Work, Energy and Power — Notes

Work is done when a force moves its point of application: $W = F s \cos\theta$ ($\theta$ = angle between force and displacement). Scalar; SI unit joule.

• $\theta = 0^\circ$: $W = Fs$. $\theta = 90^\circ$: $W = 0$ (load carried horizontally; a satellite in circular orbit). $\theta = 180^\circ$: $W = -Fs$ (friction, braking).
• Work against gravity in lifting mass $m$ through height $h$: $W = mgh$.

Energy: the capacity to do work; same units as work.

Power = rate of doing work: $P = \dfrac{W}{t}$. SI unit watt (1 J s⁻¹); kW = $10^{3}$ W, MW = $10^{6}$ W, GW = $10^{9}$ W; 1 hp = 746 W.

Energy (how much work) differs from power (how fast): equal-weight students climbing the same stairs gain the same energy; the faster one has more power.

Gravitational potential energy: work done to raise $m$ through $h$ against $mg$: $U = mgh$.

Kinetic energy: a force $F$ accelerates $m$ from rest; $v^{2} = 2as$, so $W = Fs = ma \cdot \dfrac{v^{2}}{2a} = \tfrac{1}{2}mv^{2}$: $K = \tfrac{1}{2}mv^{2}$.

Kinetic energy may be translational, rotational or vibrational (only translational is examined numerically). Doubling $v$ quadruples $K$.

Forms: mechanical, heat, light, sound, chemical, electrical, magnetic, nuclear.

Principle: energy can neither be created nor destroyed; it changes form, and the total energy of an isolated system is constant.

Free fall from height $H$ (no air resistance): at any height $h$, $U = mgh$ and $K = mg(H - h)$, so $U + K = mgH$. Speed at the ground: $v = \sqrt{2gH}$.

Simple pendulum: all PE at the extremes, all KE at the mean position; total constant (air resistance slowly turns it into heat).