TOPIC: Average Depth Theorem Now, consider a Theorem I discovered a few semester

Question

TOPIC: Average Depth Theorem

Now, consider a Theorem I discovered a few semesters ago while teaching Math 182. I hesitate to name this theorem because I don’t know that someone else discovered it and gave it a name, but here goes:

Theorem: The hydrostatic force acting on a uniform laminar plate is equal to the product of the density of the fluid, the average depth of the submerged portion of the plate and the area of that submerged portion.

Proof:

Suppose we have a plate submerged or partially submerged in a fluid with weight density such that the plate is perpendicular to the surface of the fluid. Suppose further that the top of the submerged portion is at y = a from the surface of the fluid and the bottom is at y = b below the surface of the fluid.

The hydrostatic force acting on the submerged portion of the plate is:

Explanation / Answer

Ans)

The hydrostatic force acting on the submerged portion of the plate is calculated as

Given, density = 64.6 , Area = a^2 , Avg. depth = sqrt(2)a

force = density*Area* Avg. depth = 91.36 A^3

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