Power and its units, including the kilowatt hour
Power is the rate of doing work, P = W/t, measured in watts. The kilowatt hour is a unit of energy: 1 kW h = 3.6 × 10⁶ J.
Why time matters
A labourer and a crane may lift the same load to the same height. Both do the same work. The crane does it much faster, so it has more power. Power is the rate of doing work: P = W/t. If ΔW is done in time Δt, the average power is ΔW/Δt. If the rate keeps changing, the instantaneous power is P = dW/dt, the limit of ΔW/Δt as Δt becomes very small. Power is a scalar. Its SI unit is the watt (W), named after James Watt: 1 W = 1 J of work in 1 s. If a constant force F moves a body at a steady speed v along the force, then P = W/t = F × (d/t) = Fv.
Units of power and energy side by side
| Quantity | Unit | Equal to |
|---|---|---|
| Power | watt (W) | 1 J s⁻¹ |
| Power | kilowatt (kW) | 10³ W |
| Power | megawatt (MW) | 10⁶ W |
| Power | horse power (hp) | 746 W |
| Energy (work) | joule (J) | 1 N m |
| Energy (work) | kilowatt hour (kW h), the 'unit' of electricity | 3.6 × 10⁶ J |
Show that 1 kW h = 3.6 × 10⁶ J
- 11 kW h is the energy used when a power of 1 kW is used for 1 hour.
- 2Energy = power × time.
- 31 kW = 1000 W = 1000 J s⁻¹, and 1 h = 60 × 60 s = 3600 s.
- 41 kW h = 1000 J s⁻¹ × 3600 s = 3 600 000 J.
- 5★So, 1 kW h = 3.6 × 10⁶ J = 3.6 MJ.
★ key step
Worked example: power of a lift motor
A motor lifts 250 kg of bricks through a height of 12 m in 20 s at steady speed. Find its power in watts and in horse power. Take g = 9.8 m s⁻².
- Work done W = mgh = 250 kg × 9.8 m s⁻² × 12 m = 29 400 J.
- Power P = W/t = 29 400 J ÷ 20 s = 1470 W = 1.47 kW.
- In horse power: 1470 W ÷ 746 W per hp ≈ 1.97 hp.
About 1.47 kW, or nearly 2 hp
Worked example: energy used by a geyser
A 1.5 kW geyser (water heater) is used for 40 minutes every day for 30 days. Find the energy used in kW h and in joules. At an example rate of ₹7 per unit, what is the cost?
- Total time = 40 min × 30 = 1200 min = 20 h.
- Energy = power × time = 1.5 kW × 20 h = 30 kW h.
- In joules: 30 × 3.6 × 10⁶ J = 1.08 × 10⁸ J.
- Cost at the example rate = 30 units × ₹7 = ₹210.
30 kW h (30 units), which is 1.08 × 10⁸ J; ₹210 at the example rate
Kilowatt and kilowatt hour: not the same thing
The electricity meter at home counts kW h. Each kW h is one 'unit' on the bill.
The electricity meter at home counts energy in kilowatt hours, the 'units' on the monthly bill. A 1 kW iron used for one hour uses one unit, and so does a 10 W LED bulb lit for 100 hours.
2 marks: “Show that 1 kW h = 3.6 × 10⁶ J.” 1 kW h = 1 kW × 1 h = 1000 J s⁻¹ × 3600 s = 3 600 000 J = 3.6 × 10⁶ J.
2 marks: “Define power. Give its SI unit and its dimensional formula.” Power is the rate of doing work, P = W/t. SI unit: watt (1 W = 1 J s⁻¹). Dimensions: [ML²T⁻³].
3 marks: “A pump raises 600 kg of water to an overhead tank 15 m high in 2 minutes. Find the power of the pump in kW and in hp. (g = 9.8 m s⁻²)” W = mgh = 600 × 9.8 × 15 = 88 200 J; t = 120 s. P = 88 200 ÷ 120 = 735 W = 0.735 kW. In hp: 735 ÷ 746 ≈ 0.99 hp.
Watt measures how FAST; kilowatt hour measures how MUCH: 1 kW h = 1000 × 3600 J = 3.6 MJ.