Question

A tubewell pumps out 2400 kg of water per minute If water is moving with a velocity of 3 ms what is power of the pump
how much work is done if pump runsfor 10 hrs
(ii)A car weighing 1120 kg is going up an incline of 1 in 56 at rate of 20 m in 2 s find power of engine if frictional froce is 64 n
(iii)a train weighing 100 metric tonne is running on a level track with a uniform speed of 72 kmh if frictional resistance amounts to 0.5 kg per metric tonne find power of engine
(iv)a pump can throw 10 quintals of coal per hrfrom a cola mine 120 m depth calculate power of engine in watt assuming that efficiency is 80%

Answer 1

2400 kg  of water per minute is pumped. initially water is at rest. the change in energy is the work done.

speed of water = 3 meters/second = v
So 2400 kg of water comes out in 1 min.  Hence 40 kg per second.
let t = time.  so  mass/time = m/t = 40 kg/sec.

   The kinetic energy = work done = 1/2 m v²
   Power = work done / time = 1/2 (m / t) v²
           = 1/2 * 40 kg/sec * 3² m²/sec²   = 180 kg-m/sec²-m = 180 W

work done for 10 hrs = 180 * 10 * 3600 Joules
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2) m = 1120 kg
     let the  angle of slope = Ф      =>  tan Ф = 1/56
     As the tan Ф is very small, angle Ф also very small.  So Sin Ф is almost equal to tan Ф ie., 1/56.
     The car travels 20 m along the slope in 2 sec. => velocity = 10 m/s along the inclined plane.
    
  Force exerted by the engine along the slope
     = m g Sin Ф + Friction force
     = 1120 kg * 10 m/sec² * 1/56 + 60 N
     = 260 N
Power by the engine = force * velocity = 2600 N-m/s or Watts.
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3)
As the truck is moving with uniform velocity and no acceleration, the force exerted by the engine of truck is equal to the frictional force.
   frictional force = 0.5 kg force / metric tonne  * 100 metric tonnes
                         = 50 kg force = 50 * g  Newtons
                       = 500 Newtons
   Power = force * velocity in the same direction = 500 N * 72 * 5/18 m/sec
       = 10,000 W = 10 kW
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4)  height = h = 120m,   mass = m = 10 quintals = 1000 kg,
     time = t = 1 hr = 3600 sec.

   change in the potential energy of coal = m g h
     power at 100% efficiency = energy / time =  m g h / t = (m/t) g h
            = 1000 kg / 3600sec * 10 m/sec² * 120 m
           = 333.333 Watts
     power at 80% efficiency = power / efficiency = 333.333 / (80/100)
           = 416.66 Watts

     if you calculate the power  with  g = 9.8 m/sec² instead of 10 m/sec²  then you will get the power as  408.33  Watts.