Pendulum Phase Portrait Simulation

A pendulum released from a small angle swings back and forth, and T = 2π√(L/g) describes it well. Give it enough starting speed and it stops swinging altogether: it carries over the top and keeps rotating. Plotting angle against angular velocity shows both at once — a swing is a closed loop, a rotation is an open track, and the curve separating them is exactly the energy needed to balance motionless at the top. Students can change start angle and start spin and compare what happens. Use it for math inquiry with High School on a projector or student device.

Standards: Common Core CCSS.MATH.CONTENT.HSF.IF.B.4, CCSS.MATH.CONTENT.HSA.CED.A.3

• Release from rest at a small angle and watch the closed loop the swing draws
• Increase the starting spin until the loop opens and the pendulum goes over the top
• Find the starting spin that lands almost exactly on the dividing curve, and watch it creep towards the top
• Add damping and watch the loop spiral inward as energy leaves
• Compare the small-angle period with the real one as the amplitude grows

1. 1Predict how changing start angle will affect the energy, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

Questions teachers ask