Calculators/Physics/Oscillations — pendulum and spring

Oscillations — pendulum and spring

Period · frequency · angular frequency

Oscillations — pendulum and spring — illustration
Period T
2.0061
s
Frequency f
0.4985
Hz
Angular freq. ω
3.1321
rad/s

T = 2π√(l/g), g = 9.81 m/s². The pendulum period depends neither on the mass nor (for small swings) on the amplitude. Relations: f = 1/T, ω = 2πf.

The period of harmonic oscillations. For a simple pendulum it depends only on the length and the gravitational acceleration:

T=2πlgT=2\pi\sqrt{\frac{l}{g}}

(it depends neither on the mass nor, for small swings, on the amplitude). For a mass on a spring with spring constant kk:

T=2πmkT=2\pi\sqrt{\frac{m}{k}}

From the period the calculator works out the frequency f=1Tf=\tfrac{1}{T} and the angular frequency ω=2πf\omega=2\pi f.

Examples

Pendulum of length l = 1 m
→ T = 2π√(1/9.81) ≈ 2.006 s, f ≈ 0.499 Hz
Spring: m = 0.5 kg, k = 20 N/m
→ T = 2π√(0.5/20) ≈ 0.993 s

Frequently asked questions

What does the pendulum period depend on?+

Only on the length l and the acceleration g. It depends neither on the mass nor (for small swings) on the amplitude — which is why the pendulum was the basis of clocks.

How do you change the period of a spring?+

A heavier mass lengthens the period (it oscillates slower), a stiffer spring (larger k) shortens it (it oscillates faster), because T = 2π√(m/k).

Found this calculator useful? Share it.

It's free — send it to a friend who's revising.

Teaching? Embed this calculator on your site.

Free widget — paste it on your school site, blog or notes. It links to the full version.

Teaching a class or tutoring? Guide students in NexTutor →