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Simple Pendulum

High schoolPhysicsRigid massless string, point-mass bob, solved numerically

About this experiment

Pull the bob aside, let it go and measure how long one swing takes. The simulation solves the full pendulum equation step by step, so it stays accurate at large angles where the familiar textbook formula starts to drift. The angle-time graph shows the exact motion next to the small-angle prediction.

Learning goals

  • Show that the period depends on string length and gravity but not on the mass of the bob
  • Verify the relation T = 2π√(L/g) and use it to measure g
  • Discover when the small-angle approximation stops being accurate
  • See how air damping reduces the amplitude while the period stays almost the same

How to use it

  1. 1Choose a string length and release angle, then press Play.
  2. 2Read the measured period after one full swing and compare it with the two predicted values.
  3. 3Drag the bob to a new starting angle and release it to start a fresh run.
  4. 4Increase the release angle to 60° or more and watch the exact curve fall behind the dashed small-angle curve.

The model

Equation of motion with linear damping b
Small-angle period
Large-amplitude correction, θ₀ in radians

Assumptions

  • The string does not stretch and has no mass. The bob is a point mass.
  • Damping is modeled as a force proportional to angular speed.
  • The equation is integrated with a fourth-order Runge-Kutta method at 1 ms steps.

Try this in class

  1. Q1Make the string four times longer. What happens to the period?
  2. Q2At what release angle does the exact period differ from the small-angle value by more than 1%?
  3. Q3Take the pendulum to the Moon. Predict the new period before you press Play.
  4. Q4Where in the swing is the bob fastest, and where is it momentarily at rest?

Frequently asked questions

Why is there no mass slider?

For an ideal pendulum the mass cancels out of the equation of motion, so the period does not depend on it.

Why does the measured period not match T = 2π√(L/g) exactly?

That formula assumes sin θ ≈ θ, which is only true for small angles. At larger angles the real period is longer.

How can students measure g with this lab?

Measure the period for several string lengths, plot T² against L and find the slope. The slope equals 4π²/g.