Question

A tank $5m$ high is half-filled with water and then is filled to the top with oil of density $0.85\,g\,c{m^{ - 3}}$ . The pressure at the bottom of the tank, due to these liquid is:
A. $1.85\,g\,dyne\,c{m^{ - 2}}$
B. $89.25\,g\,dyne\,c{m^{ - 2}}$
C. $462.5\,g\,dyne\,c{m^{ - 2}}$
D. $500\,g\,dyne\,c{m^{ - 2}}$

Answer 1

Here, we will use the direct formula of pressure at the depth $h$ of the liquid of density $\rho$ to calculate the pressure at the bottom of the tank that is filled with water and oil. Here, we will add both the pressures due to the water and oil.

Complete step by step answer:
The height of the tank is $5\,m$ . as the tank is half filled with water and half-filled with oil therefore, the height of the water and the oil in the tank will be $2.5\,m$. Therefore, the height of water and the oil is,
$h = \,2.5 \times 100\,cm$
$\Rightarrow h = 250\,cm$

Now, as given in the question, the density of oil ${\rho _{oil}} = 0.85\,g\,c{m^{ - 3}}$
Also, the density of water ${\rho _{water}} = 1\,g\,c{m^{ - 3}}$
Now, the pressure acting at the bottom of the tank due to water is ${P_{water}}$ and the pressure acting at the bottom of the tank due to the oil is ${P_{oil}}$ .
Now, using the formula of the pressure $P$ at the depth $h$ of the liquid of density $\rho$, which is given by
$P = h\rho g$
Here, $g$ is the acceleration due to gravity.

Therefore, in case of both the oil and the water, the formula will become
${P_{air}} = h{\rho _{air}}g$
$\Rightarrow {P_{water}} = h{\rho _{water}}g$
Now, the total pressure acting at the bottom of the tank is given by
${P_{total}} = {P_{air}} + {P_{water}}$
$\Rightarrow \,{P_{total}} = h{\rho _{air}}g + h{\rho _{water}}g$
Now, putting the values of height and densities in the above equation, we get
${P_{total}} = 250 \times 0.85 \times g + 250 \times 1 \times g$
$\Rightarrow \,{P_{total}} = 212.5\,g + 250\,g$
$\therefore \,{P_{total}} = 462.5\,g$

Hence, the total pressure at the bottom of the tank is $462.5\,g\,dyne\,c{m^{ - 3}}$ .Hence, option C is the correct option.

Note: Here, the pressure on the water and the oil is summed up. This is because the pressure exerted by the oil to the water and this pressure will be transported to the bottom of the tank and it is along the individual pressure of the water at the bottom. As a result, the pressure at the bottom of the tank will be the result of both the pressures.