Work, Energy and Power

Work and its sign:

• $W = Fs\cos\theta$, where $\theta$ is the angle between the force and the displacement. Unit: the joule (1 J = 1 N m).
• Positive work: force along the motion ($\theta < 90°$), the body speeds up.
• Zero work: force perpendicular to the motion ($\theta = 90°$), or no displacement at all.
• Negative work: force against the motion ($\theta > 90°$), as friction does.
• In circular motion the string's pull points to the centre, at right angles to the motion all the way round, so it does no work.

The two forms and how they link:

• Kinetic energy $K = \tfrac{1}{2}mv^{2}$. In terms of momentum $p = mv$: $K = \dfrac{p^{2}}{2m}$, and $p = \sqrt{2mK}$.
• So at fixed mass, $K$ goes as $v^{2}$ and as $p^{2}$.
• Potential energy near the ground $U = mgh$: it depends on the height above the chosen level.
• Conservation: with no friction, $K + U$ stays constant. A body falling from height $h$ reaches $v = \sqrt{2gh}$.
• Pendulum: potential energy is least at the mean (lowest) position and greatest at the two ends; kinetic energy is the other way round.
• Work-energy theorem: the net work done on a body equals its change in kinetic energy.

Power and its units:

• $P = \dfrac{W}{t}$. Unit: the watt (1 W = 1 J/s); 1 kW = 1000 W; 1 horsepower is about 746 W.
• Lifting: the least power to raise mass $m$ through height $h$ in time $t$ is $P = \dfrac{mgh}{t}$.
• At the same work, power and time are inversely related: half the time means twice the power.
• The kilowatt hour (kWh) is a unit of energy, not power: 1 kWh = $3.6 \times 10^{6}$ J.
• If the time is given in minutes, either convert it to seconds or keep the answer in joules per minute: compare like with like.