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.

A mass on a string sweeps out a horizontal circle while the string traces a cone. The string makes angle $\theta$ with the vertical, length $L$, so the radius of the circle is $r = L \sin\theta$.

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:

Use vectors. At the top, both point down (toward centre, so add). At the bottom, tension points up (toward centre) and gravity points down (away from centre, so subtract).

Mass cancels because both the gravitational force and the required centripetal force scale with $m$.

Speed changes because gravity does work. Use conservation of energy to find the speed at different heights.