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
P = W / t = FS / t
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.
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.
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
- 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 .
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.
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
Work: Work is also measured in eV, kWh, MWh, GWh.
Power:Power is also measured in kW, MW, GW.
- 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.
- 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
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.
- 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.
- 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
- 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
- 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 105J / 10
Efficiency = Useful power ⁄ Power input (power rating)
Power rating = Useful power ÷ Efficiency
= ( 4x 104W / 40 ) x 100
= 105 WATT
- 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
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
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
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
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
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
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
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
(WORK ) ( Energy)