At a winter camp, an electric winch bolted to a post hauls firewood in from the treeline across a stretch of level frozen ground. This morning's load is a sled stacked with split wood — 500 kg on the yard scale — and the cable runs horizontally from the drum to the hitch. The operator marks the start point, engages the drum, and times the haul with a stopwatch: the sled slides 25.0 m in 40.0 s at a steady crawl, runners squeaking over the ice. A power meter on the supply line shows the winch motor drawing a steady 1.40 kW of electrical power the whole time, and the coefficient of kinetic friction between the sled runners and the frozen ground is known to be 0.30. Find the friction force, the work done against it, the useful power delivered, the electrical energy consumed, and the efficiency of the winch.
Every number in this problem is editable — change any value below and the whole chain recalculates.
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.
Rearranged for W
W=Fdcosθ
1.471 kNcarried from step 2
Your values, in your units
W=(1,471N)(25m)cos(0∘)
Answer
W=36.775kJ
Carried onward at full precision, not this rounded figure.
Now the input side of the energy balance. The motor is badged in kilowatts and the balance is kept in joules, so the 1.40 kW becomes 1,400 W before it meets the same 40.0 s.
Rearranged for E
E=Pt
Your values, in your units
E=(1.4kW)(40s)
Converted to base units
E=(1,400W)(40s)
Answer
E=56kJ
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 ÷ 1,400 W — identical, because both ran for the same 40.0 s.
Rearranged for η
η=WinWout
36.775 kJcarried from step 3
56 kJcarried from step 5
Your values, in your units
η=(56,000J)(36,774.9J)
Answer
η=656.7per mille
Carried onward at full precision, not this rounded figure.
Therefore the cable fights 1,471 N of friction, the 25.0 m haul is 36.8 kJ of work against it, the winch delivers a useful 919 W while the motor consumes 56.0 kJ, and the machine runs at 65.7% efficiency — the missing third lost to the gearbox, the windings and the drum.
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.
