Power: MDCAT Physics notes
Power notes for MDCAT: P = W/t = Fv, instantaneous vs average power, watt, horsepower, kilowatt-hour = 3.6 MJ, pumping water and zero-power cases.
Definition of power
Power is the rate of doing work (rate of transfer of energy):
$$P = \frac{W}{t}$$
- Average power: $P_{av} = \Delta W/\Delta t$ over an interval.
- Instantaneous power: the rate of doing work at a particular instant, $P = \lim_{\Delta t\to0}\Delta W/\Delta t$.
Power is a scalar. SI unit: watt, $1\ \text{W} = 1\ \text{J s}^{-1} = 1\ \text{N m s}^{-1}$. Also $1\ \text{kW} = 1000\ \text{W}$, $1\ \text{MW} = 10^6\ \text{W}$.
In the British engineering system the unit is the horsepower: $1\ \text{hp} = 746\ \text{W}$.
Power in terms of force and velocity
For a constant force, $P = \dfrac{F\,\Delta d}{\Delta t}$, so
$$P = \vec F\cdot\vec v = Fv\cos\theta$$
Power is the dot product of force and velocity. At constant velocity, doubling the force doubles the power.
Zero power
- A cord keeping a block in uniform circular motion: tension is perpendicular to velocity, so $P = 0$.
- The normal force on a block moving horizontally: perpendicular to velocity, so $P = 0$.
A force opposite to the velocity (braking) gives negative power: a 2 N braking force on a car at $50\ \text{m s}^{-1}$ has magnitude $Fv = 100\ \text{W}$.
Same work, different power
Two students each carry a 20 kg load up the same stairs. They do the same work ($mgh$), but the one who takes 10 s develops twice the power of the one who takes 20 s.
Worked examples
| Problem | Working | Answer |
|---|---|---|
| Motor lifts 2.0 N through 1.0 m in 4 s | $2 \times 1/4$ | 0.5 W |
| $64\times10^6\ \text{J}$ in 8 s | $64\times10^6/8$ | 8 MW |
| 60 W bulb for 2 s | $60 \times 2$ | 120 J |
| 60 W bulb for 2 min | $60 \times 120$ | 7200 J = 7.2 kJ |
| 1 kW pump, well 10 m deep: mass per second | $1000/(9.8\times10)$ | about 10.2 kg |
The kilowatt-hour
The kilowatt-hour is a unit of energy (power × time), not power. It is the commercial unit of electrical energy:
$$1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3.6\times10^6\ \text{J} = 3.6\ \text{MJ}$$
Key formulas
- $P = W/t$
- $P = Fv\cos\theta = \vec F\cdot\vec v$
- Pumping: $P = (m/t)gh$
- $1\ \text{hp} = 746\ \text{W}$; $1\ \text{kWh} = 3.6\times10^6\ \text{J}$
Common MDCAT traps
- kWh is energy, not power; it is the odd one out among hp, kW and N m/s.
- 1 kWh = 3.6 MJ, not 3.6 kJ or 3.6 MW.
- Convert minutes and centimetres to seconds and metres before calculating.
- Same load, same height: same work; the faster person has more power.
- Forces perpendicular to velocity deliver zero power.
Quick revision
- Power = rate of doing work.
- $1\ \text{W} = 1\ \text{J s}^{-1} = 1\ \text{N m s}^{-1}$.
- $1\ \text{hp} = 746\ \text{W}$.
- Force × velocity = power.
- Energy = power × time.