1.3 Work, energy, and conservation

Newton’s second law is a statement about instantaneous force and acceleration. Integrating it over a path rather than over time produces the second great bookkeeping tool of mechanics — energy — and the conservation law that makes so many problems solvable without ever tracking the force in detail.

The work–energy theorem

Define the work done by a force $\mathbf F$ along a path as $W = \int \mathbf F\cdot d\mathbf r$ . Newton’s second law then forces a direct relation between the work and the kinetic energy.

▶ W = ΔT, with T = ½m|v|² Derivation

Start from $\mathbf F = m\,d\mathbf v/dt$ and take the dot product with the velocity $\mathbf v = d\mathbf r/dt$ :

\mathbf F\cdot\mathbf v \;=\; m\,\mathbf v\cdot\frac{d\mathbf v}{dt} \;=\; \frac{d}{dt}\!\left(\tfrac12 m|\mathbf v|^2\right).

Integrate in time from $t_1$ to $t_2$ , and use $\mathbf v\,dt = d\mathbf r$ to turn the left side into a path integral:

\int \mathbf F\cdot d\mathbf r \;=\; \tfrac12 m|\mathbf v_2|^2 - \tfrac12 m|\mathbf v_1|^2.

The left side is the work $W$ ; the right side is the change in the kinetic energy $T = \tfrac12 m|\mathbf v|^2$ . So $W = \Delta T$ . ✓

Conservative forces and potential energy

A force is conservative if the work it does around any closed path is zero — equivalently, if it is the gradient of a scalar, $\mathbf F = -\nabla U$ , for some potential energy $U$ . For such a force the work between two points depends only on the endpoints, $W = U(A) - U(B)$ , and combining this with the work–energy theorem gives the conservation of mechanical energy,

T + U \;=\; \text{constant along the trajectory}.

Gravity, the electrostatic force, and the linear restoring force of a spring are conservative; friction and viscous drag are not, because the work they do depends on the length of the path travelled, draining mechanical energy into heat.

W along path-0.000

U(A) − U(B)0.000

path-independent?yes — conservative

Force:

Path A → B:

A conservative force has a potential U(r); the work done from A to B depends only on the endpoints, never the path. A non-conservative force like drag opposes the direction of motion regardless of path — its line integral grows with the path's *length*, so the arc costs more work than the direct path.

Toggle between a conservative radial spring force and a non-conservative drag, and toggle the path between a straight line and an arc. The conservative work is the same for both paths — it is $U(A) - U(B)$ regardless of route. The drag work grows with path length: the long way costs more. That path-independence is what “conservative” means operationally, and it is what makes potential energy a well-defined function of position at all.

Power

The rate of doing work is the power $P = \mathbf F\cdot\mathbf v = dW/dt$ . For a conservative system it is the rate at which energy moves between the kinetic and potential ledgers; for a dissipative one it is the rate at which mechanical energy is lost. Power is the natural quantity whenever the question is not “how much energy” but “how fast” — the output of an engine, the dissipation of a damped oscillator, the flux of energy carried by a wave.