Dragging a load: work against friction, power, and the machine's efficiency
SPH3U Grade 11 Physics · Energy and Society
An electric winch drags a 500 kg sled of firewood 25.0 m across level frozen ground in 40.0 s, pulling horizontally on a cable. The coefficient of kinetic friction between the sled runners and the ground is 0.30, and the winch motor draws a steady 1.40 kW of electrical power the whole time. Find the friction force, the work done against it, the useful power delivered, the electrical energy consumed, and the efficiency of the winch.
On level ground the surface has to hold up the entire weight, so mg is also the normal force — the one number friction is built from.
Carried onward at full precision, not this rounded figure.
Friction is what the cable is actually fighting. The sled is not speeding up, so the pull equals this force exactly — no more, no less.
Carried onward at full precision, not this rounded figure.
Force times distance, with θ = 0° because the cable pulls along the motion. This is the useful work — every joule of it ends up as heat in the snow, which is exactly what the job required.
Carried onward at full precision, not this rounded figure.
The same work spread over the 40.0 s haul. This is the winch's useful output, not what it draws from the battery.
Carried onward at full precision, not this rounded figure.
Now the input side of the ledger. The motor is badged in kilowatts and the ledger is kept in joules, so the 1.40 kW becomes 1400 W before it meets the same 40.0 s.
Carried onward at full precision, not this rounded figure.
Efficiency is the ratio of the two energies, and this is the first step that reaches back to two different earlier answers at once. Compare it with 919.4 W ÷ 1400 W — identical, because both ran for the same 40.0 s.
Carried onward at full precision, not this rounded figure.
Why this order
An efficiency is never a single measurement; it is always a ratio of two independently computed energies, which is why this chain has to build both sides before it can say anything. The output side takes three steps — weight, friction, work — because the useful work is not something you read off a gauge: you have to know what force the job actually demanded. The input side takes one, because electrical energy is simply what the meter says. The classic error is to compare a force with a power, or the work done against friction with the motor's rated power, and the fix is dimensional: efficiency is joules over joules, so both sides must be energies.
Note that the 25.0 m and the 40.0 s make no difference to the answer. Drag the same sled twice as far in twice the time and both energies double, leaving η at 66%. That is the honest way to read the number: it says nothing about how hard the winch worked and everything about where the missing 34% went — gearbox friction, resistive heating in the motor windings, and the cable creaking over the drum. Efficiencies also multiply, so a 66% winch fed by a generator that is itself 30% efficient delivers about 20% of the fuel's energy to the firewood. That compounding is the whole reason the Energy and Society strand exists: a slogan about conservation becomes an argument only once it is a chain of numbers.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.