Question

A U-tube differential manometer connects two pressure pipes A and B. Pipe A contains carbon tetrachloride having a specific gravity 1.594 under a pressure of 11.772$N/c{m^2}$ and pipe B contains oil of sp. gr.0.8 under a pressure of 11.772 $N/c{m^2}$. The pipe A lies 2.5m above pipe B. Find the difference of pressure measured by mercury as fluid filling U-tube.

Answer 1

In this question, we will discuss the basics of pressure and then use the relation between pressure, height, density and gravity. By substituting the given values in the equation, we will get the required result. Also, we will study about Archimedes' principle and Bernoulli’s theorem, for our better understanding.
Formula used:
$P = \rho gh$

Complete step by step solution:
As we know that, pressure is defined as the force exerted on an object per unit area. Unit of pressure is Pascal which is represented as P. The unit of pressure can also be written as Newton per meter square.
Here, we have the formula for pressure, which is given as:
${P_A} = \rho gh$
${P_B} = \rho g(h + 2.5)$
Now, equating the above two equations, we get:
${\rho _A}gh = {\rho _B}g(h + 2.5)$
$\Rightarrow {\rho _A}h = {\rho _B}(h + 2.5)$
\eqalign{& \Rightarrow 1.594h = 0.8(h + 2.5) \cr & \Rightarrow 0.0794h = 2 \cr}
Further by solving the equation for height h, we get:
\eqalign{& \Rightarrow h = 0.296518cm \cr & \Rightarrow h = 29651\% \cr & \therefore h = 29.65cm \cr} Of mercury
Therefore, we get the required result.

Additional information:
As we know, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.
Also, Archimedes principle states that an object immersed in a fluid experiences some buoyant force that is equal in magnitude to the force of gravity on the displaced fluid. It is also known as the law of buoyancy. This principle was discovered by the ancient Greek mathematician and inventor Archimedes
The weight of the displaced fluid is equal to the subtraction of weight of object in vacuum and weight of object in fluid.
The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density).
The buoyant force on a body floating in a liquid or gas is also equivalent in magnitude to the weight of the floating object and is opposite in direction. So, in this situation the object neither rises nor sinks.
We should also know about the buoyant force. This is an upward force exerted by a fluid which opposes the weight of a partially or fully immersed object in the fluid. Here, in the fluid, pressure increases with depth as a result of the weight of the overlying fluid. So, the pressure at the bottom of a column of fluid is greater than at the top of the column.

Note:
One should notice that the Archimedes principle is only valid for fluids, where buoyant force can be observed. Also, the displaced fluid is equal to the weight of the object immersed in the water. In the displaced fluid, gravity also plays an important role.