Chapter 6.8, Problem 22ES

(a) Let $D=D_a$ denote a disk of radius $a$ submerged in a fluid of weight density $\rho$ such that the center of $D$ is h units below the surface of the fluid. For each value of $r$ in the interval $\left(0,a\right]$ , let $D_r$ denote the disk of radius $r$ that is concentric with $D$ . Select a side of the disk $D$ and define $P\left(r\right)$ to be the fluid pressure on the chosen side of $D_r$ . Use $\left(5\right)$ to prove that

$\underset{r\to 0^{+}}{\lim }P\left(r\right)=\rho h$

(b) Explain why the result in part (a) may be interpreted to mean that fluid pressure at a given depth is the same in all directions. (This statement is one version of a result known as Pascal’s Principle.)