Calculation of Pressure Required to Maintain Flow in Terms of Height of Mercury Column

Given:
- Length of the tube = 4 km = 4000 m
- Radius of the tube = 4 cm = 0.04 m
- Rate of water flow = 20 liters/second

Step 1: Convert the given measurements
- Convert the length of the tube from kilometers to meters: 4 km = 4000 m
- Convert the radius of the tube from centimeters to meters: 4 cm = 0.04 m

Step 2: Calculate the cross-sectional area of the tube
- The cross-sectional area (A) of a tube can be calculated using the formula: A = πr^2, where r is the radius of the tube.
- Substituting the given values, A = π(0.04)^2 = 0.0016π m^2

Step 3: Calculate the volume of water flowing per second
- The volume of water flowing per second can be calculated using the formula: Volume = flow rate × time
- Substituting the given values, Volume = 20 liters/second = 0.02 m^3/second

Step 4: Calculate the velocity of water flow
- The velocity (v) of water flow can be calculated using the formula: v = Volume / A
- Substituting the calculated values, v = (0.02 m^3/second) / (0.0016π m^2) ≈ 3.98 m/second

Step 5: Calculate the pressure required to maintain the flow
- The pressure required to maintain the flow can be calculated using Bernoulli's equation: P + 1/2ρv^2 + ρgh = constant
- Since the tube is horizontal, the height difference (h) between the two ends is zero. Thus, the equation simplifies to: P + 1/2ρv^2 = constant
- Rearranging the equation, P = constant - 1/2ρv^2
- The density of water (ρ) is approximately 1000 kg/m^3
- Substituting the values, P = constant - 1/2 × 1000 kg/m^3 × (3.98 m/second)^2

Step 6: Convert pressure to the height of the mercury column
- The pressure required can be converted to the height of the mercury column using the equation: P = ρgh
- Rearranging the equation, h = P / (ρg)
- The density of mercury (ρ) is approximately 13,600 kg/m^3
- The acceleration due to gravity (g) is approximately 9.8 m/s^2
- Substituting the values, h = (P / (13,600 kg/m^3 × 9.8 m/s^2)) × (1 m / 100 cm)

Step 7: Calculate the final pressure in terms of the height of the mercury column
- Substitute the value of P from step 5 into the equation obtained in step 6 to calculate the final pressure in terms of the height of the mercury column.