(L/g) and you get T = 2.006 s. A pendulum about 0.994 m long has a period of exactly 2 s, ticking once a second each way, which is why it was once called a "seconds pendulum" and used in clocks.

What value of g should I get?

The accepted value at the Earth's surface is about 9.81 m/s². It varies slightly with place, from about 9.78 m/s² at the equator to 9.83 m/s² at the poles. A careful school pendulum practical usually lands within 2% of 9.81. The same g decides how far a ball flies: see the best angle for maximum range, and every equation is on the Physics formulas page.

Try it in the Pendulum Lab

The Pendulum Lab lets you do the whole pendulum practical on screen: drag the bob to set the angle, time 10 swings with the stopwatch, record your results and watch the T² against L graph work out g as you add points. Then try things you can't do in a school lab: set gravity to the Moon or Jupiter, make the mass four times bigger to prove it doesn't matter, or push the angle to 70° and see the formula start to fail.