Potential Energy
1. It belongs to a system
A system has potential energy when the objects inside it interact through a conservative force. It takes two. The familiar phrase, the ball has potential energy, is shorthand at best: the energy belongs to the ball and the Earth together, and what changes it is a change in how they are arranged.
That is not pedantry. It is the reason a single asteroid drifting in deep space, far from everything, can only have kinetic energy. There is nothing for it to have potential energy with.
2. Where zero sits is your choice
You decide where the potential energy is zero, and you decide it to make the arithmetic easy. Put the zero at the floor, at the table top, or at the ball itself and you get three different values. All three are allowed, and all three give the same change, which is the quantity the physics actually uses.
One ball, three perfectly good choices of where zero sits. Move the ball and watch what each observer would write down, then watch what happens to the change when it falls.
3. Near the ground
Close to the surface the gravitational field is very nearly constant, so the change in gravitational potential energy is the weight times the change in height.
ΔU_g = m g Δywritten as a change, on purpose
Write it as a change. Going down makes it negative, and that sign does real work later on when you are balancing an energy equation and wondering why the numbers refuse to come out.
4. The spring
A stretched or compressed spring stores elastic potential energy of one half the spring constant times the square of the displacement from its relaxed length.
U_s = ½ k (Δx)²Δx measured from the relaxed length
Drag the stretch. The graph is force against stretch, a straight line through the origin, and the shaded triangle under it is the stored energy. That is where the one half comes from, and it is worth seeing once rather than memorizing.
Because the displacement is squared, stretching by ten centimeters and compressing by ten centimeters store exactly the same energy, and doubling the stretch stores four times as much.
5. Far from the ground
For two spherical bodies a long way apart, such as a planet and a moon, the gravitational potential energy of the pair is
U_g = − G m₁ m₂ / rzero taken at infinite separation
It is negative everywhere, and that is a consequence of the choice, not a strange physical fact: zero was placed at infinite separation, and bringing two attracting masses together releases energy, so every finite separation sits below zero. Near a planet surface this same expression flattens into the straight line that gives mg Δy. They are one result at two scales.
Finally, when a system holds more than two objects, add up the potential energy of every pair. Three objects make three pairs, four objects make six. Count the interactions, not the objects.
Check yourself
1. A 0.50 kg ball is held 1.2 m above the floor, over a table 0.80 m tall. Find the potential energy with the zero at the floor and with the zero at the table top, then find the change when the ball falls to the floor.
Floor zero: 0.50 × 10 × 1.2 = 6.0 J. Table zero: the ball is 0.40 m above it, so 2.0 J. Falling to the floor: ΔU = −6.0 J either way. The values disagree, the change does not.
2. A spring with k = 200 N/m is stretched 0.10 m. Find the energy stored, then say what happens to it if the stretch is doubled.
U = ½ × 200 × 0.10² = 1.0 J. Doubling the stretch gives 4.0 J, four times as much, because the displacement is squared. Compressing by 0.10 m stores the same 1.0 J, for the same reason.
3. Explain why the general gravitational potential energy is negative for every finite separation, without using the word formula.
Because the zero was put at infinite separation. Two masses that attract each other release energy as they approach, so any arrangement closer than infinitely far apart has less energy than the zero point, and less than zero is negative.
4. Three masses sit at the corners of a triangle. State how many potential energy terms the system has, and say what happens if a fourth mass is added.
Three terms, one per pair. A fourth mass makes six pairs, so six terms. The count grows faster than the number of objects because it is counting interactions.
Topic 3.4, Conservation of Energy. Kinetic and potential are the two accounts. The next topic is the bookkeeping that connects them, and the choice that decides whether the books balance.