Question

Figure shows a container filled with a liquid of density $\rho$. Four points A, B, C and D lie on the vertices of a vertical square. Points A and C lie on a vertical line and points B and D lies on a horizontal line. Choose the correct statement(s) about the pressure at the four points

• A. $P_D=P_B$
• B. $P_A<P_B=P_D<P_C$
• C. $P_D=P_B=\frac{P_C-P_A}{2}$
• D. $P_D=P_B=\frac{P_C+P_A}{2}$

Answer 1

Answer: (A), (B), (D)

$P_D=P_B$
$P_A<P_B=P_D<P_C$
$P_D=P_B=\frac{P_C+P_A}{2}$

Lets say depth of A, B, C and D are $h_A$, $h_B$, $h_C$ and $h_D$, then we have
$P_A=P_o+h_A\rho g{P}_B=P_o+h_B\rho g{P}_C=P_o+h_C\rho g{P}_D=P_o+h_D\rho g\because h_B=h_D$
$\Rightarrow P_D=P_B$ $\Rightarrow$ option A is correct.
$\because h_A<h_B=h_D<h_C$
$\Rightarrow P_A<P_B=P_D<P_C$ $\Rightarrow$ option B is correct.
Since ABCD is a square, So we can have
$h_D=h_B=\frac{h_C+h_A}{2}\Rightarrow h_D\rho g=h_B\rho g=\frac{h_C+h_A}{2}\rho g\Rightarrow P_o+h_D\rho g=P_o+h_B\rho g=P_o+\frac{h_C+h_A}{2}\rho g\Rightarrow P_D=P_B=\frac{P_o}{2}+\frac{P_o}{2}+\frac{h_C}{2}\rho g+\frac{h_A}{2}\rho g\Rightarrow P_D=P_B=\frac{P_o}{2}+\frac{h_C\rho g}{2}+\frac{P_o}{2}+\frac{h_A\rho g}{2}\Rightarrow P_D=P_B=\frac{(P_o+h_C\rho g)+(P_o+h_A\rho g)}{2}$
$\Rightarrow P_D=P_B=\frac{P_C+P_A}{2}$ $\Rightarrow$ option D is correct.
Lets say option 'C' is also correct
$\Rightarrow P_D=P_B=\frac{P_C+P_A}{2}=\frac{P_C-P_A}{2}\Rightarrow \frac{P_C+P_A}{2}=\frac{P_C-P_A}{2}\Rightarrow \frac{P_C}{2}+\frac{P_A}{2}=\frac{P_C}{2}-\frac{P_A}{2}\Rightarrow \frac{P_A}{2}=-\frac{P_A}{2}$
$\Rightarrow P_A=0$, (this is not true) $\Rightarrow$ option C is incorrect.