Work, energy and power: study notes

1. Work done by a constant force

Work is the energy transferred by a force. For a constant force $F$ whose line of action makes an angle $\theta$ with the displacement $s$, $$W = F s \cos\theta$$ (in the booklet). Work is a scalar measured in joules ($1\ \text{J} = 1\ \text{N m}$). Only the component of the force along the displacement does work. Three cases to recognise instantly: $\theta = 0^\circ$ gives $W = Fs$ (force pulls along the motion, positive work); $\theta = 90^\circ$ gives $W = 0$ (a normal reaction on a fixed surface, the tension in a string swinging a bob in a circle, or any force perpendicular to the motion); $\theta = 180^\circ$ gives $W = -Fs$ (friction or air resistance opposing the motion). Sign convention (learn it, not in the booklet): work done by a force is positive when the force has a component along the displacement; when we say work is done against a force such as friction, we quote the magnitude $Fs$ and describe it as energy removed from the mechanical store. State which force you are calculating work for and use the angle between the force and the displacement, not the angle a slope makes with the ground unless that is the same thing.

2. Work from a force-displacement graph and variable forces

When the force changes with position, $W = Fs\cos\theta$ cannot be used directly. Instead the work done equals the area between the force-displacement graph and the displacement axis (learn it, not in the booklet). For a rectangle (constant force) the area is $F \times s$; for a triangle (force rising linearly from zero, as for a spring) the area is $\frac{1}{2} F_{\text{max}} \Delta x$; for a curve, count squares or split the shape into trapezia. Areas below the axis count as negative work. The spring is the standard example: the force needed to stretch an ideal spring is $F = k\Delta x$, so the graph is a straight line through the origin with gradient $k$, and the area up to extension $\Delta x$ is $\frac{1}{2} k \Delta x^2$, which is exactly the elastic potential energy stored. In a Paper 1 item the trap is to read the force at the end of the motion and multiply by the whole distance; always ask whether the force was constant over the interval.

3. The energy principle: forms, transfers and conservation

Energy is stored in named forms: kinetic (a moving mass), gravitational potential (a mass raised in a field), elastic potential (a deformed spring or material), thermal (random motion of particles, raised temperature), chemical (fuel, food, batteries), nuclear (binding of nucleons) and electrical (charges moved through a potential difference). A transfer moves energy between objects; a transformation changes its form. The principle of conservation of energy (learn the statement) says that in a closed system the total energy is constant: energy is neither created nor destroyed, only transferred or transformed. The work-energy theorem (learn it) is the mechanical version: the net work done on a body by all forces equals its change in kinetic energy, $$W_{\text{net}} = \Delta E_k = \frac{1}{2} m v^2 - \frac{1}{2} m u^2 .$$ Two equivalent ways of attacking a problem follow. Either add up the work done by every force (gravity, tension, friction, normal reaction) and set the total equal to $\Delta E_k$, or write the mechanical energy $E_k + E_p$ at two points and let the difference equal the work done against resistive forces. Both routes give the same answer; the second is faster for tracks and projectiles because gravity is handled through $E_p$ without resolving forces along the path.

4. Kinetic, gravitational potential and elastic potential energy

Kinetic energy: $$E_k = \frac{1}{2} m v^2 = \frac{p^2}{2m}$$ (in the booklet). The momentum form is useful when two particles share the same momentum, for example after an explosion from rest: the lighter one takes more kinetic energy because $E_k \propto 1/m$ at fixed $p$. Note that doubling the speed quadruples the kinetic energy, a favourite Paper 1 ratio. Gravitational potential energy near the surface: $$\Delta E_p = m g \Delta h$$ (in the booklet), valid where $g$ is uniform. Only changes in height matter, so choose a reference level (usually the lowest point) and measure every $h$ from it. The force of gravity does work $mg\Delta h$ on a falling body, which is why a free-fall speed comes from $\frac{1}{2} m v^2 = m g \Delta h$, independent of mass. Elastic potential energy stored in a spring of constant $k$ at extension or compression $\Delta x$: $$E_H = \frac{1}{2} k \Delta x^2$$ (in the booklet). The symbol $E_H$ is the booklet symbol for elastic energy. When a spring launches a trolley, $\frac{1}{2} k \Delta x^2$ becomes $\frac{1}{2} m v^2$ (plus any work against friction); when a block drops onto a vertical spring, gravitational energy becomes elastic energy at maximum compression, and the height fallen includes the compression itself.

5. Power

Power is the rate of doing work or transferring energy: $$P = \frac{\Delta W}{\Delta t}$$ (in the booklet), measured in watts ($1\ \text{W} = 1\ \text{J s}^{-1}$). For a constant force acting on a body moving at speed $v$ along the force, $$P = F v$$ (in the booklet), which follows from $W = Fs$ divided by $t$. Two exam settings dominate. A motor or crane lifting a load at constant speed: the lifting force equals the weight (zero acceleration), so $P = m g v$. A car or cyclist at constant speed on a level road: the driving force equals the total resistive force, so $P = F_{\text{resist}} v$; if the speed doubles and the whole of the resistance is air resistance proportional to $v^2$, the power needed rises by a factor of eight. Always state what a power value refers to (the useful power to the load, or the power supplied to the motor) and the speed or time interval it belongs to. Watch prefixes: $3.5\ \text{kW} = 3500\ \text{W}$.

6. Efficiency, friction and Sankey diagrams

Efficiency is the fraction of the input that appears as useful output: $$\text{efficiency} = \frac{\text{useful work out}}{\text{total work in}} = \frac{\text{useful power out}}{\text{total power in}}$$ (in the booklet). It has no unit and is quoted as a decimal or a percentage; it can never exceed 1 because energy is conserved. Friction and air resistance transform mechanical energy (kinetic plus potential) into thermal energy of the surfaces and the air. Energy is still conserved, but the thermal energy is spread out and cannot be recovered to do useful work, which is why we call it wasted. The work done against friction over a stretch of track is the loss of mechanical energy: $$W_{\text{friction}} = (E_k + E_p)_{\text{start}} - (E_k + E_p)_{\text{end}},$$ and dividing by the distance travelled gives the average frictional force. A Sankey diagram (learn how to draw and read one) shows energy or power flow as arrows whose widths are proportional to the quantity: a single input arrow on the left, the useful output continuing straight to the right, and each wasted form branching off, usually downwards. The widths must add up: the input width equals the sum of all output widths. To read efficiency from a Sankey diagram, divide the width of the useful arrow by the width of the input arrow.

Formulas

Worked examples

Can you...

• Before the exam, can you write $W = Fs\cos\theta$ and state, with reasons, which forces do positive, negative and zero work on a block pulled across a floor?
• Before the exam, can you find the work done by a variable force from the area under a force-displacement graph, including a triangular spring graph that gives $\frac{1}{2} k \Delta x^2$?
• Before the exam, can you set $E_k + E_p$ at one point equal to $E_k + E_p$ at another point to find a speed, then use the difference of mechanical energies to find the work done against friction and the average frictional force?
• Before the exam, can you calculate the power of a motor lifting a load at constant speed with $P = mgv$ and convert correctly between W and kW?
• Before the exam, can you compute efficiency from useful and total power, draw a Sankey diagram with widths that add up, and explain why energy transferred to thermal stores is wasted even though total energy is conserved?