Conservation & dissipation
Conservation of energy is not a new source of energy. It is a complete account: energy at the start plus energy supplied must equal energy at the finish plus energy dissipated.
Choose the system and count every transfer once
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The system is the body or group of bodies whose energy you include. An external force is exerted by something outside that system. Dissipated energy has been transferred from the useful mechanical stores, usually to thermal energy by resistance.
- A useful mechanical-energy version is initial KE + initial GPE + driving work = final KE + final GPE + work against resistance.
- If you include GPE in the account, do not also add the work done by weight: that would count the same transfer twice.
- If the only force doing work is gravity, KE + GPE remains constant.
Common mistake
A smooth curved path depends on endpoints, not its shape
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Worked example
Smooth curved slide
A 2 kg particle starts from rest and descends through a vertical height of 4 m on a smooth curved slide.
The mass cancels. The detailed curve and the reaction do not enter because the slide is smooth and the reaction is perpendicular to motion.
Key idea
For connected particles, include the energy change of the whole system
When particles move together, choose both as the system. Add the KE of both particles and include the GPE change of every particle whose height changes. Internal string tension then need not appear.
Worked example
Two masses connected over a smooth pulley
A 3 kg particle descends 1.5 m while a connected 2 kg particle rises 1.5 m. They start from rest. The string and pulley are ideal.
A 2 kg particle starts from rest and descends through a vertical height of 3 m along a rough curved path of length 5 m. A constant resistance of 3 N acts throughout. Find its speed at the bottom.