Work and power
W = F × d × cos θ
The work of a force along a displacement, with the angle that makes the whole lesson: motor before 90°, zero at 90° (carrying a suitcase does no work), resistive beyond, negative, the one of friction. Power follows, in its two forms: W/t for a work spread over time, F·v for a traction in steady state, and the tool shows the two agree. The verdicts name the regime instead of leaving a bare sign, and resistive work is treated as a full citizen, negative rate included.
500 J
- Calculation
- W = F·d·cos θ = 50 N × 10 m × cos 0° = 500 J
- Regime
- motor work
The force provides a motor work of 500 J.
The cos θ has three regimes and that is the whole lesson: before 90°, the force helps the motion (motor work); at 90°, it does no work; beyond, it opposes it (resistive work, negative). The sign of the work says who gives energy to whom.
A watt is a joule per second: power is a flow of energy, not an energy. The metric horsepower is 735.5 W; a 100 hp car can deliver 73.5 kJ every second, what a cyclist (about 200 W in endurance) provides in six minutes.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasW = F × d × cos θ
W = F × d × cos θ
P = W / t
P = F × v (steady state)
Only the component of the force along the displacement works: hence the cos θ. Power is the rate of work; in steady state, d = v·t turns it into F·v, the formula of drivetrains, and the two writings return the same number.
- Work (J)
- · the energy the force gives to the motion (motor, positive) or takes from it (resistive, negative). One joule: one newton over one metre.
- Power (W)
- · the rate of work: one watt is one joule per second. An energy per unit time, never an energy.
- Motor work
- · θ < 90°: the force accompanies the motion and feeds it energy.
- Resistive work
- · θ > 90°: the force opposes the motion and drains its energy. Friction is the pure case, at 180°.
Validity domainW = F·d·cos θ assumes a constant force along a straight displacement: for a varying force (a spring) or a curved path, the work is the integral of F·dl, and this formula is only its special case.
W = F·d·cos θ assumes a constant force along a straight displacement: for a varying force (a spring) or a curved path, the work is the integral of F·dl, and this formula is only its special case. P = F·v holds at the instant the speed is v; in steady state (constant speed), it equals W/t over the whole interval. Angles run from 0 to 180°: the angle between two directions never exceeds a half-turn.
Reading the resultThe sign of the work says who gives energy to whom: the work-energy theorem sums all these works into the change of speed.
The sign of the work says who gives energy to whom: the work-energy theorem sums all these works into the change of speed. The zero case is the most instructive: the centripetal force of a turn, the weight of a suitcase carried flat, the normal reaction of the ground never work, they deflect or carry without exchanging energy. On the power side, P = F·v explains why the available traction force drops with speed at constant power: at 90 km/h, 50 kW leave only 2000 N.