# Power and Efficiency

### Power and Efficiency

#### Lessons

In this lesson, we will learn:

• Meaning of power and efficiency in physics
• How to solve problems involving power and efficiency

Notes:

• Power is a quantity that describes the rate that work can be done. If you have some heavy books that you need to carry up a flight of stairs, it takes more power to run up the stairs than it does to go slowly, even though you do the same amount of work in either case. Power is a scalar quantity measured in watts (W), which are equal to one joule per second (J/s).
• Efficiency tells you how much of the work that is done is "useful work", and how much wasted. For example, incandescent light bulbs are only about 10% efficient: for every 100 J of energy an incandescent bulb uses, 10 J of light energy (useful) and 90 J of heat (waste) are produced. A newer 80% efficient lightbulb could take the same 100 J of energy and convert it to 80 J of light and 20 J of heat. Efficiency is expressed as a percentage of the total energy used that results in useful work.

• Power

$P = \frac{W}{t}$

$P:$ power in watts (W)

$W:$ work, in joules (J)

$t:$ time, in seconds

Efficiency

$\mathrm{efficiency} = \frac{W_{output}}{W_{input}} \centerdot 100 = \frac{P_{output}}{P_{input}} \centerdot 100$

$\mathrm{efficiency:}$ percentage of work that is transformed to mechanical energy in a process

$W_{input}:$ total energy used in a process, in joules (J)

$W_{output}:$ useful work done in a process, in joules (J)

$P_{input}:$ total power used in a process, in watts (W)

$P_{output}$ useful power used in a process, in watts (W)

Work

$W = F_\parallel d = \Delta E_{mech} = (E_{kf} + E_{pf}) - (E_{ki} + E_{pi})$

$W:$ work, in joules (J)

$d:$ displacement, in meters (m)

$F_\parallel:$ component of force parallel to $d$ in newtons (N)

$\Delta E_{mech}:$ change in mechanical energy

$(E_{kf} + E_{pf}):$ total final mechanical energy, in joules (J)

$(E_{ki} + E_{pi}):$ total initial of force parallel to $d$, in newtons (N)

• 1.
Power and efficiency of accelerating a car
A 1210 kg sports car can reach 100 km/h in 2.70 s.
1. Find the power output of the engine.
2. If the fuel used in this time produces a total of 718 kJ of energy, what is the efficiency of the engine?

• 2.
Power of pulling an object up a ramp
An 87.0 kg cart is pulled up a ramp at a constant speed by an electric motor. The motor outputs 375 W and the ramp's incline 25.0°. What is the speed of the cart? Assume friction is negligible.