Chapter 7: Work, Energy, and Simple Machines

In science, work is done when a force makes an object move in the direction of the force. For a constant force F and a displacement s along the force, W = F × s. The SI unit of work is the joule (J), where 1 J = 1 N × 1 m. Work has magnitude only, with a positive or negative sign. If there is no displacement, there is no work. On a force-displacement graph, the work done is the area under the graph.

The work done by a force can be positive, negative or zero. If the displacement is in the direction of the force, the work is positive and the energy of the object increases. If the displacement is opposite to the force, the work is negative and the energy decreases. If the object does not move, or the force is at right angles to the displacement, the work is zero. Pushing a wall that does not move does no work on the wall, though the muscles still use up energy.

Energy is the capacity to do work, and its SI unit is the joule. The work-energy theorem says that the work done on an object equals the change in its energy. Positive work raises the energy and negative work lowers it. Energy exists in many forms such as kinetic, potential, heat, light, sound, chemical and electrical energy, and it can change from one form to another. Energy can pass from one object to another by doing work, by heat flow from a hotter to a colder body, or by radiation such as sunlight.

Kinetic energy is the energy a body has because of its motion. For a body of mass m moving with speed v, K = ½ m v². A body at rest has zero kinetic energy, and kinetic energy has no direction. It is proportional to the mass but to the square of the speed, so doubling the speed makes the kinetic energy four times as large. By the work-energy theorem, the work done on a body equals its change in kinetic energy.

Potential energy is the energy stored in an object or a system because of its position or shape. Near the Earth's surface, the gravitational potential energy of a mass m raised to a height h is U = m g h. It is proportional to both mass and height, and it depends only on the height, not on the path taken to reach it. A stretched or compressed spring and a bent bow store potential energy because of their changed shape. When the object or spring is released, the stored energy changes into kinetic energy.

The mechanical energy of an object is the sum of its kinetic and potential energies, E = K + U. If no external force such as friction or air resistance acts, the mechanical energy stays constant. This is the conservation of mechanical energy. In a fall, the potential energy decreases while the kinetic energy increases by the same amount. At the top of a thrown ball's path the kinetic energy is zero and the potential energy is maximum. If friction or air resistance acts, some mechanical energy changes to heat, so the motion dies out, but the total energy of the whole system stays constant.

Power is the rate of doing work: P = W ÷ t. Its SI unit is the watt (W), and 1 W = 1 J per second. Larger units are the kilowatt (1 kW = 1000 W) and the horsepower (1 hp ≈ 746 W). The runner and the walker do the same work, but the runner does it in less time and so has greater power. The work done can be found from W = P × t.

A simple machine makes work easier by changing the size or the direction of the force. The force we apply is the effort and the force we overcome is the load. The mechanical advantage is MA = load ÷ effort. A fixed pulley only changes the direction of the effort, so its MA is 1. An inclined plane of length L and height h has MA = L ÷ h, so a longer ramp needs less force but a longer path. A lever turns about a fixed point called the fulcrum and balances when effort × effort arm = load × load arm. A machine cannot create energy, so it does not reduce the total work.