Work and Power

Definition: Work is the energy transferred when a force moves an object over a distance, and power is the rate at which that work is done.

How It Works

Work equals force times displacement in the direction of the force, measured in joules (J); no work is done if there is no displacement, no matter how large the force.
Only the component of force parallel to the displacement contributes to work; a force perpendicular to motion (like gravity on a person walking on flat ground) does zero work.
Work can be positive (force and displacement point the same way, energy is added to the object) or negative (force opposes displacement, energy is removed, as with friction slowing a sliding block).
The work-energy theorem connects work directly to kinetic energy: the net work done on an object equals its change in kinetic energy.
Power equals work divided by time, measured in watts (W); the same task done faster requires more power even though total work is unchanged.
One watt equals one joule of energy transferred per second; larger units like the kilowatt (1000 W) and horsepower (about 746 W) are common in engineering.
Power can also be expressed as force times velocity, useful for continuously varying situations like a car accelerating or a motor under load.
Efficiency compares useful work output to total energy input, since real machines always lose some energy to friction, heat, or other forms.

Illustration

Under the Hood

W = F·d·cos(θ)                    (work, θ = angle between force and displacement)

P = W / t                         (average power)

P = F·v                           (instantaneous power)

W_net = ΔKE = ½mv_f² - ½mv_i²     (work-energy theorem)

Efficiency = (useful output / total input) × 100%

Worked Example 1: Work done lifting a load Given: a 20 kg box is lifted straight up 1.5 m. Step 1: find the force needed, F = mg = 20 × 9.8 = 196 N. Step 2: apply W = F·d·cos(θ), with θ = 0° (force and displacement both upward), cos(0°) = 1. Step 3: W = 196 × 1.5. Answer: W = 294 J.

Worked Example 2: Power required for a task Given: the same 294 J of work is done in 3.0 seconds. Step 1: apply P = W/t = 294 / 3.0. Answer: P = 98 W.

Worked Example 3: Work at an angle Given: a 100 N force pulls a sled 10 m across the ground, applied at 30° above horizontal. Step 1: apply W = F·d·cos(θ) = 100 × 10 × cos(30°). Step 2: cos(30°) ≈ 0.866. Answer: W ≈ 866 J.

Worked Example 4: Power from force and velocity Given: a car engine delivers 3000 N of driving force while moving at 25 m/s. Step 1: apply P = F·v = 3000 × 25. Answer: P = 75,000 W = 75 kW, roughly 100 horsepower.

Worked Example 5: Efficiency of a motor Given: an electric motor consumes 500 W of electrical power and produces 425 W of useful mechanical output. Step 1: apply Efficiency = (useful output / total input) × 100%. Step 2: Efficiency = (425 / 500) × 100%. Answer: Efficiency = 85%, with the remaining 15% lost mostly as heat.

Work-Energy Theorem in Practice

If the net work on an object is positive, its kinetic energy increases, it speeds up.
If the net work is negative (like braking friction), kinetic energy decreases, it slows down.
If net work is zero, kinetic energy is unchanged, even if individual forces are doing positive and negative work that cancel out.
This theorem lets you solve many problems without ever calculating acceleration or time directly, purely from force, distance, and the resulting speed change.

Why It Matters

These concepts let engineers size motors and engines correctly, since a machine must supply enough power, not just energy, to perform a task within a useful time.
Electric utility billing is based on energy (kilowatt-hours), while equipment ratings (like a hair dryer’s 1500 W) describe power, the rate that energy is used.
Vehicle performance specs (horsepower, torque) directly describe how quickly a car can convert fuel energy into kinetic energy.
Understanding efficiency helps engineers minimize wasted energy in engines, motors, and power transmission systems.
Human metabolism and exercise science measure power output (watts) to quantify athletic performance, such as a cyclist’s sustained wattage.
Elevator and crane designers must calculate both work (total energy to lift a load) and power (motor rating needed to do it in an acceptable time).
Renewable energy systems like wind turbines and solar panels are rated by peak power output, while their practical value depends on total energy produced over time.

Common Pitfalls

Assuming a bigger force always means more work. Work requires displacement in the force’s direction; holding a heavy weight stationary does zero physics work, however tiring it feels.
Forgetting the cos(θ) term when force and displacement aren’t parallel, leading to overestimated work calculations.
Confusing work (energy transferred, joules) with power (rate of transfer, watts); they’re related but not interchangeable.
Believing more power always means more total energy used. A high-power device used briefly can use less total energy than a low-power device used for a long time.
Mixing up horsepower and watts without converting; 1 horsepower ≈ 746 watts, a common source of unit errors in engineering problems.
Ignoring negative work from friction or air resistance when applying the work-energy theorem, leading to energy “appearing from nowhere” in a flawed calculation.

Comparison

	Work	Power	Energy
Definition	Force × displacement	Rate of doing work	Capacity to do work
SI unit	Joule (J)	Watt (W)	Joule (J)
Depends on time?	No	Yes	No
Formula	W = Fd·cos(θ)	P = W/t	Various forms (KE, PE, etc.)
Everyday billing unit	Rarely billed directly	Device rating (e.g., 1500 W heater)	Kilowatt-hour (kWh) on utility bills

Example

Carrying a box up a flight of stairs takes the same total work whether you walk or run, since the force and vertical displacement are identical either way, but running requires more power because that same work is done in less time. A 100 W light bulb and a 100 W space heater consume energy at the same rate, but a 2000 W space heater running for 5 minutes uses far more total energy than a 100 W bulb running for an hour, since energy depends on both power and time.

History

The formal concept of “work” in physics emerged in the early 19th century, with Gaspard-Gustave de Coriolis credited with the modern definition (force times distance) around 1826.
James Watt introduced the horsepower unit in the late 18th century to market his steam engines by comparing their output to the working capacity of a horse.
The joule, the SI unit of energy and work, was named in honor of James Joule, whose experiments in the 1840s established the mechanical equivalent of heat.
The watt, the SI unit of power, was adopted in 1889 and named after James Watt, cementing his connection to the concept he helped popularize commercially.

FAQ

Why does holding a heavy object still count as zero work, even though it’s tiring? Physics work strictly requires displacement in the direction of the applied force; your muscles do expend biological energy fighting fatigue and maintaining tension, but no mechanical work is done on the object itself since it doesn’t move.

Is horsepower an outdated unit that’s no longer used? It’s still widely used in the automotive and engine industries, even though the SI unit is the watt; 1 horsepower equals approximately 746 watts, and converting between the two is common in engineering practice.

Why do electric bills use kilowatt-hours instead of joules? A kilowatt-hour is simply a more convenient, human-scaled unit of energy for household consumption (3.6 million joules), avoiding the need to work with the very large raw joule numbers involved.

Does more power always mean a machine is “better”? Not necessarily. Power measures speed of energy delivery, but efficiency, reliability, and how well power output matches the actual task requirement often matter more for real-world performance.

Why can a small electric motor lift a heavy load slowly but not quickly? Its power rating fixes the maximum rate at which it can do work; it can still generate enough force to lift a heavy load, but only by trading off speed, since P = Fv limits how fast it can move that load.

Referenced by

Dig deeper

Spot an error in this note?