Work and energy on curves, connected particles and rough planes

Last time

Before we start: last time, in Work, energy and power, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

The shortcut around Newton

Newton's second law gives you accelerations, and then you need the suvat equations to get speeds. Energy skips the middle. It links the start and the end directly, and it does not care about the shape of the path in between. Wherever a question asks for a speed after a distance, ask yourself whether energy gets there faster.

Conservation of energy

If the only forces doing work are gravity and nothing else, then kinetic plus potential energy stays constant. So whatever potential energy is lost becomes kinetic energy. Dropping through a height h gives a half m v squared equals m g h. The mass cancels, so v squared is two g h. A speed from rest after a drop of 1.8 metres is 6 metres per second.

A smooth curved path

On a smooth curve, the normal reaction is always perpendicular to the motion, so it does no work. Only gravity does work. So the speed depends only on the vertical drop, not on the shape of the curve. Do not try to resolve forces along a curve. Use energy.