Conservation of Energy
1. What a system can have
A system made of one object can only have kinetic energy. Put two or more objects inside the boundary, let them interact through conservative forces, and the system can have potential energy as well. So the first question in any energy problem is not what the numbers are. It is what is inside the system.
Mechanical energy is the kinetic plus the potential, and nothing else. Thermal energy and sound are real energy and are not mechanical energy. That one distinction is why mechanical energy can fall while total energy stays exactly constant, which is otherwise the most confusing sentence in the unit.
E_mech = K + Utwo accounts, and only two
2. The bookkeeping
Any change to one kind of energy inside a system has to be matched by an equal change in other kinds inside it, or by a transfer across the boundary. Energy bar charts are the way to see this: draw the bars before and after, and if they do not balance, something crossed the boundary that you have not drawn yet.
A cart released from rest, down a ramp and around a loop. The bars on the right are the energy accounts at this instant. Add friction and watch a third bar appear, which is the energy that left the mechanical account and did not leave the universe.
3. The choice that decides everything
You may choose the system so that its total energy is constant. If no work is done on the system you chose, and nothing nonconservative acts inside it, then its mechanical energy stays constant. If work is done on it, energy is transferred across the boundary.
The same block on the same rough ramp gives two different answers depending on where you drew the line, and both are correct.
One block, one rough ramp, two boundaries. Switch between them and read what each choice says about the total. Nothing about the physical situation changes when you switch.
4. Nothing is ever lost
Energy is conserved in all interactions. There is no exception and no fine print. When friction or air resistance drags the mechanical energy down, that energy has become thermal energy and sound. It is in the block, in the ramp, in the air.
So never write that energy was lost. Write where it went. A reader can tell the difference between a student who is tracking energy and a student who is hoping the term goes away.
5. The method, every single time
Name the system. Choose the zero of potential energy. Draw the bars for the start and the finish. Write one equation setting the starting energy plus any transfer equal to the finishing energy. Solve in symbols, and put the numbers in last.
K_i + U_i + W = K_f + U_fone line, once the system is named
Energy methods jump from the start to the finish and ignore everything in between. That is what makes them quicker than forces, and it is also exactly what they cannot tell you: no time, no acceleration, no force at one instant. Those still come from Unit 2.
Check yourself
1. A cart is released from rest at height h on a frictionless track and runs around a vertical loop of radius R. Find its speed at the top of the loop, in symbols, then check the expression at h = 2R.
Taking the zero at the bottom, mgh = ½mv² + mg(2R), so v = √(2g(h − 2R)). At h = 2R it gives v = 0, meaning the cart just reaches the top with nothing left. In practice it would have fallen off the track before that, because a real loop needs a nonzero speed at the top to keep the cart on it.
2. A 4.0 kg block loses 30 J of mechanical energy sliding across a rough floor. Say where those 30 J are, and what is wrong with the sentence friction destroys energy.
In the block and the floor as thermal energy, plus a little that left as sound. Both are slightly warmer. Friction destroys nothing: energy is conserved in all interactions, and friction is a converter, not a drain.
3. A block slides down a rough ramp. Take the block alone as the system, then take the block, the ramp and the Earth. Say whether the total energy is constant in each case.
Block alone: not constant, because friction and gravity both act across its boundary and do work on it. Block plus ramp plus Earth: constant, because nothing crosses that boundary. The mechanical energy still falls and exactly that much thermal energy appears inside.
4. Give one situation where the mechanical energy of a system rises, and say what paid for it.
A cart pushed along a level track speeds up, so its kinetic energy rises with no drop in potential energy. The work done by the hand paid for it, which means energy crossed the boundary inward. Nothing was created.
Topic 3.5, Power. Everything so far has been about how much. The last topic of the unit puts a clock on it.