AS 91524 · 2024 paper · simple harmonic motion (watch glass)

Question 3(d) Merit to Excellence

A ball bearing is released on a watch glass — a shallow semi-circular glass bowl of radius of curvature R — and rolls back and forth with simple harmonic motion.

released 0.0400 m from equilibrium
T	0.882 s

Name this phenomenon, and explain in detail why the ball eventually comes to rest. Support your answer with any relevant formulae.

The 2024 paper, page 7 ↗ Marking schedule, page 6 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1 What is it called when an oscillation dies away over time?
Damping.
2 As the ball swings, how does its energy change form?
Back and forth between gravitational potential energy at the ends of the swing and kinetic energy in the middle. At every moment E_total = E_gp + E_k.
3 What acts against the ball’s motion, and what does it do to that total?
Friction between the ball and the glass, and air resistance. They always act against the ball’s velocity, so they do work against it every swing, turning some of its kinetic energy into heat. E_total goes down a little each time.
4 How does less total energy show up in the swing?
At the ends of each swing all the energy is gravitational, E_gp = mgh. Less total energy means a smaller E_gp there, so the ball rises less far up the bowl: the amplitude is smaller. (For any SHM, E = ½kA², so the energy goes as the amplitude squared.)
5 So why does it stop?
Energy is taken out every swing and nothing puts any back. The amplitude keeps shrinking until the ball sits at rest at the bottom, with all its mechanical energy turned into heat. The period stays about the same throughout.

Answer Damping. As the ball rolls, its energy changes between gravitational potential and kinetic, but friction and air resistance act against its motion and turn some of its kinetic energy into heat every swing. With less total energy, there is less gravitational potential energy at the ends of each swing, so it rises less high and the amplitude shrinks. Nothing replaces the lost energy, so the ball ends at rest at the equilibrium position.

What the marker wanted

Achieved One of: energy lost to friction; the amplitude decreasing; or the word damping.
Merit Damping named AND the energy leaving the system as heat, because of friction; or less total (or kinetic) energy linked to less gravitational potential energy, and so a smaller amplitude.
Excellence The full answer: the energy changes, what causes the loss, and how the loss shrinks the amplitude.

In our words, from NZQA’s 2024 marking schedule.

Common mistake

Stopping at “damping makes the amplitude decrease”. NZQA’s 2024 report lists that among answers below Achieved; the Excellence answers covered every energy change, the cause of the loss and what it does to the swing.

How did it go?

Got it moves it further out. Review brings it back tomorrow.