AS 91524 · 2023 paper · vertical circular motion

Question 1(c) Merit to Excellence

Practises Vertical circular motion and apparent weight

Tane has a toy car track: part is a horizontal banked curve of radius 0.750 m, part is a vertical loop of radius 0.250 m. The car has a mass of 0.120 kg. Assume sideways friction on the tyres is negligible.

m	0.120 kg
banked curve r	0.750 m at 1.55 m s⁻¹
loop r	0.250 m

Explain why a person in a car on a real roller coaster feels heavier at the bottom of the loop than at the top. Begin by drawing labelled vectors for the forces on the car at the top AND at the bottom of the loop.

The 2023 paper, page 3 ↗ Marking schedule, page 2 ↗

Work it out

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

1 Which forces act on the person at the loop’s highest point, and at its lowest?
At both: their weight, F_g = mg, straight down and the same size everywhere, and the reaction force from the seat, F_R. At the bottom the seat pushes up; at the top, upside down, it pushes down, towards the centre.
2 At the bottom the centre of the loop is above. Which way is the net force, so how big must F_R be?
The net force is the centripetal force, pointing up: F_c = F_R − F_g. So F_R = F_g + F_c: bigger than the weight.
3 At the top the centre is below. Which forces point towards it now?
Both: F_c = F_R + F_g, so F_R = F_c − F_g. Gravity already supplies much of the centripetal force, so the seat pushes much less, perhaps hardly at all.
4 What do you actually feel as your weight?
The push of the seat, the reaction force, not gravity itself. A big F_R at the bottom feels heavy; a small one at the top feels light.

Answer How heavy you feel depends on how hard the seat pushes on you: the reaction force. Down at the bottom, the centre of the loop is above you, so the seat must push up harder than your weight to provide the centripetal force (F_c = F_R − F_g): you feel heavier. At the top the centre is below and your weight already points at it, so the seat pushes only a little (F_c = F_R + F_g): you feel lighter.

What the marker wanted

Achieved One of: the right force arrows at the top and bottom, only F_g and F_R, with F_g the same at both and F_R bigger than F_g at the bottom but smaller at the top (an extra F_c arrow only if it is shown as the net force); saying that how heavy you feel is set by how big the reaction force is; saying the reaction force is largest at the bottom; or the equations for F_c at the top and bottom.
Merit Two of: those force arrows with the right relative sizes; at the bottom F_R has to be bigger than F_g to provide F_c, so you feel heavier; at the top F_R is small or zero, so you feel lighter.
Excellence The full answer, with an accurate diagram and a clear reason why F_R is bigger at the bottom than at the top.

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

Common mistake

Explaining it with energy, or drawing the reaction force the wrong way round or the wrong size. NZQA’s 2023 report lists both among the answers that fell short.

How did it go?

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