Question

(a.) Calculate the absolute pressure at an ocean depth of 1 000 m. Assume the density of seawater is $\mathbf{1030}\mathbf{}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$and the air above exerts a pressure of $\mathbf{101}\mathbf{.}\mathbf{3}\mathbf{}\mathbf{}\mathit{k}\mathit{p}\mathit{a}$.

(b.) At this depth, what is the buoyant force on a spherical submarine having a diameter of 5.00 m?

Short answer

(a) The absolute pressure at an ocean depth is $P=1.00pa$

(b) The buoyant force on a spherical submarine is$B\simeq 19.5\times {10}^{9}N$

Step-by-step solution

Step 1:

The pressure in a fluid at rest varies with depth h in the fluid according to the expression:

$P=P_0+\rho gh$

Where, $P_0$ is the pressure at $h=0$and$\rho$ the density of the fluid is assumed uniform. Pascal’s law states that when pressure is applied to an enclosed fluid, the pressure is transmitted undiminished to every point in the fluid and to every point on the walls of the container.

When an object is partially or fully submerged in a fluid, the fluid exerts on the object an upward force called the buoyant force. According to Archimedes’s principle, the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object:

$B={\rho }_{fluid}g{V}_{fluid}$

Step 2:

Given:

Part(a):

Depthofocean$h=1000m$

Density of water$\rho =1030kg/m^3$

Pressure of air $P_0=101.3kpa$

By using concept and the formula from step (1), we get

$$\begin{gathered} P=P_0+\rho gh \\ P=101.3\times {10}^3+1000\times 1030\times 9.8 \\ P=1.00pa \end{gathered}$$

Step 3:

Given:

Part(b):

Depthofocean$h=1000m$

Density of water$\rho =1030kg/m^3$

Diameter of submarine $D=5m$

Radius of submarine $r=\frac{D}{2}=2.5m$

By using concept and the formula from step (1), we get

$$\begin{gathered} B={\rho }_{fluid}g{V}_{fluid} \\ B={\rho }_{fluid}g\times Ah \\ B={\rho }_{fluid}g\times \pi \times r^2\times h \\ B=101.3\times {10}^3\times 9.8\times 3.14\times {\left(2.5\right)}^2\times 1000 \\ B\simeq 19.5\times {10}^{9}N \end{gathered}$$