AS 91524 · 2023 paper · simple harmonic motion

Question 3(d) Merit to Excellence

Tanya studies a toy bouncing on a spring hanging from the ceiling. The spring constant is 24.6 N m⁻¹. She draws a graph of acceleration against displacement for the toy oscillating in simple harmonic motion.

k	24.6 N m⁻¹
a plotted against y
T	0.199 s

Once set oscillating with amplitude 0.100 m and period T = 0.199 s, the amplitude gradually decreases and the toy eventually stops. State the name of this phenomenon and explain what causes the decrease in amplitude, then draw a graph of amplitude against time for three complete oscillations.

The 2023 paper, page 8 ↗ Marking schedule, page 8 ↗

Work it out

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

1 What is the name for an oscillation that loses amplitude?
Damping.
2 What causes it here?
Air resistance and friction act against the toy’s motion and turn some of its energy into heat in the surroundings. An oscillation’s energy goes as its amplitude squared (E = ½kA²), so as energy is lost, the amplitude falls.
3 What does the period do while this happens?
It stays at 0.199 s. Damping takes away energy, not time: each cycle is smaller but just as long.
4 Now the graph. What goes on the axes, and where does the curve start?
Displacement (m) up the side, marked ±0.100 m; time (s) along the bottom, marked 0.199, 0.398 and 0.597 s for three complete oscillations. It starts at −0.100 m, because she pulled the toy down before letting go: a cosine curve, upside down.
5 How do the peaks change?
Each peak and trough is a little smaller than the one before, inside an envelope that dies away smoothly towards zero, while the crossings stay evenly spaced.

Answer Damping. Air resistance and friction take energy away from the toy as heat, and since an oscillation’s energy goes as its amplitude squared, the amplitude decreases while the period stays at 0.199 s. The graph: displacement (m) against time (s), starting at −0.100 m, three complete cycles ending at 0.597 s, each swing smaller than the last inside a decaying envelope.

What the marker wanted

Achieved One of: naming damping; energy lost to the surroundings through friction (so the amplitude falls); a cosine shape whose amplitude falls over three cycles; or three cycles with a constant period.
Merit Three of: damping named; energy lost as heat through friction, shrinking the amplitude; the cosine shape shrinking over three cycles; three cycles with a constant period; axes labelled, with values for the amplitude and the period.
Excellence All five.

In our words, from NZQA’s 2023 marking schedule.

Common mistake

Leaving the values off the graph. The Excellence column needs all five pieces, including the axes labelled with 0.100 m and the 0.199 s period; the shape on its own stops at Merit.

How did it go?

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