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7.4 Power  (Page 2/8)

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

980 W

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

P = d W d t = F → · d r → d t = F → · ( d r → d t ) = F → · v → ,

where 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.

An automobile is shown moving up along a 15 percent grade at a speed of v = 90 kilometers per hour. The car has mass m = 1200 kilograms.
We want to calculate the power needed to move a car up a hill at constant speed.

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 g → · v → = m g v sin θ , where θ is the angle of the incline. A 15% grade means tan θ = 0.15 . This reasoning allows us to solve for the power required.

Solution

Carrying out the suggested steps, we find

0.75 P = m g v sin ( tan −1 0.15 ) ,

or

P = ( 1200 × 9.8 N ) ( 90 m / 3.6 s ) sin ( 8.53 ° ) 0.75 = 58 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 ( 1 hp = 0.746 kW ). 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.

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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 d W = F → · d r → = | F → | | d r → | cos θ
Work done by a force acting along a path from A to B W A B = ∫ path A B F → · d r →
Work done by a constant force of kinetic friction W fr = − f k | l A B |
Work done going from A to B by Earth’s gravity, near its surface W grav, A B = − m g ( y B − y A )
Work done going from A to B by one-dimensional spring force W spring, A B = − ( 1 2 k ) ( x B 2 − x A 2 )
Kinetic energy of a non-relativistic particle K = 1 2 m v 2 = p 2 2 m
Work-energy theorem W net = K B − K A
Power as rate of doing work P = d W d t
Power as the dot product of force and velocity P = F → · 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.

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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