Question

A vertical $U-$ tube has two liquids $1$ and $2$. The heights of liquids columns in both the limbs are $h$ and $2h$, as shown in the figure. If the density of the liquid $1$ is $2\rho$.
(a) Find the density of liquid $2$.
(b) If we accelerate the tube towards right till the heights of liquid columns will be the same, find the acceleration of the tube

Answer 1

a. $P_A=P_B$

$h.\left(2\rho \right).g=2h.{\rho }_2.g$

${\rho }_2=\rho$

b. If accelerated with a

$P_A=P_B+(2h-x)2\rho a+x.\rho .a$

$\Rightarrow x.2\rho g-x\rho g=(2h-x)2\rho a+x\rho a$

$\frac{3h}{2}.2\rho g-\frac{3}{2}h\rho g=(2h-\frac{3}{2}h)2\rho a+\frac{3}{2}h\rho a$

$\frac{3}{2}h\rho g=\rho ah+\frac{3}{2}h\rho a$

$a=\frac{3}{5}g$