Work
In physics, work is energy transferred by a force. For a constant force F acting through a displacement d, W = F d cos θ, where θ is the angle between the force and the displacement. Work is measured in joules (1 J = 1 N·m). Work is zero when the force is perpendicular to the motion (a bag carried horizontally, a planet in a circular orbit), positive when the force has a component along the motion, and negative when it opposes the motion, like friction.
Kinetic energy and the work-energy theorem
Kinetic energy is the energy of motion: KE = ½ m v². The work-energy theorem says the net work done on an object equals its change in kinetic energy: W_net = ΔKE. A car at 20 m/s has four times the kinetic energy it has at 10 m/s, because KE depends on the square of the speed.
Potential energy and conservation
Gravitational potential energy near the surface is PE = m g h, measured from a chosen reference height. A stretched or compressed spring stores PE = ½ k x². When only conservative forces such as gravity and springs do work, mechanical energy is conserved: KE + PE stays constant. If an object slides down a frictionless hill of height h from rest, ½ m v² = m g h, so v = √(2 g h), regardless of its mass. With friction or air resistance, some mechanical energy turns into thermal energy, and W_friction = −f d tells you how much.
Power and efficiency
Power is the rate of doing work: P = W / t, measured in watts (1 W = 1 J/s). A common bill unit is the kilowatt-hour, the energy used by a 1 kW device in an hour (3.6 million joules). Efficiency is useful output divided by input, usually quoted as a percentage; it is always below 100 percent for real machines.
Worked example
A 0.80 kg block slides down a ramp from a height of 1.5 m and reaches the bottom at 4.0 m/s. Use g = 9.8 m/s².
- Energy at the top: m g h = 0.80 × 9.8 × 1.5 = 11.76 J.
- Kinetic energy at the bottom: ½ × 0.80 × 4.0² = 6.40 J.
- Energy turned into heat by friction: 11.76 − 6.40 = 5.36 J.
If the ramp were frictionless, the speed would be √(2 × 9.8 × 1.5) = 5.42 m/s.
Common mistakes
- Forgetting cos θ, or using the wrong angle.
- Using mass instead of weight in potential energy, or forgetting g.
- Squaring only part of the kinetic energy, or leaving out the ½.
- Treating kilowatt-hours as a unit of power.