Question

The area of cross-section of the wider tube shown in figure is $800c{m^2}$ . If a mass of $12kg$ is placed on the massless piston, the difference in heights h in the level of water in the two tubes is :
$(A)10cm$
$(B)6cm$
$(C)15cm$
$(D)2cm$

Answer 1

A piston is a disk or cylindrical part tightly fitting and moving within a cylinder, either to compress or move a fluid collected in the cylinder, as air or water, or to transform energy imparted by a fluid entering or expanding inside the cylinder, as compressed air, explosive gases, or steam, into a rectilinear motion usually transformed into rotary motion by means of a connecting rod.

Complete step by step solution:
Firstly, we need to understand what is the function of a Piston. In an engine, its purpose is to transfer force from expanding gas in the cylinder to the crankshaft via a piston rod or connecting rod. In a pump, the function is reversed and force is transferred from the crankshaft to the piston for the purpose of compressing or ejecting the fluid in the cylinder.
Given,
$A = 800c{m^3}$
$m = 12kg = 12000gm$
Here, the force on the piston is $F = mg$
Hence, we can say that the increase in the pressure on the liquid in the wider tube is
$P = \dfrac{F}{A}.......(1)$
We also know that $F = mg$
So, we can write equation (1) as,
$P = \dfrac{{mg}}{A}$
Since, h is the difference in the level of water in the two tubes,
$P = h\rho g$
$h = \dfrac{P}{{\rho g}}$
The above equation can also be written as,
$h = \dfrac{{mg}}{{A\rho g}} = \dfrac{m}{{A\rho }}$
Now, by putting the values, we get,
$h = \dfrac{{12000}}{{800}}.......(\because \rho = 1)$
$h = 15cm$
Therefore, the correct answer is $(C)15cm$

Note:
It is important to note that this question needs to be solved in the CGS unit system. In fluid mechanics, the symbol $\rho$ represents fluid density and the fluid density $\rho$ for water is always $1\dfrac{{gm}}{{c{m^3}}}$ . But this value changes with the change in temperature or if some other substance is dissolved in the water.