# 55 7.7 Power

### Summary

- Calculate power by calculating changes in energy over time.
- Examine power consumption and calculations of the cost of energy consumed.

# What is Power?

*Power*—the word conjures up many images: a professional football player muscling aside his opponent or a dragster roaring away from the starting line.

These images of power have in common the rapid performance of work, consistent with the scientific definition of **power **(** P**) as the rate at which work is done.

### POWER

Power is the rate at which work is done.

The SI unit for power is the **watt **(**W**), where 1 watt equals 1 joule/second (**1 W = 1 J/s**).

Because work is energy transfer, power is also the rate at which energy is expended. A 60-W light bulb, for example, expends 60 J of energy per second. Great power means a large amount of work or energy developed in a short time. For example, when a powerful car accelerates rapidly, it does a large amount of work and consumes a large amount of fuel in a short time.

# Calculating Power from Energy

### Example 1: Calculating the Power to Climb Stairs

What is the power output for a 60.0-kg woman who runs up a 3.00 m high flight of stairs in 3.50 s, starting from rest but having a final speed of 2.00 m/s? (See Figure 2.)

**Strategy and Concept**

The work going into mechanical energy is ** W = KE + PE**. At the bottom of the stairs, we take both

**KE**and

**PE**initially zero; thus, [latex]\boldsymbol{W=\textbf{KE}_{\textbf{f}}+\textbf{PE}_{\textbf{g}}=\frac{1}{2}{mv_{\textbf{f}}}^2+mgh},[/latex] where

_{gas}*is the vertical height of the stairs. Because all terms are given, we can calculate*

**h***and then divide it by time to get power.*

**W****Solution**

Substituting the expression for * W* into the definition of power given in the previous equation,

**yields**

*P*=*W*/*t*Entering known values yields

**Discussion**

The woman does 1764 J of work to move up the stairs compared with only 120 J to increase her kinetic energy; thus, most of her power output is required for climbing rather than accelerating.

It is impressive that this woman’s useful power output is slightly less than 1 **horsepower **(**1 hp = 746 W**)! People can generate more than a horsepower with their leg muscles for short periods of time by rapidly converting available blood sugar and oxygen into work output. (A horse can put out 1 hp for hours on end.) Once oxygen is depleted, power output decreases and the person begins to breathe rapidly to obtain oxygen to metabolize more food—this is known as the *aerobic* stage of exercise. If the woman climbed the stairs slowly, then her power output would be much less, although the amount of work done would be the same.

### MAKING CONNECTIONS: TAKE-HOME INVESTIGATION—MEASURE YOUR POWER RATING

Determine your own power rating by measuring the time it takes you to climb a flight of stairs. We will ignore the gain in kinetic energy, as the above example showed that it was a small portion of the energy gain. Don’t expect that your output will be more than about 0.5 hp.

# Section Summary

- Power is the rate at which work is done, or in equation form, for the average power
for work**P**done over a time**W**,*t*.*P*=*W*/*t* - The SI unit for power is the watt (W), where
**1 W = 1 J/s**.

### Problems & Exercises

**1: **A person in good physical condition can put out 100 W of useful power for several hours at a stretch, perhaps by pedalling a mechanism that drives an electric generator. Neglecting any problems of generator efficiency and practical considerations such as resting time: (a) How many people would it take to run a 4.00-kW electric clothes dryer? (b) How many people would it take to replace a large electric power plant that generates 800 MW?

**2: **(a) What is the average useful power output of a person who does 6.00 × 10^{6} J of useful work in 8.00 h? (b) Working at this rate, how long will it take this person to lift 2000 kg of bricks 1.50 m to a platform? (Work done to lift his body can be omitted because it is not considered useful output here.)

## Glossary

- power
- the rate at which work is done

- watt
- (W) SI unit of power, with
**1 W = 1 J/s**

- kilowatt-hour
- (
**kW • h**) unit used primarily for electrical energy provided by electric utility companies

### Solutions

**Problems & Exercises**

**1: **(a) 40 (b) 8 million

**2: **(a) 208 W (b) 141 s