Work Formula
What is the Work Formula?
In physics, work is done when a force causes an object to move — but only the component of the force in the direction of motion counts. If force and displacement are perfectly aligned (θ = 0°), cos(0°) = 1, and the formula simplifies to the familiar W = Fd. This is a stricter, more specific definition than the everyday sense of "work" — physically holding a heavy bag motionless for an hour is exhausting, but it does zero work in the physics sense, since nothing actually moves.
If a force acts but causes no displacement at all — like pushing against an immovable wall — the work done is exactly zero, no matter how hard you push, because d = 0. Similarly, if a force acts entirely perpendicular to the direction of motion (θ = 90°), cos(90°) = 0, so that force does no work either — this is why gravity does no work on a satellite in a perfectly circular orbit, since the gravitational pull is always perpendicular to the direction the satellite is moving.
Work is directly tied to energy: doing positive work on an object increases its kinetic energy, doing negative work (like friction) removes it — this relationship, the work-energy theorem, is one of the most useful shortcuts in mechanics for solving problems that would otherwise require tracking forces at every instant.
What Each Variable Means
When to Use It
- Calculating the energy transferred by a force over a distance
- Analyzing situations where force and motion aren't perfectly aligned
- As part of the work-energy theorem relating work to change in kinetic energy
Step-by-Step Example
Problem: A force of 50 N pushes a box horizontally over 10 m. What is the work done?
θ = 0°, so cos(0°) = 1.
cos(0°) = 1Multiply force, distance, and cos(θ).
W = 50 × 10 × 1Interactive Calculator
Common Mistakes
Mistake: Forgetting the cos(θ) term when force and motion aren't parallel.
Fix: If the force acts at an angle to the direction of motion, only the cos(θ) component contributes — leaving it out overestimates the work done.
Mistake: Assuming pushing hard always means doing work.
Fix: If there's no displacement (d = 0), the work done is zero regardless of how much force is applied — work requires actual motion in the force's direction.
Practice Questions
A force of 20 N moves an object 5 m in the same direction as the force. Find the work done.
A 30 N force acts at 60° to a 4 m displacement. Find the work done.
Hint: cos(60°) = 0.5.
Frequently Asked Questions
Why is work zero when pushing against a wall?
Because the wall doesn't move — d = 0, and multiplying anything by zero gives zero, regardless of how much force is applied.
Can work be negative?
Yes — if the force has a component opposite to the direction of motion (θ between 90° and 180°), cos(θ) is negative, meaning the force removes energy rather than adding it (like friction slowing something down).