AS 91524 · 2025 paper · rotational motion

Question 2(d) Merit to Excellence

Rīhari and Jane are on a playground roundabout of radius 1.30 m. Rīhari stands on it while Jane pushes; from rest she steadily accelerates it so it takes 12.0 s to make 4.00 rotations.

r	1.30 m
4.00 rotations in 12.0 s
I (roundabout + Rīhari)	430 kg m²

Later, Rīhari and Jane both stand on the edge while the roundabout turns at a constant angular velocity. Rīhari moves inwards, towards the centre.

Explain the effect this has on the rotational energy of the system.

The 2025 paper, page 6 ↗ Marking schedule, page 4 ↗

Work it out

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

1 Rīhari moves closer to the axis. What happens to the rotational inertia?
A person on the roundabout counts roughly as a point mass, I = mr². As his r shrinks, his share of I shrinks, so the system as a whole has less rotational inertia.
2 Is anything outside the system applying a torque? So what is conserved?
No outside torque acts, so the system’s angular momentum, L = Iω, is conserved.
3 If L stays the same and I gets smaller, what must ω do?
ω goes up, in the same proportion as I goes down: if I fell by a third, ω would rise by half.
4 Now the energy. Which way of writing E_k(rot) makes the answer clear?
E_k(rot) = ½Iω². I falls, but ω rises by the same factor and is squared, so the product gets bigger. Or write it as E_k(rot) = L² / 2I: L is fixed and I is smaller, so the energy is larger.
5 Where does the extra energy come from?
From Rīhari. To move inwards he has to pull himself towards the centre, and the work he does becomes the extra rotational kinetic energy.

Answer The rotational kinetic energy increases. Moving in lowers Rīhari’s I = mr², so the system’s rotational inertia falls. With no outside torque, angular momentum L = Iω is conserved, so ω rises in proportion. Because E_k(rot) = ½Iω² = L² / 2I, the same L with a smaller I means more rotational energy, supplied by the work Rīhari does pulling himself in.

What the marker wanted

Achieved One of: I goes down; ω goes up because angular momentum is conserved; or moving mass inwards makes the angular velocity or the rotational energy go up.
Merit Two of the ideas explained correctly and linked to each other.
Excellence All four, explained and linked: I falls; L is conserved; so ω rises in proportion; and because ω is squared in ½Iω² (or E = L² / 2I), the rotational energy goes up.

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

Common mistake

Saying the rotational energy stays the same because “energy is conserved”. Here angular momentum is what is conserved, and the rotational energy rises. NZQA’s 2025 report says candidates were often unclear about when angular momentum is conserved and when rotational kinetic energy is, and wrote “constant” where they meant “conserved”.

How did it go?

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