Pendulum period calculator
For small swings the period depends only on length and gravity — not on mass and not, to first order, on amplitude.
m/s2
Period, small-angle
2.006 409s
- Frequency
- 0.498 403 Hz
- Period corrected for amplitude
- 2.007 365 s
- Small-angle error
- 0.0476 %
- Length for a one-second half period
- 0.993 621
Show the working
- Length = 1
- Gravity = 9.806 65
- Amplitude = 5
- Period, small-angle = 2 * pi * sqrt(1 / 9.806 65) = 2.006 41
- Frequency = 2.006 41 > 0 ? 1/2.006 41 : 0 = 0.498 403
- theta = rad(5) = 0.087 266 5
- Period corrected for amplitude = 2.006 41 * (1 + 0.087 266 5^2/16 + 11*0.087 266 5^4/3072) = 2.007 36
- Small-angle error = 2.006 41 > 0 ? (2.007 36 - 2.006 41)/2.006 41*100 : 0 = 0.047 617 2
- Length for a one-second half period = 9.806 65 / pi^2 = 0.993 621
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- T = 2 pi sqrt(L / g)
- amplitude correction: T x (1 + theta^2/16 + 11 theta^4/3072 + ...)
A pendulum one metre long has a period very close to two seconds, which is why the metre was once nearly defined that way.
At 5 degrees the small-angle formula is already about 0.05% fast.