Physics › Mechanics › Work, energy and power
Work, energy and power
Work is energy in transit, and it has a strict definition that ignores effort entirely: only force along the motion counts. Power is how fast the transfer runs, and one rearrangement of it, P = Fv, explains the top speed of everything with an engine.
Builds on Scalars and vectors.
IN THIS TOPIC
- Calculate work done as W = Fs cos θ, including recognising forces that do no work or negative work.
- Find work from the area under a force-displacement graph when the force varies.
- Use P = ΔW/Δt = Fv, and efficiency as the ratio of useful output power to input power.
WHAT YOU PROBABLY THINK
Work means effort.
What counts as work
In physics, work is done on an object when a force moves it: work is energy transferred by a force acting through a displacement,
where θ is the angle between the force and the displacement. The cos θ is the honest part: only the component of the force along the motion transfers energy.
The definition produces results that offend everyday English. Carry a heavy box across a room at constant height and you do no work on the box: the force you apply is vertical, the motion horizontal, cos 90° = 0. Your muscles burn energy and you are genuinely tired, but none of it went into the box. And a force with a component opposing the motion, friction, drag, does negative work: it takes energy out, which is why friction warms things up.
WORKED EXAMPLE
Work at an angle
A sledge is pulled 12 m across level snow by a rope at 30° to the ground, with a tension of 40 N.
Only the horizontal component works: W = Fscos θ = 40 × 12 × cos 30° = 420 J to two significant figures.
The vertical component of the tension merely lightens the sledge's press on the snow. It moves nothing vertically, so it transfers no energy.
When the force varies with position, a stretching spring being the standard case, no single F exists to multiply. The area under the force-displacement graph is the work done, for any shape of force.
Power
Power is the rate of doing work, equivalently the rate of energy transfer:
measured in watts, one joule per second. The Fv form is the one exams lean on. At a vehicle's top speed the acceleration is zero, so the driving force exactly equals the total resistive force, and the engine's maximum power is that force times the top speed. The same relation explains why acceleration fades at speed: at constant power, a larger v leaves a smaller available F.
Efficiency
No machine turns all its input into the output you wanted. Efficiency is the useful fraction:
quotable as a decimal or a percentage, and the same ratio works with energies in place of powers; use whichever the question supplies.
Efficiency can never exceed 100%, so any answer above it is a mistake, almost always the fraction inverted. And the “wasted” share is not destroyed; it is transferred somewhere useless, nearly always ending as internal energy in the surroundings.
THE EXAM BIT
- The cos θ is not optional. A rope at an angle does W = Fs cos θ, and the perpendicular component does no work at all.
- P = Fv is the top-speed workhorse: at maximum speed, driving force equals total resistive force, so P = Fres × vmax.
- Friction and drag do negative work. When a question asks for the work done against friction, the magnitude is wanted, but keep the sign convention straight in energy audits.
- For a variable force, resist the urge to average by eye: the work is the area under the force-displacement graph, counted properly.
- Efficiency above 100% is always an error; check the fraction is useful over total, not the other way up.
CHECK YOURSELF
A car of mass 1200 kg travels at a constant 30 m s−1 against total resistive forces of 600 N. What power does the engine deliver, and why does the mass not matter?
Show a hint
Constant speed is a statement about the resultant force.
Show the answer
Constant speed means zero acceleration, so the resultant force is zero and the driving force equals the resistive force: 600 N.
P = Fv = 600 × 30 = 18 kW.
The mass is irrelevant because nothing accelerates: mass could only enter through F = ma, and a = 0. It is a distractor, and a deliberate one.
Only force along the motion works.
Power is how fast the joules move.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.