AS 91524 · 2023 paper · simple harmonic motion
Question 3(b) Merit
Practises Simple harmonic motion (SHM)
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 |
By calculating the gradient of the graph, show that the period of oscillation is T = 0.199 s, and hence determine the mass of the toy hanging on the spring.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
-
1 Read two points off the line, as far apart as you can.
It runs from (−0.020 m, +20 m s⁻²) to (+0.020 m, −20 m s⁻²). Gradient = (−20 − 20) / (0.020 − (−0.020)) = −40 / 0.040 = −1000 s⁻². -
2 From (a), what does the gradient tell you?
−ω² = −1000, so ω = √1000 = 31.62 rad s⁻¹. -
3 Turn ω into the period.
T = 2π / ω = 2π / 31.62 = 0.1987 s, which rounds to the 0.199 s asked for. -
4 Which period formula has the mass in it?
For a mass on a spring T = 2π√(m / k), so m = kT² / 4π² = k / ω² = 24.6 / 1000 = 0.0246 kg.
What the marker wanted
- Achieved Either the working that shows T, with the gradient worked out correctly, or the mass worked out correctly.
- Merit Both: the “show” for T and the right mass.
The schedule’s working gives ω = 31.2 rad s⁻¹, but √1000 = 31.62, and its own next line, T = 0.1986 s, needs 31.62 (31.2 would give 0.201 s). It looks like a digit dropped.
In our words, from NZQA’s 2023 marking schedule.
✕Common mistake
Finding the gradient and then not knowing what to do with it. NZQA’s 2023 report says many candidates could calculate the gradient but not link it to the period. The link is (a): the gradient is −ω², and T = 2π / ω.How did it go?
Got it moves it further out. Review brings it back tomorrow.