A pipe 40 cm in diameter conveying water at 3 m/s, branches into two pipes having diameter of 20 cm and 30 cm. What will the velocity in 20 cm diameter pipe if velocity in 30 cm diameter pipe is 1 m/s?a)8 m/sb)9.75 m/sc)10.5 m/sd)5.25 m/sCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Answer

Given data:
Diameter of the main pipe (D1) = 40 cm
Velocity in the main pipe (V1) = 3 m/s
Diameter of the first branch pipe (D2) = 20 cm
Velocity in the first branch pipe (V2) = ?

Using the principle of conservation of mass:
The mass flow rate of water in the main pipe must be equal to the sum of the mass flow rates in the two branch pipes.

Calculation:

Step 1: Convert the diameters to meters:
Diameter of the main pipe (D1) = 40 cm = 0.4 m
Diameter of the first branch pipe (D2) = 20 cm = 0.2 m

Step 2: Calculate the cross-sectional areas:
Area of the main pipe (A1) = (π/4) * D1^2
Area of the first branch pipe (A2) = (π/4) * D2^2

Area of the main pipe (A1) = (π/4) * (0.4)^2 = 0.125π m^2
Area of the first branch pipe (A2) = (π/4) * (0.2)^2 = 0.025π m^2

Step 3: Calculate the velocities:
Using the equation Q = A * V, where Q is the discharge, A is the cross-sectional area, and V is the velocity.

Discharge in the main pipe (Q1) = A1 * V1
Discharge in the first branch pipe (Q2) = A2 * V2

Q1 = 0.125π * 3 = 0.375π m^3/s
Q2 = 0.025π * 1 = 0.025π m^3/s

Step 4: Apply the principle of conservation of mass:
The sum of the discharges in the branch pipes must be equal to the discharge in the main pipe.

Q1 = Q2 + Q3
0.375π = 0.025π + Q3
Q3 = 0.35π m^3/s

Step 5: Calculate the velocity in the second branch pipe:
V3 = Q3 / A3

Area of the second branch pipe (A3) = (π/4) * D3^2

Diameter of the second branch pipe (D3) = ?
Let's assume the velocity in the second branch pipe (V3) is 1 m/s to find the diameter of the second branch pipe.

V3 = Q3 / A3
1 = 0.35π / A3
A3 = 0.35π m^3/s

A3 = (π/4) * D3^2
0.35π = (π/4) * D3^2
D3^2 = (0.35 * 4)
D3 = sqrt(1.4) m

Diameter of the second branch pipe (D3) = sqrt(1.4) m

Step 6: Calculate the velocity in the second branch pipe:
Using the equation

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