Work done by a force

The angle between two directions is taken from $0^\circ$ to $180^\circ$. One joule is the work done by a 1 N force acting through 1 m in its direction: $1\text{ J}=1\text{ N m}$.

• A force in the direction of motion has $\theta=0^\circ$, so $W=Fd$.
• A force perpendicular to the motion has $\theta=90^\circ$, so it does no work.
• A force directly opposite the motion has $\theta=180^\circ$, so its work is negative.

The sign tells you the direction of energy transfer. A positive work term transfers energy into the chosen body or system; a negative term transfers energy out.

• “Work done by a 12 N resistance over 4 m” is $-48$ J.
• “Work done against that resistance” is the positive magnitude, $48$ J.
• A person holding a stationary load may feel tired, but the force does zero mechanical work on the load because its displacement is zero.

1. Mark the displacement direction and distance.
2. List only the real forces acting during that displacement.
3. Find the angle between each force and the displacement.
4. Calculate each signed work term, then add if total work is required.

On a body moving along a fixed smooth surface, the normal reaction is perpendicular to each small displacement. Its work is therefore zero.

The total work is the signed sum of the work done by the real forces.