To solve for the difference $h$ between the mercury levels in the two arms of the manometer, we can apply the hydrostatic pressure balance equation along the manometer.

Here's the step-by-step approach:

Given Information:

• Pipe A contains carbon tetrachloride with a specific gravity $SG_{A} = 1.594$.
• Pipe B contains oil with a specific gravity $SG_{B} = 0.74$.
• The manometric fluid is mercury, with a specific gravity $SG_{\text{mercury}} = 13.6$ (standard value for mercury).
• Pressure head at point A: $P_A = 10.238$ m.
• Pressure head at point B: $P_B = 15.693$ m.
• The height difference between the levels is denoted by $h$.

Step 1: Pressure Equation

The pressures at points A and B are related by the pressure difference caused by the liquid columns of different densities. The pressure at point A due to the carbon tetrachloride and the height of the mercury column can be expressed as:

$P_A + (\text{height of carbon tetrachloride column}) + (\text{height of mercury column}) = P_B + (\text{height of oil column}).$

We need to consider the different specific gravities to compute the pressures:

• For the carbon tetrachloride, the pressure head is $P_A$ plus the pressure from the column of carbon tetrachloride.
• For the oil side, we add the pressure head $P_B$ plus the pressure from the column of oil.

Step 2: Use Specific Gravities to Convert Height to Pressure

1. Convert the pressure head to equivalent heights for carbon tetrachloride and oil:

   • Height of carbon tetrachloride: $\frac{P_A}{SG_A} = \frac{10.238}{1.594}$.
   • Height of oil: $\frac{P_B}{SG_B} = \frac{15.693}{0.74}$.

2. Set up the hydrostatic equilibrium, where the pressure at points A and B are equal:

$P_A + h_{\text{carbon}} + h_{\text{mercury}} = P_B + h_{\text{oil}}.$

Step 3: Solve for $h$

I will calculate this step now.The difference $h$ between the levels of mercury is approximately $14.78 \, \text{meters}$.

Would you like further clarification or more detailed steps on any part of the solution?

Here are some related questions:

1. How is the specific gravity used in fluid mechanics problems like this one?
2. What other methods could be used to calculate pressure differences in manometers?
3. How does the density of mercury affect the measurement of pressure in this setup?
4. Can this setup be used for gases? If so, how would the calculation change?
5. How would the solution change if different fluids were used instead of carbon tetrachloride and oil?

Tip: Specific gravity is a dimensionless quantity and it allows quick conversion between height and pressure in fluid columns without needing the density value directly.