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Physics Unit 3: Work, energy and power

Energy is the ability to do work. Without friction, mechanical energy is conserved, which makes many motion problems quick.

Covers: Work, kinetic and potential energy, conservation of energy, springs, power

Study guide

Work is energy transferred by a force acting through a distance: W = Fd cos θ, where θ is the angle between the force and the displacement. A force in the direction of motion does positive work, a force opposing motion (like friction) does negative work, and a force perpendicular to motion does no work. Work and energy are measured in joules (J).

Kinetic energy is the energy of motion, KE = ½mv². Doubling the speed quadruples the kinetic energy. The work-energy theorem says the net work on an object equals its change in kinetic energy.

Potential energy is stored energy due to position. Gravitational potential energy is PE = mgh, measured from a chosen reference level. Elastic potential energy in a spring is ½kx², where k is the spring constant and x is the stretch or compression. Hooke's law gives the spring force: F = −kx, proportional to the displacement and pointing back toward equilibrium.

Energy is conserved. With no friction, mechanical energy (KE + PE) stays constant, so energy converts between forms: a falling ball trades PE for KE, and a pendulum trades back and forth. Use KE₁ + PE₁ = KE₂ + PE₂ to find speeds or heights without tracking forces; a ball dropped from height h lands with v = √(2gh). With friction, some mechanical energy becomes thermal energy, and the work done by friction accounts for the loss.

Power is the rate of doing work or transferring energy: P = W/t, measured in watts (1 W = 1 J/s). Power can also be written P = Fv. Efficiency is useful output energy divided by input energy.

An energy bar chart shows each kind of energy (K, U_g, U_s) at one moment as a bar. With no friction or outside work, the bars add up to the same total at every moment. Choose where U_g = 0 first; the bars before and after then show where the energy went. On a level surface, friction does work W = −μₖmgd. A block released from height h on a frictionless ramp that then slides to a stop on a rough floor satisfies mgh = μₖmgd, so d = h/μₖ, which doesn't depend on mass.

When the force varies, the work it does is the area under the force-vs-position graph. For a spring that area is ½kx². Whether energy counts as "conserved" depends on the system you choose. If Earth is part of the system, gravity changes potential energy; if it isn't, gravity does external work.

Key ideas

• Work W = Fd cos θ; only force along the motion does work.
• Kinetic energy KE = ½mv²; gravitational PE = mgh; spring PE = ½kx².
• Without friction, KE + PE stays constant.
• Friction turns mechanical energy into thermal energy.
• Power = work/time, measured in watts.

Formulas and rules

Common mistakes

• Forgetting to square v in kinetic energy.
• Counting a perpendicular force (like the normal force on flat ground) as doing work.

Words to know

Kinetic energy

½mv²

Gravitational potential energy

mgh

Watt (W)

The unit of power: one joule per second.

Work by a perpendicular force

Zero: the force does no work.

Conservation of mechanical energy

KE + PE is constant with no friction.

Work

W = Fd cos θ

Joule (J)

The unit of work and energy: 1 N·m.

Work-energy theorem

Net work = change in kinetic energy.

Elastic potential energy

½kx²

Hooke's law

F = −kx

Thermal energy from friction

Mechanical energy lost to friction becomes heat.

Efficiency

Useful energy out ÷ energy in.

Power (force and velocity)

P = Fv

Spring constant (k)

Stiffness of a spring in N/m; larger k means a stiffer spring.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.