In classical mechanics, work quantifies energy transfer by a force acting through a displacement. The work–energy theorem connects net work directly to the change in kinetic energy.
Work
For a constant force:
W = \vec{F} \cdot \vec{d} = Fd\cos\theta$\theta$ is the angle between force and displacement. Only the parallel component of force contributes to work.
- $\theta = 0^\circ$: maximum positive work
- $\theta = 90^\circ$: zero work
- $\theta = 180^\circ$: negative work (force opposes motion)
SI unit: joule (1 J = 1 N·m).
Kinetic and Potential Energy
K = \tfrac{1}{2}mv^2 U_g = mgh \quad\text{(near Earth's surface)} U_s = \tfrac{1}{2}kx^2 \quad\text{(ideal spring)}Work–Energy Theorem
W_{\mathrm{net}} = \Delta K = K_f - K_iIf only conservative forces do work, mechanical energy $K + U$ is conserved. Nonconservative work (e.g. friction) appears as:
K_i + U_i + W_{\mathrm{nc}} = K_f + U_fPower
Average power is work per unit time: $P = W/t$. Instantaneously, $P = \vec \cdot \vec$. Unit: watt (W = J/s).
Worked Example
A 2.0 kg block starts from rest and is pushed 3.0 m by a 10 N force parallel to the displacement (no friction).
$W = Fd = 10 × 3.0 = 30$ J
W = \Delta K = \tfrac{1}{2}mv^2 \implies v = \sqrt{2W/m} = \sqrt{30} \approx 5.5 m/s