AS 91524 · 2024 paper · rotational motion (flywheel)

Question 2(d) Merit to Excellence

Practises Energy methods for rolling and rotating bodies

A golf ball is placed in a light cup attached to a flywheel of diameter 0.220 m, and begins to accelerate downwards at a constant 1.83 m s⁻².

m ball	0.0460 kg
flywheel diameter	0.220 m
a	1.83 m s⁻²

Explain what happens to the final velocity of the ball and cup if the flywheel is replaced by a solid disc of the same radius and mass.

The 2024 paper, page 5 ↗ Marking schedule, page 4 ↗

Work it out

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

1 Where is the mass of a flywheel, compared with a solid disc of the same mass and radius?
A flywheel is built with most of its mass out near the rim. A solid disc spreads the same mass evenly, so much more of it sits close to the axle.
2 So which has the smaller rotational inertia, and why?
The disc. Each bit of mass counts in proportion to r² (I ∝ mr²), so moving mass towards the axis lowers I. The mass and radius are the same, so only that distribution has changed.
3 How does a smaller I change how fast the wheel, and so the cup, speeds up?
τ = Iα: the string’s pull gives the disc a bigger angular acceleration, and a = rα means the cup accelerates faster too. (The tension, and so the torque, drops a little as the cup accelerates faster, but the acceleration still rises: mg = a(m + I / r²).)
4 Check it with energy. Where does the gravitational potential energy go as the cup falls 0.750 m?
mgh = ½mv² + ½Iω², shared between the ball and cup and the spinning wheel. With a smaller I, less of it goes into the wheel and more is left for the ball and cup.

Answer The ball and cup end up moving faster. A solid disc of the same mass and radius has more of its mass near the axle, so its rotational inertia is smaller than the flywheel’s. The same pull then gives it a bigger angular acceleration (τ = Iα), so the cup accelerates faster (a = rα) and is moving faster after the same 0.750 m. In energy terms, less of the released energy goes into turning the wheel and more into the ball and cup.

What the marker wanted

Achieved Says I goes down, or that the final velocity goes up.
Merit One link explained: the mass sitting closer to the axis makes I smaller (I ∝ mr²); or a smaller I gives a bigger angular acceleration (τ = Iα).
Excellence The whole chain to a faster ball and cup. An energy argument counts, and saying the torque stays constant is not marked down.

The schedule’s second method ends by saying the final velocity will decrease. That contradicts its first method and its own Achieved column, which both say it increases; it is a slip.

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

How did it go?

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