Work (Physics)
In physics, work is done on an object when a force causes a displacement of that object in the direction of the force. Work is a scalar quantity equal to the product of the force, the displacement, and the cosine of the angle between them. It represents the transfer of mechanical energy to or from an object and is measured in joules (J); no work is done if the force is perpendicular to the motion or if there is no displacement.
Key Formula
W = F × d × cos(θ)
LaTeX: W = F d \cos\theta
| Symbol | Meaning | Unit |
|---|---|---|
| W | Work done | J (Joule) |
| F | Magnitude of the applied force | N (Newton) |
| d | Displacement of the object | m |
| \theta | Angle between force and displacement vectors | degrees or radians |
Worked Example
Problem
A person pushes a box with a force of 50 N at an angle of 30° below the horizontal. The box moves 4 m along the floor. Calculate the work done by the person.
Solution
Step 1 — Identify values: F = 50 N, d = 4 m, θ = 30°. Step 2 — Apply the formula: W = F × d × cos θ = 50 × 4 × cos 30° = 200 × 0.866 = 173.2 J.
Answer
Work done by the person = 173.2 J.
Work done for a 50 N force over 4 m at various angles
| Angle θ (°) | cos θ | Work W (J) | Interpretation |
|---|---|---|---|
| 0 | 1.000 | 200.0 | Force fully along displacement — maximum work |
| 30 | 0.866 | 173.2 | Partial component along motion |
| 60 | 0.500 | 100.0 | Half component along motion |
| 90 | 0.000 | 0.0 | Force perpendicular — zero work done |
| 120 | −0.500 | −100.0 | Negative work — force opposes motion |
| 180 | −1.000 | −200.0 | Force opposite to motion — maximum negative work |
Interactive Tools
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Related Terms
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its state of motion. It depends on both the mass of the object and the square of its speed, meaning that doubling the speed quadruples the kinetic energy. Kinetic energy is transferred to objects through work and is a key quantity in collision analysis, transport safety, and the work-energy theorem.
Power (Physics)
Power is the rate at which work is done or energy is transferred per unit time. It quantifies how quickly a system can perform work, making it essential for comparing engines, motors, and other energy-conversion devices. In practical applications, power determines the performance capacity of machines and biological systems alike.
The scientific use of "work" to mean force times displacement was introduced by French mathematician Gaspard-Gustave de Coriolis in 1829, in his paper "Du Calcul de l'effet des machines". The word "work" itself comes from Old English "weorc" (deed, action), cognate with Greek "ergon" (work) — which also gives the unit "erg".