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Dumb_as_a_Duck

An engine can pump 30000 litres of water to a vertical height of 45 m in 10 minutes. Calculate work done by the machine and its power.

  • 2 years ago
  • 2 years ago

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  1. JamesJ
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    The work done is the equal to the change in gravitational potential energy of the water. Calculate that, using PE = mgh. Then Power = (Work)/(Time)

    • 2 years ago
  2. Dumb_as_a_Duck
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    given - [g = 9.8ms ^{-2} \], \[density of water = 1000 kg m ^{-3}\] and \[1000L = 1 m^{3}\]

    • 2 years ago
  3. Dumb_as_a_Duck
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    (that was a part of the question)

    • 2 years ago
  4. Dumb_as_a_Duck
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    how do you convert the given litres of water to kg?

    • 2 years ago
  5. JamesJ
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    This is the beauty of SI units and you can deduce this from the relation you wrote down above: 1 litre of water has a mass of 1 kg.

    • 2 years ago
  6. Dumb_as_a_Duck
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    how did you get that/derive that 1 litre of water has a mas of 1 kg from the relation given?

    • 2 years ago
  7. eashmore
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    He got it from density. Note that 1000 litres are equal to 1 m^3.

    • 2 years ago
  8. Dumb_as_a_Duck
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    and then? (im not seeing the relation or the beauty here, i'm sorry.)

    • 2 years ago
  9. eashmore
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    No problem. Let's use dimensional analysis. \[1000 \left [\rm kg \over m^3 \right ] \cdot \left [ \rm 1 m^3 \over 1000L \right ] = 1 \left [\rm kg \over L \right ]\]

    • 2 years ago
  10. Dumb_as_a_Duck
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    WAIITTTT

    • 2 years ago
  11. eashmore
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    Waiting.

    • 2 years ago
  12. Dumb_as_a_Duck
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    i didnt get a word of that. the thing you wrote under dimensional analysis, i mean. t

    • 2 years ago
  13. Dumb_as_a_Duck
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    if you were gonna explain that, then continue.

    • 2 years ago
  14. Dumb_as_a_Duck
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    i havent learnt about dimensional analysis either.

    • 2 years ago
  15. eashmore
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    It is a fact that 1 m^3 equals 1000 L. Can you accept that? Dimensional analysis is just a method of converting units. So the above relation can be written as \[\rm 1 m^3 \over 1000 L\] If we multiply this by the density that is given\[1000 \left [\rm kg \over m^3 \right ] \cdot \left [\rm 1 m^3 \over 1000 L \right ]\]Notice that \(m^3\) cancel out, and that 1000/1000 = 1

    • 2 years ago
  16. eashmore
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    Therefore, we are left with \[1 \left [ \rm kg \over L \right ]\]

    • 2 years ago
  17. JamesJ
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    Or put another way: if 1 cubic meter of water is 1000 kg, and 1 cubic meter of water holds 1000 liters, it must be that 1 liter has a mass of 1 kg.

    • 2 years ago
  18. Dumb_as_a_Duck
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    Oh! I see :)

    • 2 years ago
  19. Dumb_as_a_Duck
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    that makes the rest of the problem obvious. thanx! :)

    • 2 years ago
  20. eashmore
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    Great! Now, we know that, as James said, \[PE = mgh\]and density is defined as\[\rho = {m \over V}\]We can write mass in terms of density as\[m = \rho V\]Therefore, the potential energy can be written as\[PE = \rho V gh\]

    • 2 years ago
  21. eashmore
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    We know that\[W = \Delta PE\] in this case, and that\[P = W \cdot t\]Therefore, \[P = \rho V ght\]where \(\rho\) is the density of water, \(V\) is the volume of water pumped, \(g\) is the acceleration due to gravity, \(h\) is the height that the pump raises the water, and \(t\) is the time that it takes to pump the given volume of water. Let me know what you get.

    • 2 years ago
  22. Dumb_as_a_Duck
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    isn't P=W/t? (Power = Work/Time)?

    • 2 years ago
  23. JamesJ
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    Yes ... eashmore made a small mistake

    • 2 years ago
  24. eashmore
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    Whoops! Sorry about that. I was just testing you. :-P

    • 2 years ago
  25. Dumb_as_a_Duck
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    oh. is my solution here right? (Given) height= 45 m time = 10 minutes = 600 seconds density of water = 1000 kg m^-3 volume of water = 30,000 litres of water = 30 m^3 (as 1000 L = 1 m^3) (solution) mass of water = density of water * volume of water = 30 * 1000 = 30,000 kg. P.E. = mgh = 30,000 * 9.8 * 45 = 1323 * 10^4 Joule Work done= P.E. = 1323 * 10^4 Joule Power = W/t = 1323 * 10^4/600 = 2205 * 10^3 watts. (or 2205 kW).

    • 2 years ago
  26. eashmore
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    Looks good. Assuming you typed it into your calculator right.

    • 2 years ago
  27. Dumb_as_a_Duck
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    I calculated it myself lol. I'll double check with a calculator.

    • 2 years ago
  28. Dumb_as_a_Duck
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    Thanks a bunch! :)

    • 2 years ago
  29. eashmore
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    Whoops! I got that it should be 22.05 kW.

    • 2 years ago
  30. Dumb_as_a_Duck
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    uhoh! did i make a conversion mistake?

    • 2 years ago
  31. eashmore
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    Nope. Everything is good up until you calculate power. Double check that guy and you should be good.

    • 2 years ago
  32. Dumb_as_a_Duck
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    I got your answer (22.05 kW) on the calculator, but that's not what i got when I quadruple-checked my calculation.

    • 2 years ago
  33. Dumb_as_a_Duck
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    ><

    • 2 years ago
  34. eashmore
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    I hate when this happens. The math gods of WolframAlpha say that it is 22.05 kW. http://www.wolframalpha.com/input/?i=30000*45*9.8%2F600+W+in+kW

    • 2 years ago
  35. Dumb_as_a_Duck
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    i mean, i got 22.05 kW when i first multiplied 1323 * 10^4 and THEN divided by 600....but when i divided 1323 by 600 AND then multiplied by 10^4, its the other wrong answer. lol i suck at math too. :P

    • 2 years ago
  36. Dumb_as_a_Duck
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    well, maybe i should just listen to the math gods in silence and not be an atheist. :P

    • 2 years ago
  37. eashmore
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    You should get the same answer with both ways that you have described. Maybe you are misinterpreting the results you get.

    • 2 years ago
  38. Dumb_as_a_Duck
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    I found my mistake! I should have divided 1323 by 600, not just 6...i forgot about that :P *sheepish look*.

    • 2 years ago
  39. Dumb_as_a_Duck
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    I salute you for sticking with me til the end. :D *high five* :P

    • 2 years ago
  40. eashmore
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    No problem. That is why we are here. I salute you for doing the same. Most users simply post all their homework questions, go eat dinner, and return to copy answers. Thank you for being engaged and striving to learn. Glad I could help.

    • 2 years ago
  41. supercrazy92
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    I salute both of you.. Great work!

    • 2 years ago
  42. Dumb_as_a_Duck
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    LOL! :D

    • 2 years ago
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