24 Inch Pendulum - Period, Frequency and Swing Rate

A pendulum 24 inches long swinging on Earth has one period, and it does not depend on what the bob is made of. Mass cancels out of the equation entirely, so a lead weight and a cork of the same size on the same string keep the same time. Only two things set the answer: the distance from the pivot to the center of the bob, which is the 24 inches, and the local gravity, taken here as the standard 9.80665 m/s2.

The square root in the formula is what surprises people. Doubling the length does not double the period; it multiplies it by about 1.41. To make a pendulum swing half as often you have to make it four times longer, which is why a grandfather clock is the height it is and why the pendulum that ticks once per second comes out at roughly 39 inches. That length very nearly became the definition of the meter.

The figure below is the small-angle result, the one every textbook quotes. It is exact only for a vanishing swing: at a 10 degree amplitude the true period is 0.19 percent longer, at 20 degrees 0.77 percent, at 30 degrees 1.74 percent. Fill in the amplitude field and the calculator works out the exact period instead, and tells you how much the textbook formula was understating it.

A one-meter pendulum, in four numbers

2.006 s

full there-and-back period

29.9

swings per minute

99.4 cm

length that ticks once a second

times the mass appears in the formula

What the formula says, and what it quietly leaves out

T = 2π√(L/g) has only two ingredients, and the one people expect to find is missing. Mass is not in it. A lead bob and a cork bob on the same string keep the same time, because gravity pulls harder on the heavier one in exactly the proportion that it is harder to move. Everything a pendulum does follows from a length and a local gravity.

The square root is the second surprise. Doubling the length does not double the period, it multiplies it by 1.414214. To make a pendulum swing half as often you have to make it four times longer, which is why grandfather clocks are the height they are and why a short pendulum is such a poor way to measure a long interval.

What the formula leaves out is that it is an approximation. It holds exactly only for an infinitely small swing. Real pendulums swing through real angles, and the true period is longer: 0.19% longer at a ten-degree swing, 0.77% at twenty, 1.74% at thirty. This calculator will give you the small-angle answer if you leave the amplitude blank, and the exact answer if you fill it in. It computes the correction from the arithmetic-geometric mean rather than from a truncated series, so the exact column is exact at every amplitude on the form, not just the gentle ones.

Three things to type

Length - from the pivot to the center of the bob, not to the top of it, in meters, centimeters, millimeters, feet or inches. This is the measurement people get wrong, and on a short pendulum the difference is large.
Where it is swinging - six bodies from the Moon to the surface of the Sun, or type a gravity of your own. Earth is taken as 9.80665 m/s², the standard value, which is also what the other physics tools here use.
Amplitude - optional, measured from vertical, so 90° means the string starts out horizontal. Leave it empty for the textbook answer; fill it in and you get the exact period and the size of the error you were about to accept.

What a bigger swing actually costs

Columns three and four are for a one-meter pendulum on Earth. The pattern is the point: the error is negligible where a physics problem lives and substantial where a real pendulum lives.

Amplitude	T / T₀	Formula runs short by	Exact period	Error in ms
1°	1.000019	0.002%	2.006 s	0.0
5°	1.000476	0.048%	2.007 s	1.0
10°	1.001907	0.191%	2.010 s	3.8
15°	1.004301	0.430%	2.015 s	8.6
20°	1.007669	0.767%	2.022 s	15.4
30°	1.017409	1.741%	2.041 s	34.9
45°	1.039973	3.997%	2.087 s	80.2
60°	1.073182	7.318%	2.153 s	146.8
90°	1.180341	18.034%	2.368 s	361.8

The same pendulum, six places

A one-meter pendulum carried around the solar system, small-angle figures throughout. Nothing about the pendulum changes, only what is underneath it.

Where	g (m/s²)	Period	Swings per minute	Seconds pendulum
Moon	1.62	4.937 s	12.2	16.4 cm
Mars	3.72	3.258 s	18.4	37.7 cm
Venus	8.87	2.110 s	28.4	89.9 cm
Earth	9.80665	2.006 s	29.9	99.4 cm
Jupiter	24.79	1.262 s	47.5	251.2 cm
Sun	274	0.380 s	158.1	2,776.2 cm

That last column is the length a clock pendulum would need in each place to tick once a second, and it is the reason clocks were once shipped with an adjustable bob. Gravity varies by about 0.53% between the equator and the poles, and because the period depends on the square root of it, that is 0.265% in rate. On a clock that is close to four minutes a day, which is why a pendulum clock carried to another latitude has to be regulated again rather than merely re-set.

Six things readers write in about

How long is a seconds pendulum, exactly?

On Earth, 99.4 cm. The name means it ticks once a second, so a full there-and-back cycle takes two seconds and the length is g/π². It is worth being careful with this one, because a pendulum whose full period is one second is a different object entirely at 24.8 cm, and the two get confused constantly. The long one is the historically important one: it came close to being the definition of the meter, which is why it lands within a percent of one.

Does a heavier bob really swing at the same rate?

Yes, and it is not an approximation. Mass cancels out of the equation of motion completely, so it does not appear in the answer. Swap a lead bob for a cork one of the same size and the period is unchanged. What does change is how long it keeps swinging: the cork loses its energy to air resistance far faster, so it stops sooner. It just does not run slow while it lasts.

Why does doubling the length not double the period?

Because the length is under a square root. Doubling it multiplies the period by √2 = 1.414214, so a two-meter pendulum takes 2.837 s against 2.006 s for a one-meter one. To genuinely double the period you need four times the length. This is also the practical rule for adjusting a clock: a small fractional change in length gives half that fractional change in rate.

How small does the swing have to be before the formula is safe?

It depends what you are doing. Below 5° the error is under half a tenth of a percent, which nothing in a school lab will detect. At 10° it is 0.191%, or about 2.8 minutes a day if you were running a clock on it. By 30° it is 1.741% and the small-angle answer is simply the wrong number. Fill in the amplitude and this calculator gives you the exact value instead of asking you to judge.

Where do I measure the length from and to?

From the pivot to the center of mass of the bob. Measuring to the top of the bob is the most common error, and it matters most where the bob is large compared to the string. Strictly, an idealized simple pendulum assumes all the mass sits at a single point and the string weighs nothing; a real rod with distributed mass is a compound pendulum and swings slightly differently, which is a separate calculation from this one.

Can I use a pendulum to measure local gravity?

That is exactly what it was used for, for two centuries. Rearranged, g = 4π²L / T², and since you can time a hundred swings and divide, the period is measurable to a precision that is hard to reach any other way with simple equipment. Two cautions: use a small amplitude or apply the exact correction, because the period error feeds straight into g, and measure the length carefully, because it enters directly while the period enters squared.

Related tools

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Kinetic Friction Calculator

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Potential Energy Calculator

What the bob carries at the top of its swing, which is the whole store the pendulum spends and recovers twice per period. See calculator

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The same energy at the bottom of the swing, where all of it has turned into speed. See calculator

Force Calculator

The weight of the bob and the restoring force behind the whole motion, for when the question moves from timing to tension. See calculator

Momentum Calculator

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Work Calculator

The energy it takes to lift the bob to its starting angle in the first place, before anything is released. See calculator

Mechanical Power Calculator

How much power a mechanism has to feed a pendulum to keep it swinging against the losses it never escapes. See calculator

How long does a pendulum 24 inches long take to swing?

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