Work, Energy & Power Notes

Last reviewed 16 Sept 2026 · 4 min read

Work

Work is done when a force moves its point of application in the direction of the force.

W = F s cos θ

$θ$ = angle between force and displacement. Unit: joule (1 J = 1 N·m).

$θ = 0°$ : maximum positive work. $θ = 90°$ : no work (e.g. normal reaction on a sliding block, centripetal force in uniform circular motion). $θ = 180°$ : negative work (friction, braking).
Variable force: $W = \int F d x$ = area under the force–displacement graph.
Spring (stiffness $k$ ) stretched from $x_{1}$ to $x_{2}$ : $W = \frac{1}{2} k (x_{2}^{2} - x_{1}^{2})$ .
Weight: work done against gravity in raising $m$ by $h$ is $m g h$ , independent of path.
Torque rotating through $θ$ : $W = T θ$ .

Energy is the capacity to do work.

FormulaMechanical energy

Kinetic energy (translation): $K E = \frac{1}{2} m v^{2}$
Kinetic energy (rotation about a fixed axis): $K E = \frac{1}{2} I ω^{2}$
Rolling body (no slip, $v = r ω$ ): $K E = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} = \frac{1}{2} m v^{2} (1 + \frac{k ^{2}}{r ^{2}})$
Gravitational potential energy: $P E = m g h$
Strain energy of a spring: $U = \frac{1}{2} k x^{2}$

FormulaWork–energy principle

The net work done by all forces on a particle equals the change in its kinetic energy:

\sum W = K E_{2} - K E_{1} = \frac{1}{2} m (v_{2}^{2} - v_{1}^{2})

It avoids finding acceleration and time — ideal when forces, distances and speeds are involved (stopping distances, spring buffers, pile driving).

If only conservative forces (gravity, spring force) do work, total mechanical energy stays constant:

K E_{1} + P E_{1} = K E_{2} + P E_{2}

Friction is non-conservative; with friction, $K E_{1} + P E_{1} = K E_{2} + P E_{2} + work against friction$ .

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