Quick questions on Non-uniform circular motion (banked tracks, conical pendulums, vertical circles) explained: HSC Physics Module 5

On a frictionless banked track angled at $\theta$ to the horizontal, the normal force $N$ acts perpendicular to the road surface. Its horizontal component provides the centripetal force; its vertical component balances gravity.

For an object moving in a vertical circle (a ball on a string, a roller coaster loop), speed is not constant because gravity does work as the object rises and falls. At any point, the net force toward the centre still equals $\frac{m v^2}{r}$, but the contributions of tension and gravity vary around the loop.

Torque is the rotational equivalent of force, the tendency of a force to cause rotation about a pivot:

Distinguish between uniform and non-uniform circular motion in one sentence each, and give one physical example of each. [2 marks]

A car of mass $1200 \text{ kg}$ crests a hill of radius $40 \text{ m}$ at $18 \text{ m/s}$. Calculate the normal force from the road on the car at the crest. [3 marks]

A $0.30 \text{ kg}$ ball on a $0.80 \text{ m}$ string swings in a vertical circle. (a) Find the minimum speed at the top so the string stays taut. (b) Find the string tension at the bottom when the ball's speed is $5.0 \text{ m/s}$.