Power is the rate of doing work

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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 m2 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 kilowatt-hour 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 kilowatt-hour (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 kilowatt-hour (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 Watt-hour of energy which is 0.1 kWh of energy.
  • Watt and Kilowatt-hour (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 kgm2 s-2 

    Power: SI Base unit of power is kgm2s-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 weight-lifting 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 105J

  • Power of the motor = P  = W/ t = 4 x 105 / 10

          =4x 104W

Efficiency =  Useful power  ⁄  Power input (power rating)

Power rating  =  Useful power  ÷ Efficiency

= (  4x 104W  /  40 )  x  100

= 105  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  105  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 = 105 x 10 x360 /36000

∴                                 = 105 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 106 J

∴ 15  kWh = 15 x 3.6 x 106 J  = 5.4  x 107 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 s-1 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 = V1= Displacement / Time = 20 / 300  = 0.66 m/s

Velocity of father  =V2= Displacement / Time = 20 / 120  = 0.166 m/s

Work done by boy =W1 = mgh  = 20 x 9.8 x 50 =9800 J

Work done by father =W2= mgh  = 20 x 9.8 x 50 =9800 J

Ratio = = 1: 1

 Power required by boy =P1 = F x V1 = 20 x 9.8 x 0.66 =129.36

Power required by father  =P2 = F x V2 = 20 x 9.8 x 0.166 = 32.536

P1/P2 =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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