A simple pendulum's swing time depends only on its length and gravity. Enter the length to get the period and frequency, or a target period to find the length needed.
Enter the length.
Keep g = 9.8 m/s² for Earth.
Read the period, or enter a target period to find the length.
Period: T = 2π √(L ÷ g)
Length: L = g (T ÷ 2π)²
A simple pendulum is a small heavy bob on a light string, swinging back and forth. This calculator gives its time period, the time for one full back-and-forth swing, and its frequency from the length and the value of g. It also works backwards: enter a target period and it tells you what length of string you need.
The pendulum is a Class 9 and Class 11 practical in almost every Indian school, often used to measure g in the lab. Pendulum clocks, metronomes used by music students and seismometers all depend on its steady swing. The calculator makes it easy to plan an experiment, check a reading, or see how the same pendulum would behave on the Moon.
1. Measure the effective length L in metres, from the point of suspension to the centre of the bob, not just the string length.
2. Choose g, usually 9.8 m/s² on Earth.
3. Compute the period: T = 2π √(L ÷ g).
4. Compute the frequency: f = 1 ÷ T, in hertz.
5. To find the length for a desired period, rearrange: L = g × (T ÷ 2π)².
6. In the lab, time 20 or more oscillations with a stopwatch and divide, to reduce reaction-time error.
When the bob is displaced by a small angle θ, gravity pulls it back along its arc with a force mg sin θ. For small angles, sin θ ≈ θ in radians, and the displacement along the arc is x = Lθ. So the restoring force is about −(mg ÷ L)x, proportional to displacement, which is the condition for simple harmonic motion. For SHM with force −kx, the period is 2π√(m ÷ k). Putting k = mg ÷ L, the mass cancels and T = 2π√(L ÷ g).
Mass cancels because a heavier bob feels a larger pull but also has more inertia, in exact proportion. For small swings the amplitude also drops out, so a pendulum swinging 5° and one swinging 10° keep nearly the same time. Galileo noticed this, and it made pendulum clocks possible. The period does depend on length and on g. Four times the length doubles the period, and on the Moon, where g is about one sixth of Earth's, the period is about 2.45 times longer.
The small-angle approximation adds less than 0.5% error below about 15°, but at 45° the true period is about 4% longer than the formula gives. A real bob is not a point, the string stretches a little, and air slows the swing. For precise work, physicists use a compound pendulum formula that includes the bob's size. The simple formula remains excellent for school experiments as long as you keep swings small and measure length to the bob's centre.
Aditya sets up a Class 11 practical with a pendulum 0.64 m long and also wants to know how long it would need to be to tick with a 1.5-second period.
T = 2π√(L/g): = 2π × √(0.64 ÷ 9.8) = 1.6057 s
Frequency: f = 1 ÷ T = 0.6228 Hz
Length for the target period: L = g (T/2π)² = 9.8 × (1.5 ÷ 2π)² = 0.5585 m
Answer: Time period 1.6057 s; Frequency 0.6228 Hz; Length for target period 0.5585 m
Measuring only the string and leaving out the distance to the centre of the bob.
Timing half a swing, one side to the other, instead of a full oscillation there and back.
Using large swing angles, which make the period longer than the formula predicts.
Timing a single swing rather than many, so reaction time dominates the error.
Expecting a heavier bob to swing faster or slower.
Measuring g in the school physics lab.
Class 9 and Class 11 SHM numericals.
Designing pendulum clocks and metronomes for music practice.
Planning swings and hanging structures that should not resonate with footsteps or wind.
Comparing gravity on other planets in classroom demonstrations.
Is the formula exact?
It is accurate for small swing angles, under about 15°.
What is a seconds pendulum?
One with a period of 2 seconds, so each swing takes 1 second.