Instruction

The phenomenon. On a sunny spring afternoon California’s solar farms can produce more electricity than the state is using. Some of that midday surplus goes into pumped-storage hydroelectric plants such as Helms in the Sierra Nevada east of Fresno and Castaic north of Los Angeles. Each plant has two reservoirs at different heights. At midday, electric pumps push water up to the upper reservoir. After sunset, when demand peaks and solar output falls, the water runs back down through the same machines working as turbines and generators. No fuel is burned. The question for this unit: where is the energy while the water sits uphill, and why do we get back less than we put in?

Work. A force transfers energy into or out of a system by doing work: W = Fd cos θ, where F is the force, d the displacement and θ the angle between them. Work is measured in joules (1 J = 1 N·m). Only the part of the force along the motion counts.

Worked example. You pull a rolling suitcase 20 m through an airport with a 60 N force on the handle at 35° above the floor. W = (60 N)(20 m) cos 35° = 983 J. Had you pulled horizontally, all 1,200 J would count; pulling at an angle wastes part of your force lifting on the handle, which does no work along the floor.

Three cases to remember. If the force points along the motion (θ = 0°), W = Fd. If it is perpendicular (θ = 90°), W = 0: carrying a box across a room at steady height, your upward push does no work on it. If it opposes the motion (θ = 180°), the work is negative and energy leaves the object: 12 N of friction on a box sliding 3 m does W = −36 J.

Kinetic energy. A moving object has KE = ½mv². The work-energy theorem says the net work on an object equals its change in kinetic energy. A 1,200 kg car going from rest to 15 m/s gains ½(1,200)(15)² = 135,000 J. If that happens over 100 m, the average net force was 135,000 J ÷ 100 m = 1,350 N. Speed is squared, so doubling speed quadruples kinetic energy.

Gravitational potential energy. Lifting a mass m a height h at steady speed takes a force mg through distance h, so the work done is mgh, and the Earth-object system stores ΔPE = mgh. Pumped storage runs on this. Take a head (height difference) of 500 m (illustrative numbers). One cubic meter of water has a mass of 1,000 kg, so lifting it stores (1,000)(9.8)(500) = 4.9 × 10⁶ J = 1.36 kWh. An upper reservoir holding 1 × 10⁷ m³ above that head would store about 4.9 × 10¹³ J, or 13.6 GWh (illustrative numbers). If one kilogram fell the full 500 m with nothing in the way, it would reach √(2gh) ≈ 99 m/s. In the plant, the turbine takes that energy instead, and the water leaves slowly.

Energy belongs to a system. Potential energy is not stored “in the water”. It belongs to the water-Earth system, because it depends on their separation. Always name your system first. If the system is water + Earth, the pump’s push is external work that adds energy. If the system is water + Earth + pump + grid, the energy has only changed form. Energy is conserved: it is transferred or transformed, never created or destroyed.

Energy bar charts. A bar chart shows the energy accounting. The chart below uses illustrative pump and turbine efficiencies of 87% and 86%. Of 100 units of electrical energy used at midday, 87 end up as gravitational potential energy and 13 become thermal energy in the motors and water. In the evening, the 87 units return as 74.8 units of electricity. The round trip is 74.8%. Published estimates for pumped storage are commonly in the 70-80% range.

Common misconceptions. (1) “Holding something heavy is work.” Your muscles use energy, but no work is done on the object if it does not move. (2) “Energy is used up.” It is transformed, mostly into thermal energy that spreads out and is hard to use again. (3) “Pumped storage makes energy.” It only stores energy that was generated elsewhere, and returns less of it.

Formative check

Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.