Work of a Force. Power

Work of a force measures the action of a force through a displacement, while power measures the rate at which work is done.

Elementary work

For an infinitesimal displacement $d\vec r$:

$$dA=\vec F\cdot d\vec r=F\,ds\cos\alpha,$$

where $\alpha$ is the angle between the force and displacement direction.

Work over a finite displacement

From point 1 to point 2:

$$A_{1\to2}=\int_1^2\vec F\cdot d\vec r.$$

For a constant force over a straight displacement $s$, $A=Fs\cos\alpha$.

Sign of work

Work is positive for an acute angle between force and displacement, negative for an obtuse angle, and zero when the force is perpendicular to the instantaneous displacement.

Work of gravity

Near Earth's surface, the work of gravity depends only on the change in height: $A_g=mg(h_1-h_2)$. It is positive for downward motion and negative for upward motion.

Work of a spring force

For a spring with $F_x=-kx$:

$$A_s=\frac{kx_1^2}{2}-\frac{kx_2^2}{2}.$$

Power

Instantaneous power is:

$$P=\frac{dA}{dt}=\vec F\cdot\vec v.$$

The SI unit is the watt: $1\,\text{W}=1\,\text{J}/\text{s}$.

Example

A constant 50 N force moves a point 3 m in the force direction. The work is 150 J. If this occurs over 5 s at a uniform average rate of doing work, the average power is 30 W.

Common mistakes

• using $Fs$ without accounting for the angle;
• confusing work and power;
• assuming work is always positive;
• using average power where instantaneous $\vec F\cdot\vec v$ is required.