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Work, Energy & Power
The currency of nature: work, kinetic and potential energy, conservation and power. 48 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Energy and work give a second way to analyse motion, often easier than tracking forces. Work is force applied over a displacement, and kinetic and potential energy store that work in motion and position. The principle of conservation of energy and the definition of power tie these ideas together and explain how machines and engines transfer energy.
How the topics connect
Energy and work give a second way to solve motion problems, often more quickly than forces. Work measures energy transferred by a force, kinetic and potential energy store it, and conservation tells you what is possible. The idea of power, the rate of doing work, connects machines and engines to everyday use.
Real-world uses
Compare the power output of a motor, the energy stored in a battery and the energy used by an appliance, using joules and watts.
Predict the speed at the bottom of a drop by converting gravitational potential energy into kinetic energy.
Estimate how much energy a wind turbine or hydroelectric dam can deliver from the height of water or wind speed.
Understand how the work done in climbing stairs relates to the energy you burn and the power you develop.
Households compare appliances by power in watts and estimate costs from energy in kilowatt-hours.
Engineers use energy conservation to estimate how much power a wind turbine or hydroelectric dam can deliver.
Work done by a constant force
Force component along displacement times displacement.
W = Fd cosθWork is energy transferred by a force over a displacement: W = Fd cosθ, where θ is the angle between force and displacement. Only the component along the motion does work.
Pulling a sled 10 m with 50 N at 30°: W = 50·10·0.866 = 433 J.
Carrying a bag horizontally at constant speed: the force (up) is perpendicular to the motion, so work is zero.
Units of work and energy
The joule is one newton-metre.
1 J = 1 N·mThe joule equals 1 N·m = 1 kg·m²/s². Other units: calorie (4.184 J), kilowatt-hour (3.6 MJ), electron volt (1.6×10⁻¹⁹ J).
Lifting a 1 kg book by 1 m takes about 9.8 J.
Torque also has units N·m but is not energy.
Positive, negative and zero work
Work is negative if the force opposes motion and zero if perpendicular.
Work is positive when force has a component along the motion (energy added), negative when it opposes it (energy removed), and zero when perpendicular. Friction does negative work on a sliding body.
Gravity does −mgh on a rising ball, +mgh on a falling one.
Negative work isn't "less than nothing", it's energy taken from the body.
Work from a force–displacement graph
Area under the F–x graph equals work.
On an F–x graph the area under the curve is the work. Triangles and rectangles give simple results; areas below the axis count as negative.
F rising from 0 to 10 N over 2 m: W = ½·10·2 = 10 J.
Plot against displacement, not time.
Work done by a variable force
Integrate force over displacement.
W = ∫F·dxWhen the force varies, W = ∫F dx. For a spring this gives ½kx², for gravity far from Earth −GMm(1/r₂ − 1/r₁).
F = 3x² N from 0 to 2 m: W = 8 J.
Use F's component along the path.
Work done by a spring
Stretching a spring stores energy proportional to x².
W = ½kx²Stretching a spring from 0 to x needs W = ½kx², the area of the triangle in the F–x graph. The spring does −½kx² on you in return.
k = 100 N/m, x = 0.2 m: W = 2 J.
Going from x₁ to x₂ the work is ½k(x₂² − x₁²), not ½k(x₂ − x₁)².
Kinetic energy
Energy of motion.
K = ½mv²The energy of motion K = ½mv². It is a scalar, never negative, and grows with the square of speed.
A 1000 kg car at 20 m/s: K = 200 kJ; at 40 m/s: 800 kJ.
Doubling the speed quadruples the kinetic energy and the braking distance.
Work–energy theorem
Net work equals the change in kinetic energy.
W_net = ΔKThe net work on a body equals its change in kinetic energy, W_net = K_f − K_i. It gives speeds without solving for time or acceleration.
A 2 kg block pushed 5 m by a net 8 N from rest: ½·2·v² = 40, v = 6.3 m/s.
Use the net work, including friction and gravity.
Potential energy
Stored energy due to position or configuration.
Potential energy is energy stored in a configuration, such as height, a stretched spring or separated charges. Only changes matter, so the zero level can be chosen freely.
A raised hammer has gravitational potential energy it converts into motion when released.
Potential energy belongs to the system, not a single body.
Gravitational potential energy
Energy due to height in a uniform field.
U = mghNear Earth U = mgh relative to a chosen level. The work done by gravity is −ΔU independent of path.
Lifting a 5 kg box 2 m: ΔU = 98 J.
This formula is only valid when g is nearly constant.
Elastic potential energy
Energy stored in a deformed spring.
U = ½kx²A spring stores U = ½kx² when stretched or compressed by x. It is zero at the natural length and the same for compression and extension.
k = 500 N/m compressed 4 cm stores 0.4 J.
x must be measured from the natural length.
Conservative forces
Work done is path independent and zero over a closed loop.
A force is conservative if the work it does between two points is independent of path, or zero around a closed loop. Gravity, springs and electrostatic forces qualify and have potential energies.
Gravity does the same work whichever ramp you lift a block along.
Friction depends on path length, so it's not conservative.
Non-conservative forces
Friction and drag dissipate mechanical energy.
Friction, drag and many human forces do path-dependent work and convert mechanical energy to heat. Total energy is still conserved but mechanical energy isn't.
A block sliding 3 m with 5 N friction loses 15 J to heat.
Don't assign potential energy to friction.
Force from potential energy
Force is the negative gradient of potential.
F = −dU/dxIn one dimension F = −dU/dx: the force points downhill on the potential curve, and its size is the slope. In 3-D, F = −∇U.
U = ½kx² gives F = −kx.
Mind the minus sign.
Potential energy curves
Minima are stable equilibria, maxima are unstable.
A plot of U versus position shows the motion: the particle is confined between turning points where U = E, the speed is high where U is low, and minima are equilibrium points.
A ball in a bowl rolls back and forth between turning points.
The total energy line is horizontal.
Equilibrium types
Stable, unstable and neutral equilibrium from the curvature of U.
At an equilibrium dU/dx = 0. If d²U/dx² > 0 it is stable (minimum), if < 0 unstable (maximum), if = 0 neutral (flat). Small displacements from stable equilibrium produce oscillations.
A ball on a hilltop is unstable; in a valley, stable.
Zero force alone doesn't tell stability.
Conservation of mechanical energy
Kinetic plus potential stays constant when only conservative forces act.
K + U = constIf only conservative forces do work, K + U stays constant. Use ½mv₁² + mgh₁ = ½mv₂² + mgh₂ for quick speed calculations.
A roller coaster from rest at 40 m reaches 28 m/s at the ground, ignoring friction.
If friction acts, mechanical energy is not conserved.
Energy conservation in free fall
Potential energy converts to kinetic energy as an object falls.
v = √(2gh)During a fall potential energy converts to kinetic: mgh = ½mv², so v = √(2gh), independent of mass.
Dropped from 20 m: v ≈ 19.8 m/s.
This ignores air resistance.
Energy on a pendulum
Energy swaps between kinetic at the bottom and potential at the top.
pendulumAt the top of the swing the bob has maximum U and zero K; at the bottom maximum K. The speed at the bottom is v = √(2gL(1 − cosθ)).
L = 1 m, θ = 60°: v ≈ 3.13 m/s.
The tension does no work because it's perpendicular to motion.
Energy in spring-mass systems
Total energy ½kA² is shared between kinetic and spring potential.
E = ½kA²The total energy E = ½kA² is constant. At displacement x the kinetic energy is ½k(A² − x²), and speed is max at x = 0.
k = 50 N/m, A = 0.2 m: E = 1 J.
Speed depends on amplitude and position, not gravity (for a horizontal system).
Roller coaster and loops
Energy conservation gives the speed at any height on a track.
v = √(2g(h−y))Using energy conservation the speed at any height is v = √(2g(h₀ − h)) (frictionless). To pass the top of a loop of radius r, start from at least h = 2.5r.
Loop radius 10 m needs a start height of 25 m.
Starting height at the same level as the loop top isn't enough.
Energy loss to friction
Mechanical energy lost equals friction work, converted to heat.
ΔE = −f dFriction converts mechanical energy into heat: ΔE_mech = −f·d. A sliding block's final speed follows from W_fric = f d in the work–energy theorem.
A 2 kg block at 6 m/s stops after 6 m: friction = 6 N.
Friction depends on path length, not displacement.
General energy conservation
Total energy including heat and other forms is conserved.
Energy is never destroyed, only transferred or transformed: mechanical, thermal, chemical, electrical and so on. The total in an isolated system stays constant.
A car's fuel energy goes into motion, heat in the engine and sound.
"Energy loss" means conversion into forms we can't use.
Power
Rate of doing work.
P = W/tPower is the rate of doing work or transferring energy, P = W/t, in watts. A 100 W lamp uses 100 J every second.
Lifting 50 kg 3 m in 5 s needs about 294 W.
Don't confuse power with energy: kWh is energy.
Instantaneous power
Force dotted with velocity.
P = F·vThe power at an instant is P = F·v = Fv cosθ, useful for engines and moving loads.
A car resisting a 600 N drag at 30 m/s needs 18 kW.
If a force is perpendicular to velocity it does no work and supplies no power.
Units of power
Watt equals joule per second; horsepower ≈ 746 W.
1 W = 1 J/s1 watt = 1 J/s. A horsepower is about 746 W. Human sustained output is about 100 W.
A 2 kW kettle uses 2 kJ per second.
kW is power, kWh is energy.
Efficiency
Useful output divided by input energy.
η = P_out/P_inEfficiency is useful output divided by total input: η = P_out/P_in ≤ 1. Losses are mostly heat. Real devices are always below 100%.
An engine using 300 J of fuel to do 75 J of work has η = 25%.
Efficiency doesn't tell you how much power is delivered.
Energy and power bills
Kilowatt-hours measure household energy use.
1 kWh = 3.6 MJUtility companies charge for energy in kWh. Cost = power (kW) × time (h) × price.
A 1.5 kW heater for 4 h at 12 c/kWh: 6 kWh, 72 cents.
A kWh is not a unit of power.
Power of engines and climbers
Power needed to lift a body at constant speed.
P = mgvClimbing steadily at speed v needs P = mgv (plus losses). Engine power at constant speed balances drag: P = Fv.
An 80 kg climber ascending 1 m/s needs 784 W of mechanical power.
Hill climbing power depends on vertical speed only.
Escape from a potential well
A particle escapes if its energy is positive relative to infinity.
A particle with total energy E ≥ 0 relative to infinity escapes, since gravitational energy is −GMm/r. Setting ½mv² = GMm/R gives the escape speed √(2GM/R).
Earth: about 11.2 km/s.
Escape speed doesn't depend on the direction or mass of the object.
Work in rotation
Torque times angle turned.
W = τθFor a torque τ turning through an angle θ, the work is W = τθ (θ in radians), analogous to F·d. Power P = τω.
A 10 N·m torque through 2 rad does 20 J.
Use radians.
Rotational kinetic energy
Energy of spinning bodies.
K = ½Iω²A spinning body has K = ½Iω², where I is its moment of inertia. For the same mass, mass farther from the axis stores more energy at the same ω.
A flywheel of I = 2 kg·m² at 100 rad/s: 10 kJ.
Total energy of a rolling body is translation plus rotation.
Rolling without slipping energy
Total energy combines translation and rotation.
K = ½mv² + ½Iω²Rolling bodies have K = ½mv² + ½Iω² with ω = v/R. For a solid sphere, rotation takes 2/7 of the total.
A 1 kg ball at 2 m/s rolling: K = 2 + 0.8 = 2.8 J.
Static friction does no work in pure rolling.
Work done by gravity on incline
Gravity's work depends only on vertical height change.
inclineGravity's work is mg·Δh regardless of slope length: W = mg·d·sinθ = mgh. Steeper slopes just reach the same height faster.
A 5 kg block sliding down 10 m of a 30° slope: W = 5·9.8·5 = 245 J.
The normal force does no work on a fixed slope.
Heat as energy transfer
Heat is energy flowing due to temperature difference.
Heat is energy transferred due to a temperature difference, not a stored substance. It can be converted to work only partially.
Rubbing hands warms them: work becomes thermal energy.
Objects contain internal energy, not heat.
Mechanical equivalent of heat
Joule showed work can be converted to heat at a fixed rate.
1 cal = 4.186 JJoule's paddle-wheel experiment showed a fixed amount of work yields a fixed amount of heat: 1 cal = 4.186 J. This established energy conservation including heat.
Falling 427 m raises water 1 K per kg.
Calories on food labels are kilocalories.
Mass–energy equivalence
Mass is a form of energy.
E = mc²Mass is a form of energy: E = mc². A tiny mass change releases huge energy: 1 g is 9×10¹³ J, about 21 kilotons of TNT.
The Sun loses about 4 million tonnes of mass per second.
Not "mass turns into energy"; the system's mass changes when its energy changes.
Sources of energy
Fossil, nuclear, solar, wind, hydro and geothermal sources.
Primary sources: fossil fuels, nuclear, solar, wind, hydro, geothermal and biomass. Sources differ in energy density, cost, emissions and availability.
Gasoline holds about 46 MJ/kg, a battery about 0.7 MJ/kg.
Electricity is a carrier, not a source.
Renewable energy
Energy from sources replenished naturally.
Renewables (solar, wind, hydro, tidal, geothermal) are replenished naturally. Solar panels convert about 20% of sunlight, wind turbines up to 59% (Betz limit) of wind's power.
A 1 m² panel at 1000 W/m² and 20% gives 200 W.
Intermittency needs storage or grids.
Conservation of energy in collisions
Kinetic energy is conserved only in elastic collisions.
Total energy is always conserved, but kinetic energy only in elastic collisions. Inelastic ones convert some into heat, sound or deformation.
Two equal clay balls collide at 2 m/s each and stick: all kinetic energy is lost.
Momentum is conserved in both types; kinetic energy isn't.
Virial theorem
For power-law potentials, average kinetic and potential energies are related.
2⟨K⟩ = n⟨U⟩For a bound system with potential U ∝ rⁿ, the long-time averages obey 2⟨K⟩ = n⟨U⟩. For gravity (n = −1) ⟨K⟩ = −½⟨U⟩, so E = −⟨K⟩.
A star that loses energy heats up: negative heat capacity.
It applies to time averages in stable systems.
Principle of virtual work
In equilibrium, virtual work of applied forces vanishes.
ΣF·δr = 0A system is in equilibrium if the virtual work of the applied forces is zero for any small virtual displacement compatible with constraints: ΣF·δr = 0.
A lever balances when F₁δ₁ = F₂δ₂.
Constraint forces drop out, which makes this powerful.
Work–energy with variable mass
Energy bookkeeping with changing mass needs care for internal energy.
With changing mass, you must define the system and include internal energy changes. Applying ½mv² naively to a leaking cart or rocket gives wrong results.
In a rocket the exhaust gains kinetic energy too.
Energy conservation can still be used if all bodies are tracked.
Hamiltonian as total energy
The Hamiltonian is the energy expressed in coordinates and momenta.
H = p²/2m + UThe Hamiltonian H(q, p) = p²/2m + U(q) is the total energy written in terms of position and momentum. Hamilton's equations q̇ = ∂H/∂p, ṗ = −∂H/∂q give the motion.
For a spring H = p²/2m + ½kx².
H equals energy only if the constraints are time independent.
Lagrangian mechanics intro
L is kinetic minus potential energy; motion extremises the action.
L = T − UThe Lagrangian L = T − U depends on coordinates and velocities. The equations of motion follow from the Euler–Lagrange equation, which works in any coordinates, ideal for constrained systems.
For a pendulum L = ½mL²θ̇² + mgL cosθ.
L is T − U, not T + U.
Euler–Lagrange equations
Equations of motion from L, valid in any coordinates.
d/dt(∂L/∂q̇) = ∂L/∂qd/dt(∂L/∂q̇) = ∂L/∂q gives one equation per coordinate. For the pendulum it reproduces θ̈ = −(g/L) sinθ.
Applied to a free particle it gives constant velocity.
Treat q and q̇ as independent when differentiating partially.
Principle of least action
Actual paths make the action stationary.
S = ∫L dtThe actual path of a system makes the action S = ∫L dt stationary compared with neighbouring paths. This unifies mechanics, optics (Fermat) and field theories.
A thrown ball follows the parabola that extremises S.
It is "stationary", not always a minimum.
Noether's theorem
Every continuous symmetry corresponds to a conserved quantity.
Every continuous symmetry of the action implies a conserved quantity: time translation → energy, space translation → momentum, rotation → angular momentum.
Since physics doesn't change with time, energy is conserved.
Conservation laws come from symmetries, not the other way round.
Frequently asked questions
Is work done when I hold a heavy object still?
In physics, no. Work requires a force to act through a displacement in the direction of the force. Holding something still involves no displacement, so the work done on it is zero, although your muscles are using energy.
What is the difference between energy and power?
Energy is the capacity to do work, measured in joules. Power is the rate at which work is done or energy is transferred, measured in watts, where one watt is one joule per second.
Why is energy conserved if friction seems to remove it?
Friction converts mechanical energy into internal (thermal) energy of the surfaces and air. Total energy stays the same, but the useful mechanical part decreases.
When should I use energy instead of forces?
Use energy when you know speeds and heights but not the time, or when a force changes during the motion. Energy avoids having to track every force at every moment.
Does energy conservation always hold?
Total energy is always conserved, but mechanical energy is not if friction or drag acts. The lost mechanical energy turns into heat and sound.
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