l = 1 m, default g
l = 1 m, blank g.
Period of a Simple Pendulum
Enter string length l. Blank g = 9.81. School model: point mass, massless rod, small angle.
Practical version (swing, clock): pendulum period. Then f = 1/T on frequency from period. ω: angular velocity.
Result
Enter values. The result shows up here.
How it works
A simple (mathematical) pendulum is a model: point mass on a massless string (or rod) of length l, planar motion, no friction, small angle. Then T = 2π√(l/g).
Same numbers as the practical page: l = 1 m, g = 9.81 m/s² → T ≈ 2.01 s. The difference is emphasis: here we state the school assumptions.
At large angles the formula is no longer exact. Mass and amplitude do not enter this T. Blank g = 9.81.
Compare with the practical pendulum when you want playground or clock examples.
How to use
- Enter length l > 0.
- Optionally enter g (blank = 9.81).
- Read T = 2π√(l/g).
- Keep the assumptions in mind: small angle, point mass, massless rod.
Formula
T = 2π √(l/g)
Assumptions: small angle, point mass, massless string, no drag.
Real-life examples
l = 0.5 m
l = 0.5 m.
l = 2 m
l = 2 m.
Exercise l = 0.8 m
l = 0.8 m.
Moon g = 1.62
l = 1 m, g = 1.62.
l = 0.25 m
l = 0.25 m.
g = 9.81 entered
l = 1 m, g = 9.81.
l = 1.2 m
l = 1.2 m.
l = 0.1 m
l = 0.1 m.
l = 3 m
l = 3 m.
Ways to use this calculator
- School problems on the simple pendulum.
- Explicit model assumptions vs the practical pendulum page.
Frequently asked questions
What does “simple/mathematical” mean?
Ideal model: point mass, massless string, small angle, no drag. Formula T = 2π√(l/g).
How does this differ from the pendulum page?
Same formula. That page is practical examples; this one stresses school assumptions. Link: <a href="pendulum-period.html">pendulum period</a>.
Why small angle?
For small θ, sin θ ≈ θ and the motion is harmonic. At large angles the period grows.
Does mass enter T?
No. In the simple model T does not depend on m.
What if g is blank?
We use 9.81 m/s².
How do I get f and ω?
f = 1/T on <a href="frequency-from-period.html">frequency from period</a>; ω = 2π/T on <a href="angular-velocity.html">angular velocity</a>.
Can the string have mass?
Not in this model. A physical pendulum uses centre of mass and moment of inertia.
Why is l ≤ 0 rejected?
Length must be positive.
What does the T(l) chart show?
Like √l at fixed g.
Spring vs pendulum?
Spring: f from k and m. Pendulum: T from l and g.
Is g different at the poles?
Slightly. You can enter your own g instead of 9.81.