Formula and Method for Work and Power
In physics, work is the energy transferred to an object when a force moves it through a distance, and power is how quickly that work gets done. This calculator finds the work done by a force acting at any angle to the direction of motion, then divides by the elapsed time to report average power in both watts and horsepower.
Work, power, and the underlying formulas
- Work: W = F · d · cos(θ), where F is the force in newtons, d is the distance moved in meters, and θ is the angle between the force and the direction of motion. The result is in joules (1 J = 1 N·m = 1 kg·m²/s²).
- Angle dependence: when the force acts entirely along the direction of motion, θ = 0° and cos(θ) = 1, so the full force contributes to the work. At θ = 90° the force is perpendicular to the motion and does zero work — for example, gravity does no work on an object sliding across a level floor.
- Power: P = W / t = F · v · cos(θ), where t is the elapsed time in seconds and v = d/t is the average velocity. Power is measured in watts (1 W = 1 J/s), and 1 mechanical horsepower ≈ 745.7 W.
Common mistakes and practical notes
- Ignoring the angle: only the force component along the displacement direction does work — a force applied at 90° to the motion (like carrying a box at constant height) contributes zero work, no matter how large the force is.
- Confusing work with power: work (joules) is a fixed amount of energy transferred; power (watts) is how fast that transfer happens. A small motor and a large motor can do the same total work, but the large motor typically does it in less time, so it has higher power.
- Negative work: when θ is greater than 90°, cos(θ) is negative, meaning the force removes energy from the object rather than adding it — this is the case for friction, air resistance, or braking forces.
- Horsepower conventions vary: mechanical/imperial horsepower is 745.7 W, metric horsepower (PS) is about 735.5 W, and electrical horsepower is defined as exactly 746 W. This calculator uses the mechanical (745.7 W) convention.
Frequently Asked Questions
What is the formula for calculating work in physics?
Work equals force times distance times the cosine of the angle between them: W = F·d·cos(θ), measured in joules (1 J = 1 N·m). Only the component of the force that acts along the direction of motion contributes to the work done.
How do you calculate power from work and time?
Power is the rate at which work is done: P = W / t, measured in watts (1 W = 1 J/s). Equivalently, average power equals force times velocity along the direction of motion: P = F·v·cos(θ).
What is the difference between work and power?
Work (in joules) is the total energy transferred by a force over a distance, while power (in watts) is how quickly that work is performed. Two machines can perform the same amount of work, but the one that finishes faster has a higher power output.
How many watts are in one horsepower?
One mechanical (imperial) horsepower equals about 745.7 watts. This calculator uses that convention; metric horsepower (PS) is slightly smaller, at about 735.5 watts.