Gravitational Potential Energy of Liquid: Why Does It Increase?

[zorro]

Why does the gravitational potential energy of a non-viscous liquid increase when a body falls through it?

 

[Dale]

Because as the body falls the liquid rises.

 

[zorro]

I missed a word in the question. Why does it increase continuously?

 

[Dale]

The center of mass of the liquid increases continuously as the body sinks through the liquid.

Consider a vertical tube of water and a cylindrical "plug" that is just barely smaller diameter than the cylinder so that you can neglect the water next to the plug and just consider the water above and below the plug. It is then fairly easy to derive an expression for the center of mass of the water as a function of the height of the plug and see that it increases continuously as the plug sinks.

If you need help, I will be glad to walk you through it.

 

[zorro]

I tried finding out the expression but didn't get through. I am ending up with many variables. Can you give some hints?

 

[Dale]

Why don't we say that A is the area of the cylinder, H is the height of the column of fluid without the plug, and h is the height of the plug, d is the distance from the bottom of the column to the bottom of the plug, and p is the density of the fluid. That is all of the variables that you need to determine the height of the center of mass.

First, split the fluid into a cylinder above the plug and a cylinder below the plug. Second, find the mass of each cylinder, m1 and m2, in terms of the above variables. Third, find the height of the center of each cylinder, h1 and h2, in terms of the above variables. Fourth, write the center of mass in terms of the above variables (eliminating m1, m2, h1, and h2).

If you get stuck on any of the steps let me know.

 

[zorro]

I got it perfectly. Thank you very much!