Question

Figure shows a container filled with a liquid of density $\rho$. Four points $A,B,C\text{and}D$ lie on the diametrically opposite points of a circle as shown. Points $A$ and $C$ lie on a vertical line and points $B$ and $D$ lie on a horizontal line. $P_A,P_B,P_C,P_D$ are the absolute pressures at the respective points. Then, the correct statement(s) is/are :

• 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)

The correct options are
A $P_D=P_B$
B $P_A<P_B=P_D<P_C$
D $P_D=P_B=\frac{P_C+P_A}{2}$

As we know,
Points at the same height in the same liquid have the same pressure.
Thus, $P_B=P_D$
Points with height difference, say ${}^{\prime }{h}^{\prime }$, will have a difference of pressure of $\rho gh$.


From figure,
$P_A=P_0+\rho gh$ ...(i)
$P_B=P_D=P_0+\rho g(h+r)$ ...(ii)
$P_C=P_0+\rho g(h+2r)$ ...(iii)
where $r$ is the radius of the circle.

By eqn (i)+(ii)
$P_C+P_A=[P_0+\rho g(h+2r\left)\right]+[P_0+\rho gh]$
$\Rightarrow P_C+P_A=P_0+\rho gh+\rho g\left(2r\right)+P_0+\rho gh$
$\Rightarrow P_C+P_A=2[P_0+\rho g(h+r\left)\right]$
$\Rightarrow \frac{P_C+P_A}{2}=P_0+\rho g(h+r)$
i.e $\frac{P_C+P_A}{2}=P_B=P_D$
i.e., option (1), (2) and (4) gives correct statement, but statement (3) is incorrect.