Work, Energy, and the Work-Energy Theorem

Work & Constant Forces

Energy bookkeeping beats force analysis: you can predict a rollercoaster's final speed without knowing the track's twists, because gravity's work only cares about height.

Work is the mechanism by which forces transfer energy to or from an object. For a constant force, Work done by a constant force is defined as:

$$W = \vec{F}\cdot\vec{d} = Fd\cos\theta$$

where $\theta$ is the angle between $\vec{F}$ and displacement $\vec{d}$. Work is measured in joules ($J = N\,m$).

Common pitfall: A force does zero work when it is perpendicular to the motion ($\theta = 90^\circ$), no matter how large it is. String tension in circular motion and normal force transfer zero energy.

The Work-Energy Theorem

The Work-Energy Theorem states that net work equals the change in kinetic energy:

$$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

For a variable force, Work done by a variable force requires integration:

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

Power is the rate of doing work:

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

Physics link: The work-energy theorem is not a new law; it is derived directly from Newton's second law via integration, reframing $F=ma$ into energy terms.