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I have a question that has been bugging for quite some time now. The quesiton is from a textbook by Bostock and Chandler on Mechanics.

It says:

Water is being raised by a pump from a storage tank $4m$ below ground and delivered at $8 m/s$ through a pipe at ground level. If the cross sectional area of the pipe is $0.12m^2$ find the work done per second by the pump (1 cubic metre of water has a mass of 1000kg).

The answer they got was $68352 \ J/s$. What's bugging me is that I think they considered the water being pumped as a single rigid body, since when calculating the P.E of the water they did not consider the distance moved by each single layer of water like the way it is calculated from the calculus perspective, and I'm confused, as I thought that non-rigid bodies like water require only calculus for questions like these (Which is which?) and I'm kind of wondering from where the water gets the energy to move in a horizonal path in the pipe at ground level. I first thought that there was a force acting horizontally, but I no longer think that is true; or is it? Sorry for the useless information. Help will be greatly appreaciated.

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  • $\begingroup$ Please, write your question properly. Separate it in paragraphs and use Mathjax (laTex notation) using dollar symbols. $\endgroup$
    – FGSUZ
    Commented Aug 3, 2018 at 21:07

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The short answer is that you only need calculus if you want to know how the water moves.

However, if you don't care about how, just the pumping rate, then a basic approach is enough.

The thing is that there is a pump raising "mass" (of water, but mass after all) from one position to another. The way that mass moves is not that relevant. However, you need some assumptions:

  • The lifting must be done so that friction is negligible.
  • That includes friction with walls, and also friction withing the water.
  • The flux goes upwards in a "beautiful way", so that energy is only invested in acceleration of the fluid and fighting the gravity, but no energy is destinated to form vortices or any other dissipation mechanism.

With those assumptions, there's no real difference with raising a solid mass.

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  • $\begingroup$ Thank you. What about the horizontal movement of the water at ground level. Where does the kinetic energy come from? $\endgroup$
    – Energy
    Commented Aug 3, 2018 at 21:38
  • $\begingroup$ Pressure difference absorbs water into the pump. $\endgroup$
    – FGSUZ
    Commented Aug 3, 2018 at 21:40
  • $\begingroup$ I'm sorry, but i am having a hard time trying to understand your last statement. How is water moving at ground level affected by the pump which is at the bottom of the tank? $\endgroup$
    – Energy
    Commented Aug 3, 2018 at 22:07
  • $\begingroup$ Oh I forgot it was at the bottom. If so, the pressure gradient adds to the gravitational attraction, main reason for which water goes downwards. $\endgroup$
    – FGSUZ
    Commented Aug 3, 2018 at 22:41

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