A manometer consists of two tubes \(A\) and \(B\), with vertical axes and uniform cross-sectional areas \(500 \mathrm{~mm}^{2}\) and \(800 \mathrm{~mm}^{2}\) respectively, connected by a U-tube \(C\) of cross-sectional area \(70 \mathrm{~mm}^{2}\) throughout. Tube \(A\) contains a liquid of relative density \(0.8\); tube \(B\) contains one of relative density \(0.9 .\) The surface of separation between the two liquids is in the vertical side of \(\mathrm{C}\) connected to tube \(A\). What additional pressure, applied to the tube \(B\), would cause the surface of separation to rise \(60 \mathrm{~mm}\) in the tube \(C\) ?

Fluid mechanics is the study of fluids (liquids and gases) and the forces acting upon them. In our manometer problem, it helps us understand how the liquid levels shift when pressure is applied.

The manometer equation relates the pressure difference to the heights and densities of the fluids involved. Here, when pressure is applied to tube B, it causes the liquid levels in tubes A and B to change. This change is a direct result of the principles outlined in fluid mechanics.

A manometer is essentially a tool that measures the difference in fluid pressure. It employs the basic principles of fluid statics, which involves the analysis of how fluids at rest influence the pressure at given points within them. These principles are critical for understanding the effects of relative densities on the liquid column heights in different sections of the manometer.

Pressure difference is the main factor that impacts the liquid levels in the manometer. It's represented by the different levels of liquid columns in the connected tubes.

To calculate the pressure difference causing the surface separation to move, we start by converting the displacement height from millimeters to meters. This height difference results directly from the different densities of the two liquids in tubes A and B: \[ h = 60 \text{ mm} = 0.06 \text{ m} \]

The pressure difference can be calculated using the hydrostatic pressure formula, which is \[ P = \rho g h \] Here, \( \rho \) represents the density of the fluid, \( g \) is the gravitational acceleration (approximately 9.81 m/s²), and \( h \) is the height difference. For the problem at hand:

For liquid in tube A with specific gravity 0.8: \[ P_A = 0.8 \times 1000 \times 9.81 \times 0.06 = 470.88 \text{ Pa} \]
For liquid in tube B with specific gravity 0.9: \[ P_B = 0.9 \times 1000 \times 9.81 \times 0.06 = 529.74 \text{ Pa} \]

The net pressure change, and the additional pressure to be applied, is: \[ P_{\text{additional}} = P_B - P_A = 529.74 \text{ Pa} - 470.88 \text{ Pa} = 58.86 \text{ Pa} \]

Specific gravity, also known as relative density, is a measure of the density of a substance compared to the density of water. It indicates whether a substance will float or sink in water.

In our manometer problem, the specific gravities of the liquids in tubes A and B are 0.8 and 0.9, respectively. This means the liquid in tube B is denser than the liquid in tube A.

Specific gravity is dimensionless, and its formula is: \[ \text{Specific Gravity} = \frac{\text{Density of Substance}}{\text{Density of Water}} \]
Knowing the specific gravity helps us determine the weights of the liquids, which is essential for calculating the pressure exerted by each liquid column:

For liquid in tube A: \[ \rho_A = 0.8 \times 1000 \text{ kg/m}^3 \]

For liquid in tube B: \[ \rho_B = 0.9 \times 1000 \text{ kg/m}^3 \]

These density values allow us to find the pressures that each liquid column exerts. In fluid mechanics, specific gravity plays a key role in understanding buoyancy, pressure changes, and fluid stability within manometers and other fluid systems.