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The period is 2 times pi times the square root of length divided by gravity (9.81 metres per second squared). A 1 metre pendulum takes about 2 seconds per full swing, and frequency is simply 1 divided by the period.
The pendulum period formula is period equals 2 times pi times the square root of length divided by gravity. Length is measured in metres from the pivot to the centre of the bob, and gravity on Earth is 9.81 metres per second squared. Frequency is 1 divided by period, so a longer period means fewer swings per second. This is the simple pendulum formula, which assumes a small swing, a point mass bob, and a light string.
Pendulum period vs length follows a square root rule, not a straight line rule. Doubling the length does not double the period: it multiplies the period by the square root of 2, about 1.41. Quadrupling the length doubles the period. A pendulum period table for Earth makes this clear: 0.25 metres gives about 1.00 seconds, 0.50 metres gives about 1.42 seconds, 1 metre gives about 2.01 seconds, 2 metres gives about 2.84 seconds, and 4 metres gives about 4.01 seconds.
A seconds pendulum takes 2 seconds for one full swing, so it ticks once per second each way, and it needs a length of about 0.994 metres on Earth. Early pendulum clock makers used this length because it was easy to count and easy to build. A pendulum clock gains time if its pendulum is too short and loses time if its pendulum is too long, and temperature changes matter because a warm rod expands and lengthens the pendulum. Old clocks used special gridiron or mercury compensated rods to fight this effect.
Gravity changes the period, so the same pendulum swings slower where gravity is weaker. A 1 metre pendulum period on the moon, where gravity is 1.62, is about 4.94 seconds. A pendulum period on Mars, where gravity is 3.71, is about 3.26 seconds for the same length. On Jupiter, where gravity is 24.79, the same pendulum takes only about 1.26 seconds. This is why a pendulum can be used to compare gravity on different worlds, and why a clock built for Earth would run wrong on the Moon.
The pendulum small angle rule says the simple formula is accurate only for small swings, generally under about 15 degrees from vertical. At wider swings the true period is longer than the formula predicts. The error is tiny at 5 degrees, about 0.43 percent at 15 degrees, about 1.7 percent at 30 degrees, and much larger at 60 degrees and beyond. For classroom work, keep the swing small, and for wide swings use a corrected formula or time the real pendulum instead.
For a simple pendulum at a small angle, mass does not change the period. A heavy bob and a light bob of the same length swing together, because gravity accelerates both at the same rate. Amplitude also does not change the period at small angles, which is why a pendulum clock keeps time even as its swing slowly shrinks. At large angles this breaks down, and the period grows with amplitude, as the small angle section explains.
A pendulum gravity calculator rearranges the same formula: gravity equals 4 times pi squared times length divided by period squared. To measure local gravity, time many swings, not one, and divide by the number of swings to get an accurate period. Timing 50 swings cuts the timing error on each swing to a fiftieth of the total error. Measure length from the pivot to the centre of the bob, keep the swing small, and repeat the timing to check the result.
Physical pendulum vs simple pendulum is a useful distinction. A simple pendulum is an ideal point mass on a light string. A physical pendulum is a real rigid body swinging on a pivot, and its period depends on its moment of inertia and on the distance from pivot to centre of mass, so it needs a different formula. Real pendulums also lose energy to air resistance and pivot friction, so they slowly stop unless driven, and a Foucault pendulum shows Earth rotation by a different effect. To use this calculator, enter length in metres, read period in seconds and frequency in hertz, and remember the small angle limit for classroom accuracy.
The period is the time for one full back and forth swing. For a 1 metre pendulum on Earth it is about 2.01 seconds.
Period equals 2 times pi times the square root of length divided by gravity. Frequency is 1 divided by period.
Divide length by gravity, take the square root, and multiply by 2 times pi. For 1 metre on Earth: 1 divided by 9.81 is 0.102, its square root is 0.319, and 2 times pi times 0.319 is 2.01 seconds.
No, not for a simple pendulum. Period depends on length and gravity only, so bobs of different mass but the same length swing together.
At small angles, under about 15 degrees, amplitude has almost no effect. At large angles the true period grows longer than the simple formula predicts.
Period is seconds per swing, and frequency is swings per second. They are reciprocals: frequency equals 1 divided by period. A 2.01 second pendulum has a frequency of about 0.50 hertz.
Rearrange the formula: length equals gravity times period divided by 2 times pi, all squared. A 2 second period on Earth needs a length of about 0.994 metres, the seconds pendulum length.
Time many swings to find the period, then use gravity equals 4 times pi squared times length divided by period squared. Measure length from pivot to bob centre and keep the swing small.
For a 1 metre pendulum on the moon, where gravity is 1.62, the period is about 4.94 seconds, much slower than the 2.01 seconds on Earth.
A simple pendulum is an ideal point mass on a light string. A physical pendulum is a real rigid body, and its period also depends on its moment of inertia and pivot to centre of mass distance.