AS 91524 · 2025 paper · simple harmonic motion

Question 3(d) Merit to Excellence

Rīhari swings a ball on a string as a pendulum. It is 1.20 m from the fixed point to the ball's centre of mass, and the ball is released 5.60° from the vertical. The string's mass is insignificant.

L	1.20 m
θ	5.60°

Rīhari holds the end of the string between his fingers and moves his hand from side to side. At a certain frequency of movement, the ball swings with a very large amplitude.

Explain why this happens, using both forces and energy.

The 2025 paper, page 8 ↗ Marking schedule, page 6 ↗

Work it out

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

1 What does Rīhari’s hand add to the pendulum?
A driving force: a push that repeats at whatever frequency he moves his hand.
2 The pendulum has a frequency of its own. What is it?
Its natural frequency, from (b): f₀ = ω / 2π = 2.86 / 2π = 0.455 Hz, about one swing every 2.2 s.
3 What happens when his hand matches that frequency? Think about when in each swing the push arrives.
That is resonance. Every push lands at the same point in the swing, in the direction the ball is already moving, so each one adds to the motion instead of fighting it.
4 Now in energy terms: what goes in, and what comes out?
His hand does work on the ball every cycle, while air resistance and friction take energy away. At resonance the hand puts energy in faster than damping drains it, so the stored energy grows from swing to swing: kinetic energy at the middle, gravitational potential energy at the ends.
5 How does more stored energy show up in the swing?
As a bigger amplitude: more gravitational potential energy at the ends means the ball rises higher, so it swings further out. It keeps growing until the losses, which rise with the amplitude, match what the hand puts in.
6 And at any other frequency?
The pushes drift out of step with the swing, so some of them land against the motion and take energy back out. Little energy builds up, and the amplitude stays small.

Answer This is resonance. His hand applies a driving force, and when he pushes at the pendulum’s own natural frequency (about 0.455 Hz), every push arrives in step with the swing and adds to it. In energy terms, the hand then does work on the ball faster than air resistance and friction remove energy, so the kinetic and gravitational potential energy build up from swing to swing and the amplitude grows very large.

What the marker wanted

Achieved One idea: naming resonance; the hand applies a driving force; the hand adds energy; friction takes energy out; pushes in time with the swing add energy or amplitude; or the driving frequency matches the natural frequency.
Merit One side explained: forces (the driving frequency equal to the natural frequency, and that tied to the bigger amplitude) or energy (at that frequency the kinetic energy goes up, so the gravitational potential energy and the amplitude do too).
Excellence A complete answer, which the question says must cover both the forces and the energy.

In our words, from NZQA’s 2025 marking schedule.

Common mistake

Naming resonance or the natural frequency and stopping there. NZQA’s 2025 report says many candidates did exactly that; the marks are for explaining how pushes in step with the swing add energy faster than it is lost.

How did it go?

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