Pendulum Period Calculator and Spring-Mass Oscillator: Period, Frequency, Maximum Speed and Energy
Pendulums and springs: simple harmonic motion with the working.
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How this calculator works
Choose a pendulum, a mass on a spring, or the length for a period.
Enter the length and gravity and the swing angle, or the mass and the spring constant and the amplitude.
The calculator gives the period, the frequency, the greatest speed and acceleration and the energy.
Formulas used
- T = 2\pi\sqrt{\frac{L}{g}}
- T = 2\pi\sqrt{\frac{m}{k}}
- L = \frac{gT^2}{4\pi^2}
- v_{max} = A\omega
Worked example
With the default numbers in the calculator above:
- Period for small swings: 2.00641 s
- Period at the amplitude you gave (the exact value from the complete elliptic integral): 2.01024 s (0.1907 % longer than the small swing value)
- Frequency (small swings): 0.498403 Hz
- Angular frequency (small swings): 3.13156 rad/s
- Greatest speed of the bob, at the bottom of the swing: 0.545866 m/s
- Greatest acceleration along the path, at the ends of the swing: 1.70291 m/s2
- Height the bob rises above its lowest point: 0.0151922 m
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: Amplitude of the pendulum: the greatest angle from the vertical 60 degrees.
- Period for small swings: 2.00641 s
- Period at the amplitude you gave (the exact value from the complete elliptic integral): 2.15324 s (7.318 % longer than the small swing value)
- Frequency (small swings): 0.498403 Hz
- Angular frequency (small swings): 3.13156 rad/s
- Greatest speed of the bob, at the bottom of the swing: 3.13156 m/s
- Greatest acceleration along the path, at the ends of the swing: 8.49281 m/s2
- Height the bob rises above its lowest point: 0.5 m
Common questions
What is the formula for the period of a pendulum?
Two pi times the square root of the length over gravity. A 1 m pendulum has a period of about 2.0 s.
Does the mass change the period?
No. The period of a pendulum does not depend on its mass, and for small swings not on the amplitude.
How long is a pendulum with a period of 2 seconds?
About 0.994 m. This is called a seconds pendulum.
How does the period of a spring depend on the mass?
The period rises with the square root of the mass and falls with the square root of the spring constant.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any physics text (simple harmonic motion).
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