How can I derive the hydraulic equation using conservation of energy and work?

[bdolle]

Hey All,

Question about hydraulics. Can't seem to find anyone videos or material to walk me through how to get the formula deltaF = rho*g*(A1+A2)d2

Any takers?

My book states: The conclusion is conservation of evergy. Work is done on the liquid by a small force pushing the liquid through a large displacement. Work is done by the liquid when it lifts the heavy weight through a small distance. a full anaylsis must consider the fact that the graviational potetial energy of the liquid is also changing, so we can't simply equate the output work to the input work. but you can see energy considerations require piston 1 to move farther than piston 2.

As a homework problem you can show that force 1 much increase by ... [insert formula above].

No I can't show it as a homework problem. Been trying for the last 2 hours. I have been searching the internet and can't find any explanations.

Please show me via work and conservation of energy how to derive this formula. Please spell out each step and go slow. Thank you!

 

[Chestermiller]

Can you please provide a diagram showing the force that this equation is supposed to be calculating?

 

[bdolle]

Can you please provide a diagram showing the force that this equation is supposed to be calculating?

Here you go. Let me know if you need more. Thanks.

 

[Chestermiller]

The solution to your homework problem does not involve energetics. This is strictly a hydrostatics problem. The starting point should be equation 5.11, or, in a slightly different form: $$F_1=F_2\frac{A_1}{A_2}+\rho g h A_1$$The question is "how much does the force F1 have to increase to achieve the new height difference d1+h+d2 (where d1 is the amount that the left level moves down)?"

 

[bdolle]

The solution to your homework problem does not involve energetics. This is strictly a hydrostatics problem. The starting point should be equation 5.11, or, in a slightly different form: $$F_1=F_2\frac{A_1}{A_2}+\rho g h A_1$$The question is "how much does the force F1 have to increase to achieve the new height difference d1+h+d2 (where d1 is the amount that the left level moves down)?"

So deltaF = F1 - F1 (where h is changed to (h+d1+d2)).

I get rho*g*A1*(d1+d2)=deltaF. we can sub "d2 = d1A1/A2" like they show in the book... But this will not help us. Instead use "d1=d2A2/A1" and sub this into d1 and we get rho*g*A1*(d2A2/A1+d2) which simplifies to rho*g*d2*(A2+A1) which is what we were looking for.

Got it.

THANKS