Pendulum period small angle:
formula and worked example

What the period of a pendulum tells you

The period of a pendulum is the time for one complete oscillation: out to one side, across to the other and back to where it started. For a simple pendulum, a small weight on a light string, that time depends on how long the string is and on how strong gravity is, and nothing else you would normally control.

That predictability is the reason pendulums have been used in clocks, and why a swinging weight is a classic way to measure g in a school lab. Change the length and you change the beat in a way you can calculate in advance.

The formula, its inputs and its limits

• Length: from the pivot to the centre of the bob in metres, not just the string.
• Gravity: 9.81 m/s² near sea level on Earth, and different on other worlds.

The formula assumes a small swing angle, conventionally under about 15 degrees from vertical, because it relies on the approximation that sin θ is close to θ. For larger swings the true period is somewhat longer than the formula says.

Mass, material of the bob and amplitude (within the small-angle limit) do not enter. A heavy bob and a light one on the same length keep the same time, provided air resistance is small.

A 1 metre pendulum on Earth

1. 1Divide length by gravity: 1 ÷ 9.81 ≈ 0.1019 s².
2. 2Take the square root: √0.1019 ≈ 0.3193 s.
3. 3Multiply by 2π (about 6.2832): 6.2832 × 0.3193 ≈ 2.006 s.
4. 4Interpret: one full swing takes about 2 s, so it ticks once per second on each pass through the lowest point.

That 2 s figure is a handy anchor: a 1 m pendulum is nearly a seconds pendulum, which is why early clocks standardised on lengths close to this.

Turning the period into frequency and checks

The frequency is one divided by the period: 1 ÷ 2.006 ≈ 0.5 swings per second. A faster tick means a shorter length or stronger gravity; a slower tick means the opposite.

• Longer string: slower swing, with the period rising by the square root of the ratio.
• Stronger gravity: faster swing, as on a planet with more gravity than Earth.
• Damping: air drag reduces amplitude over time but barely changes the period.

You can also run the formula backwards. Measure the period T and the length L, then compute g = 4π²L ÷ T². With a 1 m string and T = 2.006 s, you get back approximately 9.81 m/s².

How length controls the period

Lengthening the string by 10% to 1.1 m lifts the period from 2.006 s to 2.104 s, a rise of only about 4.9%. The square root dampens the effect, since √1.1 is about 1.049.

For tuning, that sensitivity is useful. To slow a pendulum clock by one per cent you need about a 2% increase in length, which is a small and precise adjustment.

Pitfalls when timing and calculating

• Measuring only to the top of the bob or only the string, instead of the pivot to the centre of mass.
• Timing a single swing by hand, so reaction delay of 0.2 s becomes a 10% error.
• Using a large release angle and expecting the small-angle result to hold.
• Counting a one-way swing as a full period, which halves the answer.

Setting up a quick experiment

Hang a small dense weight from a fixed point, measure the length to its centre, pull it aside a few centimetres and release it without pushing. Start timing at the lowest point of a swing, where the bob moves fastest and your timing is least ambiguous.

Compare your measured time with the formula. A difference of a percent or two usually comes from length measurement and reaction time; a larger gap suggests a big release angle or a loose pivot.

A few build details improve the match with the formula. A thin, stiff thread keeps the length fixed, and a compact bob behaves like the point mass the maths assumes. A large, light or flat bob adds air drag and shifts the effective centre of swing, which is why a small steel ball makes a better test weight than a foam one.