Question

Figure shows a container filled with a liquid of density $\rho$. Four points $A,B,C$ and $D$ lie on the diametrically opposite points of a circle as shown. Point $A$ and $C$ lie on vertical and point $B$ and $D$ lie on horizontal line. The incorrect statement is ($P_A,P_B,P_C,P_D$ are absolute pressure of the respective points).

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

Answer 1

The correct option is B $P_D=P_B=\frac{P_C-P_A}{2}$


Points at same height have same pressure and point with height difference say ${}^{\prime }{h}^{\prime }$ will have the difference of $\rho gh$.

Let the radius of circle ${}^{\prime }{r}^{\prime }$.
Pressure at point $A$,
$P_A=P_o+\rho gh...\left(i\right)$
Pressure at points $B$ and $D$,
$P_B=P_D=P_o+\rho g(h+r)...\left(ii\right)$
Pressure at point $C$,
$P_C=P_o+\rho g(h+2r)...\left(iii\right)$

Now, adding equation $\left(i\right)$ & $\left(iii\right)$
$P_A+P_C=(P_o+\rho gh)+P_o+\rho g(h+2r)$

$\Rightarrow P_A+P_C=2[P_0+(h+r\left)\rho g\right]$

$\Rightarrow \frac{P_C+P_A}{2}=P_0+(h+r)\rho g$
So, from eq. $\left(ii\right)$,
$\Rightarrow P_B=P_D=\frac{P_C+P_A}{2}$
& $P_A<P_B,P_D<P_C$

So, option (a), (c), (d) are correct but option (b) is incorrect.