Potential Energy

Energy with nowhere to be seen

In the chapter on Work, Kinetic Energy and Power, work was the transfer of energy by a force. The energy of a moving body is its kinetic energy \(K = \tfrac12 mv^2\), where \(m\) is the mass and \(v\) the speed. Now lift a brick slowly from the floor onto a shelf. You do work on it, but at the end the brick is at rest. Its kinetic energy is zero.

The energy has not vanished. Let the brick fall and it reaches the floor with exactly the speed that this energy would give it. So the energy was stored while the brick sat on the shelf. It was stored in the arrangement, in the brick being high above the Earth. Energy stored in an arrangement is called potential energy, written \(U\).

A stretched catapult, a wound clock spring and a compressed air cushion all store energy in the same way. Nothing in them moves and nothing is hot. The energy is in the shape: how far the parts are from where they would like to be.

How a change in potential energy is measured

The Earth pulls on the brick with its weight \(mg\), where \(g = 10\,\text{m/s}^2\) is the acceleration due to gravity. When the brick rises, gravity does negative work on it. When the brick falls, gravity does positive work and speeds it up. We define the change in potential energy as the negative of the work done by this force.

The minus sign carries the meaning. When gravity does positive work, as in a fall, the potential energy decreases and the body gains speed. When gravity does negative work, as in a lift, the potential energy increases. Only some forces can store energy this way; the next lesson shows which. Gravity and the force of a spring are two of them.

Gravitational potential energy

Lift a body of mass \(m\) through a height \(h\) near the Earth's surface. The weight \(mg\) points down and the displacement points up, so gravity does work \(-mgh\). The change in potential energy is therefore \(\Delta U = -(-mgh) = +mgh\). The path does not matter: gravity's work depends only on the change in height.

Now choose a level and call it \(U = 0\). This level is the reference level. A body at a height \(h\) above it then has \(U = mgh\). A body below it has a negative \(h\), and so a negative \(U\). A negative potential energy is not a mistake. It only means the body is below the level you chose.

Try the three choices in Fig 1.1. With the floor as zero, \(U\) is \(15\,\text{J}\) at A and \(5\,\text{J}\) at B. With the table top as zero, it is \(10\,\text{J}\) and \(0\). With the ceiling as zero, both values are negative, \(-5\,\text{J}\) and \(-15\,\text{J}\). In all three cases the ball loses \(10\,\text{J}\) of potential energy in the fall. That \(10\,\text{J}\) is what becomes kinetic energy, so the speed at B is the same for every choice.

The energy in a spring

A spring stretched by \(x\) from its natural length pulls back with a force of size \(kx\). Here \(k\) is the spring constant, in N/m. The force always points towards the natural length, so we write it as \(F = -kx\). In the chapter on Work you found the work done by this force from \(x = 0\) to \(x\). It is the area of a triangle under the force-extension graph, \(-\tfrac12 kx^2\).

So the potential energy stored in the spring is \(U = +\tfrac12 kx^2\), with \(U = 0\) at the natural length. The square has a plain meaning. A compression \(x = -0.2\,\text{m}\) gives the same energy as a stretch \(x = +0.2\,\text{m}\). The spring stores energy for being deformed, and it does not care in which direction.

The last chip of Fig 1.2 shows the square at work. Doubling the stretch doubles the force, and the force acts over twice the distance. So the stored energy grows four times. Tripling the stretch would store nine times the energy.

Springs in series and in parallel

Two springs joined end to end are springs in series. In the chapter Objects in Equilibrium you saw that one force \(F\) runs through both. So each spring stores \(U = F^2/2k\). The softer spring, with the smaller \(k\), stretches more and stores more energy. For two springs in series, \(U_1/U_2 = k_2/k_1\).

Two springs side by side, both fixed to the wall and to the block, are springs in parallel. They share one stretch \(x\), and their forces add: \(F = k_1x + k_2x\). So the pair acts as one spring of constant \(k = k_1 + k_2\). Each spring stores \(\tfrac12 k x^2\) with its own \(k\), so now the stiffer spring stores more, and \(U_1/U_2 = k_1/k_2\).

Cutting a spring

Cut a spring of natural length \(L\) and constant \(k\) into a piece of length \(L_1\). Under one force, every centimetre of the spring stretches by the same amount. So a piece stretches in proportion to its length, and a shorter piece is stiffer: \(k_1 L_1 = kL\). A spring cut in half has twice the spring constant.

Whose energy is it?

We often say "the potential energy of the ball". That is a convenient short form. The energy is stored in the separation between the ball and the Earth, and it takes both of them to store it. Neither body owns it. The same is true of a spring joining two blocks: the energy is in the spring between them, not in either block.

The short form is harmless near the Earth for one reason. When a ball falls, the Earth also moves towards the ball, but its enormous mass keeps its motion tiny. Almost all the released energy goes to the ball. Two bodies of similar mass, such as two stars or two blocks joined by a spring, share the energy. Then the potential energy must be assigned to the pair.