Work is not effort. In physics, work has a precise definition: it is the amount of energy transferred into or out of a system by a force acting over a displacement. You can push against a wall all day and do zero work — because the wall doesn't move.
Three things are required for work to be done: a force, a displacement, and the force must have a component along the displacement. All three conditions must be met. Missing any one of them means zero work.
Work is measured in Joules (J) — the same unit as energy, because work is energy transfer. Work can be positive (energy added to the system), negative (energy removed), or zero (no transfer).
The work done by a force is the product of the component of force parallel to the displacement and the magnitude of the displacement:
When a force is applied at an angle θ to the direction of motion, the parallel component is F cos θ. So the formula becomes:
Where θ is the angle between the force vector and the displacement vector. This is the form you'll see on the AP equation sheet — but understanding it as "only the parallel component" is more useful than memorizing the cosine.
Set the force, displacement, and angle. Watch the parallel component (the only part that does work) change as the angle increases toward 90°.
Drag angle to 90° — work drops to zero. The force is still there, but none of it is aligned with motion. No energy transferred.
A person pushes a 20 kg box across the floor with a force of 80 N at 35° below horizontal. The box moves 6 m horizontally. How much work does the person do?
When a force acts exactly perpendicular to the displacement — θ = 90° — cos 90° = 0, so W = 0. The force transfers no energy. This is one of the most important results in the unit.
The work-energy theorem states that the net work done on an object equals its change in kinetic energy:
This is one of the most powerful results in classical mechanics. It connects what forces do (work) directly to what motion becomes (kinetic energy), bypassing the need to track force and acceleration at every instant.
Set an object's initial state and the net work done on it. Watch the work-energy theorem connect work directly to the change in kinetic energy.
Try setting W_net negative — it removes energy. At some point KE hits zero and the object stops. Kinetic energy can't go below zero.
A 3 kg object starts at rest. A net force does 96 J of work on it. What is its final speed?
Not all forces behave the same way when an object moves along a path. This distinction matters enormously for how we account for energy.
The work done depends only on the starting and ending positions — not on the path taken between them.
The work done depends on the path taken. A longer or rougher path means more energy lost.
When the force is constant, W = F‖ · d is straightforward. But when force varies with displacement — like a spring — you need a different approach. On a force vs. displacement graph, the total work done is the area under the curve.
A force vs. displacement graph shows a straight line from (0, 0) to (4 m, 24 N). What is the total work done over this displacement?