A.3 Work, energy and powerIB Physics SL: Revision notes

The principle of conservation of energy: energy cannot be created or destroyed, only transferred from one form to another. The total energy of an isolated system is constant.

Work done by a force is a transfer of energy. For a constant force F at angle θ to the displacement s:

$W = Fs\cos\theta$

Only the component of force along the displacement does work. A force perpendicular to the motion (such as the normal force on a level surface, or the centripetal force in circular motion) does no work. A force opposing the motion, such as friction, does negative work.

The work done by the resultant force on a body equals the change in its kinetic energy (and in general the change in the energy of the system).

A Sankey diagram represents energy transfers as arrows whose widths are proportional to the amount of energy. The input arrow splits into a useful output arrow and one or more wasted (dissipated) arrows. Because energy is conserved, the widths of the output arrows add up to the width of the input arrow. They make efficiency easy to see at a glance.

Mechanical energy is the sum of kinetic, gravitational potential and elastic potential energy:

• kinetic: $E_k = \frac{1}{2}mv^2 = \frac{p^2}{2m}$
• gravitational potential (near Earth's surface): $\Delta E_p = mg\Delta h$
• elastic potential: $E_H = \frac{1}{2}k(\Delta x)^2$

In the absence of friction and other resistive forces, total mechanical energy is conserved: work simply transfers energy between these forms (a pendulum, a spring launcher, a roller coaster). When resistive forces act, mechanical energy decreases and the loss equals the work done against them, usually becoming internal (thermal) energy.

Power is the rate of doing work, or rate of energy transfer:

$P = \frac{\Delta W}{\Delta t} = Fv$

The form $P = Fv$ applies to a force F acting along the direction of a velocity v — for example, the driving force of a vehicle moving at constant speed, where the driving force equals the total resistive force. Power is measured in watts (1 W = 1 J s⁻¹).

Efficiency compares useful output with total input:

$\eta = \frac{\text{useful work out}}{\text{total work in}} = \frac{\text{useful power out}}{\text{total power in}}$

It has no unit and is always less than 1 (100%) for real devices. For devices in series, the overall efficiency is the product of the individual efficiencies.

The energy density of a fuel is the energy released per unit mass (J kg⁻¹). Fossil fuels have high energy densities (about 45 MJ kg⁻¹ for petrol or diesel), far higher than batteries, which is one reason they remain common in transport. To find the mass of fuel needed: mass = energy required from fuel ÷ energy density, remembering to divide the useful energy by the efficiency first.