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.

lists 5 drawing instruments and their uses
that is a question you can find on Google, anyway of top of my head, compass, ruler, protractor, try square, triangles.
Rongfang
A force F is needed to break a copper wire having radius R. The force needed to break a copper wire of radius 2R will be
2F
Jacob
The difference between vector and scaler quantity
vector has both magnitude & direction but scalar has only magnitude
Manash
my marunong ba dto mag prove ng geometry
ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
Jacob
define moment of inertia
what is Euler s theorem
what is thermocouple?
joining of two wire of different material forming two junctions. If one is hot and another is cold the it will produce emf...
joining of two metal of different materials to form a junction in one is hot & another is cold
Manash
define dimensional analysis
mathematical derivation?
Hira
explain what Newtonian mechanics is.
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
SO ASK What is Newtonian mechanics in physics? Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
tq u.
Prema
these are the best physics books one can fond both theory and applications.
can any one suggest best book for maths with lot of Tricks?
Vivek
what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here... . 1. Newton's laws of Motion, are they applicable to motions of all speeds? 2.state the speeds which are applicable to Newtons laws of Motion
Derek
mechanics which follows Newtons law
Manash
The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
jyotirmayee
explain the rule of free body diagram
The polar coordinates of a point are 4π/3 and 5.50m. What are its Cartesian coordinates?
application of elasticity
good
Anwar
a boy move with a velocity of 5m/s in 4s. What is the distance covered by the boy?
What is the time required for the sun to reach the earth?
anthony
24th hr's, your question is amazing joke 😂
Arzoodan
velocity 20 m, s
Ahmed
the sun shines always and the earth rotates so the question should specify a place on earth and that will be 24hrs
Opoku
20m
Gabriel
good nice work
Anwar
20m
Evelyn
why 20?.
Arzoodan
v =distance/time so make distance the subject from the equation
Evelyn
20m
Olaide
exatly
Arzoodan
what is differemce between principles and laws
plz
Anwar
how can a 50W light bulb use more energy than a 1000W oven?
That depends on how much time we use them
Phrangsngi
Define vector law of addition
It states that, " If two vectors are represented in magnitude and direction by the two sides of a triangle, then their resultant is represented in magnitude and direction by the third side of the triangl " .
Nabin
thanks yaar
Pawan
And it's formula
Pawan
Manash