Question

A U-tube containing a liquid is accelerated horizontally with a constant acceleration
${a_0}$ . If the separation between the vertical limbs is $l$ , find the difference in the heights of the liquid in the two arms.

Answer 1

Here, we will use the formula of the pressure to calculate the height of the liquid in the two arms of the container. Pressure is the force that is applied at the right angle to the surface. Firstly, we will calculate the pressure difference when the fluid is accelerated, then we will equate this equation to the common formula of the pressure difference.

FORMULA USED:
The formula used for the pressure in the liquid accelerated with an acceleration $a$ is given by
${P_2} - {P_1} = \rho la$
Here, ${P_2} - {P_1}$ is the difference in pressures, $\rho$ is the density of the liquid, $l$ is the distance between the limbs of the container and $a$ is the acceleration of the liquid.
Also, the common formula of the pressure is given by

$P = {P_2} - {P_1} = h\rho g$

Here, $\rho$ is the density of the liquid, $h$ is the height of the liquid in the two arms of the container and $g$ is the acceleration due to gravity.

COMPLETE STEP BY STEP ANSWER:
When any liquid will be accelerated with an acceleration $a$ , then the difference in the pressures acting on the liquid is given by
${P_2} - {P_1} = \rho la$
Here, $\rho$ is the density of the liquid, $l$ is the distance between the limbs of the container and
$a$ is the acceleration of the liquid.

Now, as given in the question, the liquid is accelerated with an acceleration ${a_0}$ , then the difference in pressures is given by
${P_2} - {P_1} = \rho l{a_0}$

Now, the common formula used for the pressure difference is given by
${P_2} - {P_1} = h\rho g$

Now, equating the above equation of the pressure difference in the liquids to find the height of the liquid, we get
$\rho l{a_0} = h\rho g$
$\Rightarrow \,hg = l{a_0}$
$\Rightarrow \,h = \dfrac{{l{a_0}}}{g}$
Therefore, the difference in the heights of the liquid in the two arms is $\dfrac{{l{a_0}}}{g}$

NOTE: Now, you might think that from where the formula ${P_2} - {P_1} = \rho la$ has come. This is nothing but the common formula of pressure in which height $h$ is replaced with length $l$ and the acceleration of gravity $g$ is replaced with the acceleration $a$ . Therefore, you might not get confused about the formula.