# 7.4 Power  (Page 2/8)

 Page 2 / 8

Check Your Understanding Estimate the power expended by a weightlifter raising a 150-kg barbell 2 m in 3 s.

980 W

The power involved in moving a body can also be expressed in terms of the forces acting on it. If a force $\stackrel{\to }{F}$ acts on a body that is displaced $d\stackrel{\to }{r}$ in a time dt , the power expended by the force is

$P=\frac{dW}{dt}=\frac{\stackrel{\to }{F}·d\stackrel{\to }{r}}{dt}=\stackrel{\to }{F}·\left(\frac{d\stackrel{\to }{r}}{dt}\right)=\stackrel{\to }{F}·\stackrel{\to }{v},$

where $\stackrel{\to }{v}$ is the velocity of the body. The fact that the limits implied by the derivatives exist, for the motion of a real body, justifies the rearrangement of the infinitesimals.

## Automotive power driving uphill

How much power must an automobile engine expend to move a 1200-kg car up a 15% grade at 90 km/h ( [link] )? Assume that 25% of this power is dissipated overcoming air resistance and friction.

## Strategy

At constant velocity, there is no change in kinetic energy, so the net work done to move the car is zero. Therefore the power supplied by the engine to move the car equals the power expended against gravity and air resistance. By assumption, 75% of the power is supplied against gravity, which equals $m\stackrel{\to }{g}·\stackrel{\to }{v}=mgv\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ where $\theta$ is the angle of the incline. A 15% grade means $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =0.15.$ This reasoning allows us to solve for the power required.

## Solution

Carrying out the suggested steps, we find

$0.75\phantom{\rule{0.2em}{0ex}}P=mgv\phantom{\rule{0.2em}{0ex}}\text{sin}\left({\text{tan}}^{-1}\phantom{\rule{0.2em}{0ex}}0.15\right),$

or

$P=\frac{\left(1200\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}9.8\phantom{\rule{0.2em}{0ex}}\text{N}\right)\left(90\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}3.6\phantom{\rule{0.2em}{0ex}}\text{s}\right)\text{sin}\left(8.53\text{°}\right)}{0.75}=58\phantom{\rule{0.2em}{0ex}}\text{kW,}$

or about 78 hp. (You should supply the steps used to convert units.)

## Significance

This is a reasonable amount of power for the engine of a small to mid-size car to supply $\left(1\phantom{\rule{0.2em}{0ex}}\text{hp}=0.746\phantom{\rule{0.2em}{0ex}}\text{kW}\text{).}$ Note that this is only the power expended to move the car. Much of the engine’s power goes elsewhere, for example, into waste heat. That’s why cars need radiators. Any remaining power could be used for acceleration, or to operate the car’s accessories.

## Summary

• Power is the rate of doing work; that is, the derivative of work with respect to time.
• Alternatively, the work done, during a time interval, is the integral of the power supplied over the time interval.
• The power delivered by a force, acting on a moving particle, is the dot product of the force and the particle’s velocity.

## Key equations

 Work done by a force over an infinitesimal displacement $dW=\stackrel{\to }{F}·d\stackrel{\to }{r}=|\stackrel{\to }{F}||d\stackrel{\to }{r}|\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$ Work done by a force acting along a path from A to B ${W}_{AB}=\underset{\text{path}AB}{\int }\stackrel{\to }{F}·d\stackrel{\to }{r}$ Work done by a constant force of kinetic friction ${W}_{\text{fr}}=\text{−}{f}_{k}|{l}_{AB}|$ Work done going from A to B by Earth’s gravity, near its surface ${W}_{\text{grav,}AB}=\text{−}mg\left({y}_{B}-{y}_{A}\right)$ Work done going from A to B by one-dimensional spring force ${W}_{\text{spring,}AB}=\text{−}\left(\frac{1}{2}k\right)\left({x}_{B}^{2}-{x}_{A}^{2}\right)$ Kinetic energy of a non-relativistic particle $K=\frac{1}{2}m{v}^{2}=\frac{{p}^{2}}{2m}$ Work-energy theorem ${W}_{\text{net}}={K}_{B}-{K}_{A}$ Power as rate of doing work $P=\frac{dW}{dt}$ Power as the dot product of force and velocity $P=\stackrel{\to }{F}·\stackrel{\to }{v}$

## Conceptual questions

Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zero-watt device.) Explain in terms of the definition of power.

Appliances are rated in terms of the energy consumed in a relatively small time interval. It does not matter how long the appliance is on, only the rate of change of energy per unit time.

what's the answer? I can't get it
what is the question again?
Sallieu
What's this conversation?
Zareen
what is catenation? and give examples
sununu
what's the si unit of impulse
The Newton second (N•s)
Ethan
what is the s. I unit of current
Amphere(A)
imam
thanks man
Roland
u r welcome
imam
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
what is head to tail rule?
kinza
what is the guess theorem
viva question and answer on practical youngs modulus by streching
send me vvi que
rupesh
a car can cover a distance of 522km on 36 Liter's of petrol, how far can it travel on 14 liter of petrol.
Isaac
yoo the ans is 193
Joseph
whats a two dimensional force
what are two dimensional force?
Where is Fourier Theorem?
what is Boyle's law
Boyle's law state that the volume of a given mass of gas is inversely proportion to its pressure provided that temperature remains constant
Abe
how do I turn off push notifications on this crap app?
Huntergirl
what is the meaning of in.
In means natural logarithm
Elom
is dea graph for cancer caliper experiment using glass block?
Bako
identity of vectors?
what is defined as triple temperature
Triple temperature is the temperature at which melting ice and boiling water are at equilibrium
njumo
a tire 0.5m in radius rotate at constant rate 200rev/min. find speed and acceleration of small lodged in tread of tire.
hmm
Ishaq
100
Noor
define the terms as used in gravitational mortion 1:earth' satellites and write two example 2:parking orbit 3:gravitation potential 4:gravitation potential energy 5:escping velocity 6:gravitation field and gravitation field strength
can gravitational force cause heat?
SANT
yes
Kawshik
_2/3 ÷34
Isaac
does gravitation affect friction
Joseph