Potential energy

Raising a 0.500 kg cuckoo clock weight by 1 m stores 4.90 J, which the clock spends as the weight sinks.

Energy stored by lifting

Lift a book onto a shelf at a steady speed and you push up with a force equal to its weight, m g, through the height h. The work you do is W = m g h. That energy has not gone anywhere. It is stored in the book and Earth together, ready to come back as motion if the book falls. This stored energy is called gravitational potential energy, PE = m g h.

A cuckoo clock uses this directly. Winding it raises a weight; as the weight sinks slowly, gravity does work on the gears. OpenStax works out the numbers: a 0.500 kg weight raised by 1.00 m gains 0.500 × 9.80 × 1.00 = 4.90 J. The clock spends those 4.90 J over hours as the weight sinks.

Height from where? The reference level

Height has to be measured from somewhere, and that choice is up to you. You might measure from the floor, from the table top or from sea level. The value of PE changes with the choice, but the difference between two positions does not, and only the difference matters. Lifting a box from the second rung of a ladder to the third takes the same energy as lifting it from the eighth to the ninth. Choose the reference level that makes your sums easiest, usually the lowest point in the problem.

The path does not matter

Gravity has a special property. The change in potential energy between two points depends only on the change in height, not on the route. Walk up a long gentle ramp or climb straight up a ladder to the same landing: the change is m g h either way. The ramp needs a smaller force over a longer distance; the product comes out the same. Forces with this property are called conservative forces. Friction is not one of them, because a longer path means more energy lost to heat.

From potential to kinetic

Let a lifted object fall and gravity does work m g h on it, turning the stored energy into kinetic energy. OpenStax uses this to show why landing stiffly is dangerous. A 60.0 kg person jumps down from 3.00 m. Just before landing, his kinetic energy equals the potential energy lost, 60.0 × 9.80 × 3.00 = 1,764 J. If his knees bend only 0.500 cm, the floor must remove 1,764 J over 0.005 m, which takes an average force of about 3.5 × 10⁵ N, roughly 600 times his weight. Bending the legs over 0.5 m instead brings that down a hundredfold. Kangaroos land this way all the time.

Elastic potential energy

Springs store energy too. A spring that obeys Hooke's law needs a force k x to stretch or squeeze it by x, where k is the spring constant in newtons per metre. The stored energy is PE = ½ k x². OpenStax's toy car example uses a spring with k = 250 N/m squeezed by 4.00 cm: ½ × 250 × 0.0400² = 0.200 J. Released, the spring launches a 0.100 kg car up a 0.180 m slope, which costs 0.100 × 9.8 × 0.180 = 0.176 J, leaving 0.024 J of kinetic energy at the top, enough for about 0.69 m/s.

Hydroelectric dams store energy the same way, just with a lot of water. Every kilogram of water that falls 100 m to the turbines gives up m g h = 1 × 9.8 × 100 = 980 J. A tonne of water (1,000 kg) gives up 980,000 J, about 0.98 MJ, and a large dam lets many tonnes through every second. The higher the dam and the more water flows through, the more power it can deliver, which is why such dams are built in deep valleys with large rivers.