Work, Energy and Simple Machines — Notes

Work is done when a force moves a body. For a force $F$ along the displacement $s$, $W = Fs$; in general $W = Fs\cos\theta$, where $\theta$ is the angle between the force and the displacement. The unit is the joule (J), $1\ \text{J} = 1\ \text{N m}$. Work is a scalar.

• $\theta = 0^\circ$: positive work (the force helps the motion).
• $\theta = 90^\circ$: zero work (for example the lift on a trunk carried along a level road).
• $\theta = 180^\circ$: negative work (friction or brakes).
• No displacement means no work. On a force–displacement graph the area under the line is the work done.

Energy is the capacity to do work, measured in joules. Kinetic energy is the energy of a moving body. From $v^2 = u^2 + 2as$ with $u = 0$ and $F = ma$, the work done in giving a body speed $v$ is $W = Fs = ma \cdot v^2/2a = \tfrac{1}{2}mv^2$, so $\text{KE} = \tfrac{1}{2}mv^2$ Doubling the speed makes the KE four times as large; doubling the mass doubles it.

Gravitational potential energy is the energy a body has because of its height. Lifting a body of mass $m$ at steady speed needs a force $mg$, so the work done in raising it by $h$ is $W = mgh$, stored as $\text{PE} = mgh$ (with $g = 9.8\ \text{m s}^{-2}$, measured from a chosen reference level).

Work–energy theorem: the net work done on a body equals the change in its kinetic energy, $W_\text{net} = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mu^2$. Brakes do negative work and reduce the KE to zero.

Energy can neither be created nor destroyed; it changes from one form to another. Mechanical energy = KE + PE. For a freely falling body, a frictionless slide, a roller-coaster or an ideal pendulum, the mechanical energy stays constant: as PE falls, KE rises by the same amount. From rest at height $H$, the speed at height $h$ is $v = \sqrt{2g(H - h)}$, which does not depend on the mass.

Examples of transformations: a fan (electrical to kinetic), a solar cell (light to electrical), a dam (potential to kinetic to electrical), burning gas (chemical to heat and light). In real systems some energy becomes heat.

Power is the rate of doing work: $P = W/t$. The SI unit is the watt (W), $1\ \text{W} = 1\ \text{J s}^{-1}$; $1\ \text{kW} = 1000\ \text{W}$. The commercial unit of electrical energy is the kilowatt hour (kWh), also called a unit: $1\ \text{kWh} = 3.6 \times 10^6$ J.

Energy in kWh = power in kW × time in hours. A 1.5 kW geyser used for 20 minutes uses $1.5 \times \tfrac{20}{60} = 0.50$ kWh.

A simple machine changes the size or direction of a force. $\text{MA} = \text{load}/\text{effort}$ (no unit). A machine never gives more work than it receives.

• Lever: $\text{MA} = \text{effort arm}/\text{load arm}$. Class I: fulcrum in the middle (seesaw). Class II: load in the middle (wheelbarrow), MA > 1. Class III: effort in the middle (tongs), MA < 1.
• Pulleys: single fixed MA = 1 (changes direction only); single movable MA = 2; for an ideal block and tackle MA = number of strands supporting the load.
• Efficiency $\eta = \dfrac{\text{work output}}{\text{work input}} \times 100\,\%$, and $\eta = \text{MA}/\text{VR} \times 100\,\%$, where VR is the distance moved by the effort divided by that moved by the load. Real machines have $\eta < 100\,\%$.