Power is the rate of doing work
Power is the rate of doing work. It is measured the amount of work done in one second.
Power, P = Work done / Time Taken
or
P = W / t = FS / t or P = F .v

 Power is a scalar quantity.
 The SI unit of power is watt (W). If 1 joule of work is done in 1 second, the power is said to he 1 watt,
1 watt = 1 joule ÷ 1 second = 1 Js^{1}
 The MKS unit of power is kg m^{2} s^{2 } and its CGS unit erg s^{1}
 Horsepower is another unit of power
which is largely used in mechanical engineering.
1 HP = 746 W = 0.746 kW  1 watt is the power used when 1 joule of work is done in 1 second.1 kilowatthour is equal to the work done when a power of 1 kW is used for 1 h.
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Define
Kilowatt hour : It is a measure of electrical energy equivalent to a power consumption of one thousand watts for one hour.
The kilowatthour (kWh) is a unit of energy. But how much?
We have seen that: energy (kWh) = power (kW) × time (h)
Normally we want energy in joules (J) and time in seconds (s). The kW above is 1000 W (1000 J/s). The hour is 3,600 seconds, so:
1 kWh = 1,000 J/s × 3,600 s
1 kWh = 3,600,000 J
1 kWh = 3·6 MJ mega joules
The kWh is a large unit of energy used to measure how much energy is used in homes.
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Difference between watt and Watt hours :
 Watt is the unit of power while Watt hour or kilowatthour (kWh) is the unit of energy
 Watt indicates the rate of using energy in J/s. This is related to how fast water is flowing out of a water pipe. If you have a Light bulb that has a rating of 100 watt, it means that the light bulb consumes 100 J per second.kWh is the unit of energy. This relate to the volume of water that comes out of the pipe. A 100 watt light bulb when used for an hour will consume 100 Watthour of energy which is 0.1 kWh of energy.

Watt and Kilowatthour (kWh) practical example
 Your electricity provider charges you 11 cents ($.11) per kWh.
 You have an Electric Heater that consumes 1500 watts (1.5kw). You use the Electric Heater for 3 hours everyday.
 For 3 hours it will cost you 1.5kw x 3hours = 4.5kWh x .11 = .495 cents (50 cents)
 Over a month it will cost you .495 x 30days = $14.85
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 Power spent by source depends upon two factors
i) Amount of work done by source
ii ) the time taken by the source to do the said work .eg . If person is performing a work at a faster rate , more power is needed to do so.
Example : If a labor A takes 5 minuets to lift a load to truck , while another labor B takes 10 minutes to upload a same load to truck . Here work done by both labors are same but a power spent by labor A is twice the power spent by labor B .
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Difference Between Work and Power
Definition of Work and Power:
Work: Work is the amount of energy transfer when an object is moved by an external force.
Power: Power is the rate of work done.
Nature of the quantity:
Work: Work is a scalar quantity.
Power: Power is also a scalar quantity.
SI Unit:
Work: Work is measured by Joule (J).
Power: Power is measured by Watt (W).
In SI Base Units:
Work: SI Base unit of work is kgm^{2} s^{2 }
Power: SI Base unit of power is kgm^{2}s^{3}
Other Units:
Work: Work is also measured in eV, kWh, MWh, GWh.
Power:Power is also measured in kW, MW, GW.
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 Numerical :
A boy of mass 45 kg climbs a ladder 10 m high at a constant velocity in 20 s. Calculate the power used by him. Take g = 10 m s^{2}.
Ans : Given: m = 45 kg, h = 10 m, g = 10 m s^{2} and t = 20 s.
Power, P = Work / Time
Mgh/t = 45 x 10 x 10 /20 = 225 W
Hence, the power used by boy is 225 W.
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 Numerical :
What is the power of a pump which takes 10 s to lift 100 kg of water to a water tank situated at a height of 20 m? (Take g = 10 m s^{2})
Ans : Given: m = 100 kg, g = 10 m s h = 20 m and t= 10s
Applying the formula work, W = m x g x h
we get
Work. W= 100 x 10 x 20 = 20000 J
.•. Power, P = work / time =20000/10 = 2000 W
Hence, the power of the pump is 2000 W.
A bag of wheat weighs 200 kg.
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 Numerical :
A man of mass 60 kg runs up a flight o 30 steps in 15 s. If each step is 20 cm high, calculate the power developed by the man. Take g= 10 ms^{2}.
Ans : Given: mass (m) = 60 kg, time (t) = 15 s, height (h) = 30 x 0.20 = 6 m and g= 10 ms^{2}
Power = Work / Time
= Increase in potential energy / Time
= mgh / t = 60 x 10 x 6/15
= 240 W
Hence, the power developed by mass is 240 W.
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 Numerical :
The world record for weightlifting is held by Serge Didyk of USSR. He lifted 261 kg to a height of 2.3 m in 4 s. Assuming that g= 10 ms^{2}, find the:
(a) weight lifted by Didyk,
(b) work done by him, and
(c) power developed by him.
Sol.: Given: mass (m) = 261 kg, height (S) = 2.3 m, g= 10 m s^{2} and time (t) = 4 s.
 Weight lifted by Didyk = mg= 261 x 10 = 2610 N
 Work done, W= F x S = 2610 x 2.3 = 6003 J
 Power developed, P =W / t = 6003 / 4
= 1500.75 W
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 Numerical :
A man exert a force of 200 N in pulling a cart at a constant speed of 16 m/s . Calculate the power used by man
Ans : Given force = 200 N , Velocity = 16 m / s
Power = Force x velocity = 3200 W
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 Numerical :
A pump is used to lift 500 kg of water from a depth of 80 m in 10 second . Calculate the work done by the pump , power rating of the pump if its efficiency is 40 % ( Take g = 10 m s^{2} Ans : The work needed in lifting a mass m to a height h against the gravitational force, W= mgh
Given: m = 500 kg, h = 80 m, t = 10 s and g = 10 m s^{2}
 Work done by the pump,
W = mgh = 500 x 10 x 80 = 4,00,000
= 4 x 10^{5}J
 Power of the motor = P = W/ t = 4 x 10^{5}J / 10
=4x 10^{4}W
Efficiency = Useful power ⁄ Power input (power rating)
Power rating = Useful power ÷ Efficiency
= ( 4x 10^{4}W / 40 ) x 100
= 10^{5 WATT }
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 Numerical :
An electric motor of power 100 W is used to drive the stirrer in a water bath . If 50 % of the energy supplied to the motor is spent in stirring the water . Calculate work done on water in 60 second
Ans : Given : Power supplied = 100 W , time = 60 second
Power used in stirring the water = 50 % of the power supplied
= ( 50 / 100 ) x 100 W
= 50 W
Work done on water = power used X time
50 W x 60 sec. = 3000 J
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Numerical :
Calculate the power of an engine required to lift 10^{5 } kg of wood per hour up to 360 m height of tower ( Take g = 10 m s^{2} )
Ans : Given Mass of wood = m = 10 ^{5} kg , g = 10 m s^{2} , Height = h = 360 m ,
Time = t = 1 hour = 60 x 60 s 3600 s
Work required for lifting a mass m to a height h against the force due to gravity = W
= mg x h = mgh
Power = P = Work done / Time taken = mgh /t = 10^{5} x 10 x360 /36000
∴ = 10^{5} Watt = 100 kW
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Numerical :
A man of mass 50 kg runs up a flight of 100steps each 10 cm . high in 20 second
Find ( i ) the workdone by the man ( ii ) the power developed ( g = 9.8 m s^{2})
Ans : While climbing the man has to do work against the force of gravity
Force of gravity on the man = F = mg = 50 x 9.8 = 490 N
Total distance coverd in 100 steps in vertical direction = S = 100 x 10 = 1000 cm = 10 m
(i) Workdone by the by the man in climbing = F x S = 490 x 10 = 4900 J
(ii) Power developed = 245 W
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Numerical : Express 15 kWh into joules
1 kilowatt hour (kWh) = 1000 W x (60 x 60 s ) = 3.6 x 10^{6} J
∴ 15 kWh = 15 x 3.6 x 10^{6} J = 5.4 x 10^{7} J
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Numerical :
The workdone by the heart is 1 J per beat . Calculate the power of heart if it beats 72 times in 1 minute
Ans : Given : Number of beats in I minute = 72
Number of beats per second 72 / 60 = 1.2
Power of heart = Work done per second = workdone per beat x number of beats in 1 second = 1 J x 1.2 s^{1}
= 1.2 Js^{1} = 1.2 W
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Numerical :
Horse can apply a maximum force of 1000 N .It is taking part in a race and is able to pull the cart at a constant speed of 50 m s1 while making its best effort . Calculate the power developed by the Horse
Ans : Power = Force x Average speed
= 1000 x 50 = 50000 watt = 50000 / 1000 = 50 kW
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Numerical :
A boy takes 5 minutes to lift a 20 litre water cans from 50 m deep well and at the same time his father require 2 minutes .Compare the workdone by both and power developed by both . Find out work done by both ? Density of water = 10 kg m^{3} and g = 9.8 N kg^{1}
Ans : Time required to lift water by Boy = 5 minutes = 5 x 60 = 300 sec.
Time required to lift water by father = 2minutes =2 x 60 = 120 sec.
Velocity of boy = V_{1}= Displacement / Time = 20 / 300 = 0.66 m/s
Velocity of father =V_{2}= Displacement / Time = 20 / 120 = 0.166 m/s
Work done by boy =W_{1} = mgh = 20 x 9.8 x 50 =9800 J
Work done by father =W_{2}= mgh = 20 x 9.8 x 50 =9800 J
Ratio = = 1: 1
Power required by boy =P_{1} = F x V_{1} = 20 x 9.8 x 0.66 =129.36
Power required by father =P_{2 }= F x V_{2} = 20 x 9.8 x 0.166 = 32.536
P_{1}/P_{2 }=129.36/32.536
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Numerical :
Manju exerts a force of 150 N in pulling a cart at a constant speed of 10 m/s . Calculate the power exerted
Ans: Given : Force = 150 N , Speed = 10 m s^{1}
Power = Force x velocity = 150 x 10 = 1500 W
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