Work, Energy, and Power

Notes

Work as an integral

• $\displaystyle W = \int \vec F \cdot d\vec r$. In one dimension, $\displaystyle W = \int_{x_1}^{x_2} F(x)\,dx$, the area under the $F$ vs $x$ graph.
• Dot product: $\vec A \cdot \vec B = A_xB_x + A_yB_y = AB\cos\theta$.
• Work-energy theorem: $W_{net} = \Delta K$.
• Spring: $\displaystyle W_{spring} = -\frac{1}{2}kx^2$ from equilibrium, so $\displaystyle U_s = \frac{1}{2}kx^2$.

Potential energy and force

• For a conservative force, $\displaystyle \Delta U = -\int F\,dx$ and $\displaystyle F = -\frac{dU}{dx}$.
• On a potential energy curve $U(x)$, the force points "downhill." Stable equilibrium is at a minimum; unstable at a maximum. The particle can't go where $U > E$ (turning points).
• Gravitational potential energy (anywhere, not just near Earth): $\displaystyle U_G = -\frac{Gm_1m_2}{r}$.
• Escape speed: $\displaystyle v_{esc} = \sqrt{\frac{2GM}{R}}$.

Conservation of energy

$K_i + U_i + W_{nc} = K_f + U_f$, where $W_{nc}$ is work by nonconservative forces like friction.

Power

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