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Question: Two stretched membranes of area \[2\,c{{m}^{2}}\] and \[3\,c{{m}^{2}}\] are placed in a liquid at th...

Two stretched membranes of area 2 cm22\,c{{m}^{2}} and 3 cm23\,c{{m}^{2}} are placed in a liquid at the same depth. The ratio of the pressure on them is

& A.\,1:1 \\\ & B.\,2:2 \\\ & C.\,3:3 \\\ & D.\,{{2}^{2}}:{{3}^{2}} \\\ \end{aligned}$$
Explanation

Solution

We will consider the formula that represents the pressure acting a fluid, that is, the sum of the atmospheric pressure and the product of the density, the acceleration due to gravity and the height of the liquid column. Thus, the pressure on the body depends on the atmospheric pressure, the density and height of the fluid and also on the acceleration due to gravity.
Formula used:
P=P0+ρghP={{P}_{0}}+\rho gh

Complete answer:
From the given information, we have the data as follows.
The two stretched membranes have the area of 2 cm22\,c{{m}^{2}} and 3 cm23\,c{{m}^{2}} .
The condition is, the stretched membranes are placed at the same depth.
The pressure on bodies will be proportional to the height.
The pressure on the body is given by the formula as follows.
P=P0+ρghP={{P}_{0}}+\rho gh
Where P0{{P}_{0}} is the atmospheric pressure, ρ\rho is the density, g is the acceleration due to gravity and h is the height of the liquid column.
As the stretched membranes are placed at the same depth, so, the same amount of the atmospheric pressure, the height and the acceleration due to gravity act on them. Even, the membranes are of the same material, so, the density will also be equal. Thus, we have,
P1=P0+ρgh{{P}_{1}}={{P}_{0}}+\rho gh and P2=P0+ρgh{{P}_{2}}={{P}_{0}}+\rho gh
The ratio of these pressures acting on the stretched membranes is given as follows.

& \dfrac{{{P}_{1}}}{{{P}_{2}}}=\dfrac{{{P}_{0}}+\rho gh}{{{P}_{0}}+\rho gh} \\\ & \therefore \dfrac{{{P}_{1}}}{{{P}_{2}}}=\dfrac{1}{1} \\\ \end{aligned}$$ $$\therefore $$ The ratio of the pressure on stretched membranes is 1:1. **Thus, option (A) is correct.** **Note:** As the pressure on the body depends on the atmospheric pressure, the density and height of the fluid and also on the acceleration due to gravity, so, if the bodies are placed at an equal depth, then the amount of the atmospheric pressure acting on them will be equal.