In analyzing certain geological features, it is often appropriate to assume that the pressure at some horizontal level of compensation, deep inside Earth, is the same over a large region and is equal to the pressure due to the gravitational force on the overlying material. Thus, the pressure on the level of compensation is given by the fluid pressure formula. This model requires, for one thing, that mountains have roots of continental rock extending into the denser mantle (Figure). Consider a mountain of height $\mathit{H}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{km}$ km on a continent of thickness $\mathit{T}\mathbf{=}\mathbf{32}\mathbf{}\mathit{k}\mathit{m}$. The continental rock has a density of$\mathbf{2}\mathbf{.}\mathbf{9}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ , and beneath this rock the mantle has a density of $\mathbf{3}\mathbf{.}\mathbf{3}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ . Calculate the depth of the root. (Hint: Set the pressure at points a and b equal; the depth y of the level of compensation will cancel out.)

In Figure, the fresh water behind a reservoir dam has depth $\mathit{D}\mathbf{=}\mathbf{15}\mathbf{}\mathbf{m}$. A horizontal pipe $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{cm}$ in diameter passes through the dam at depth $\mathit{d}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}$. A plug secures the pipe opening.

(a) Find the magnitude of the frictional force between plug and pipe wall.

(b) The plug is removed. What water volume exits the pipe in $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{h}$ ?

A fish maintains its depth in fresh water by adjusting the air content of porous bone or air sacs to make its average density the same as that of the water. Suppose that with its air sacs collapsed, a fish has a density of $\mathbf{1}\mathbf{.}\mathbf{08}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ . To what fraction of its expanded body volume must the fish inflate the air sacs to reduce its density to that of water?

A block of wood floats in fresh water with two-thirds of its volume V submerged and in oil with $\mathbf{0}\mathbf{.}\mathbf{90}\mathbf{}\mathbf{V}$ submerged. (a) Find the density of the wood. (b) Find the density of the oil.

In Fig. 14-39a, a rectangular block is gradually pushed face-down into a liquid. The block has height $\mathbf{d}$; on the bottom and top the face area is $\mathbf{A}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{67}\mathbf{}{\mathbf{cm}}^{\mathbf{2}}$. Figure 14-39b gives the apparent weight ${\mathbf{\text{W}}}_{\mathbf{app}}$ of the block as a function of the depth h of its lower face. The scale on the vertical axis is set by ${\mathbf{W}}_{\mathbf{s}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{}\mathbf{N}$. What is the density of the liquid?

Suppose that your body has a uniform density of$\mathbf{}\mathbf{0}\mathbf{.}\mathbf{95}$ times that of water. (a) If you float in a swimming pool, what fraction of your body's volume is above the water surface? Quicksand is a fluid produced when water is forced up into sand, moving the sand grains away from one another so they are no longer locked together by friction. Pools of quicksand can form when water drains underground from hills into valleys where there are sand pockets. (b) If you float in a deep pool of quicksand that has a density $\mathbf{1}\mathbf{.}\mathbf{6}$ times that of water, what fraction of your body's volume is above the quicksand surface? (c) Are you unable to breathe?