Work and the work-energy theorem

Work as energy transfer

In mechanics, work is energy transferred by a force acting through a displacement. For a constant force, work is the dot product of force and displacement:

W=ec{F}cdotDeltaec{r}

Equivalently,

$W=FDelta rcos heta$

where $heta$ is the angle between force and displacement. Only the component of force along the displacement does work.

If force and displacement point in the same direction, work is positive. If they point in opposite directions, work is negative. If they are perpendicular, work is zero.

Work by a variable force

Many forces are not constant. A spring force changes with displacement, and gravitational force changes with distance on large scales. For a variable force in one dimension, work is

$W=int_{x_i}^{x_f} F(x),dx$

Graphically, this is the signed area under the force-position graph.

In three dimensions, work along a path is written as a line integral:

W=int_C ec{F}cdot dec{r}

This expression adds the tiny amount of work done along each small displacement of the path.

Kinetic energy

Kinetic energy is energy of motion:

K=rac{1}{2}mv^2

It is a scalar, not a vector. Although velocity has direction, kinetic energy depends only on speed. A car moving north at $20 m/s$ and the same car moving south at $20 m/s$ have the same kinetic energy.

Deriving the work-energy theorem

For one-dimensional motion with constant mass, Newton's second law gives

$F_{net}=ma$

Using the chain rule,

a=rac{dv}{dt}=rac{dv}{dx}rac{dx}{dt}=vrac{dv}{dx}

Thus

F_{net}=mvrac{dv}{dx}

Multiply both sides by $dx$ and integrate:

$int_{x_i}^{x_f} F_{net},dx=int_{v_i}^{v_f} mv,dv$

This gives

W_{net}=rac{1}{2}mv_f^2-rac{1}{2}mv_i^2

So

$W_{net}=Delta K$

This is the work-energy theorem.

Meaning of the theorem

The theorem says net work changes kinetic energy. Positive net work increases speed. Negative net work decreases speed. Zero net work means kinetic energy remains unchanged, though direction may change.

For example, centripetal force in uniform circular motion is perpendicular to displacement at every instant, so it does no work and does not change speed.

Work by friction

Kinetic friction usually does negative work:

$W_f=-f_k d$

This reduces mechanical kinetic energy and transforms it into thermal energy. The work-energy theorem still holds if all forces are included, but mechanical energy may not be conserved.

Why energy methods are useful

Energy methods can solve problems without tracking acceleration at every moment. If you know the work done over a displacement, you can find the change in speed. This is especially helpful for variable forces, curved paths, and systems where forces change but energy accounting remains simple.

The big idea

Work is the mechanical transfer of energy by force through displacement. The net work done on an object equals its change in kinetic energy. This theorem connects force-based dynamics with energy methods and provides a powerful way to solve motion problems when displacement and speed are more important than time.