Work and power
Work of a force (W = F·d·cos θ) and power (P = W/t or P = F·v), with the three regimes of cos θ named.
Result
500 J
- Calculation
- W = F·d·cos θ = 50 N × 10 m × cos 0° = 500 J
- Regime
- motor work
- Work
- 500 J
- Power
- 25 W
Note
The force provides a motor work of 500 J.
Calculation, assumptions and sources
OverviewThe work of a force along a displacement: driving below 90°, zero at 90°, resistive and negative beyond.
The work of a force along a displacement: driving below 90°, zero at 90°, resistive and negative beyond. Power, in its two forms: W/t for work spread over time, F·v for steady-state traction. The regime is named with every result; resistive work gives a negative power.
Method & formulasW = F × d × cos θ
Only the component of the force along the displacement does work, hence the cos θ. Power is the rate of work; in steady state, d = v·t gives P = F·v.
- 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, one metric horsepower 735.5 W.
- 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 corresponds to 180°.
Validity domainW = F·d·cos θ assumes a constant force and a straight displacement; for a varying force (a spring) or a curved path, the work is the integral of F·dl.
W = F·d·cos θ assumes a constant force and a straight displacement; for a varying force (a spring) or a curved path, the work is the integral of F·dl. P = F·v is instantaneous; at constant speed, it equals W/t over the whole interval. Angles run from 0 to 180°.
Reading the resultThe work-energy theorem sums these works into the change of speed.
The work-energy theorem sums these works into the change of speed. No work is done by the centripetal force of a turn, the weight of a suitcase carried flat, or the normal reaction of the ground. At constant power, the available traction force drops with speed (P = F·v): 50 kW at 90 km/h give 2000 N.