Water flows at a speed of 15 m/s through a pipe that has a radius of 0.40 m. The water then flows through a smaller pipe at a speed of 45 m/s. What is the radius of the smaller pipe?
The radius of the smaller pipe can be calculated using the equation for the continuity of flow, which states that the flow rate (Q) is equal to the product of the cross-sectional area (A) and the velocity (V).
Q = A * V
Therefore, the radius of the smaller pipe can be calculated as follows:
r = (Q / V) / π
r = (15 m/s * 0.40 m) / (45 m/s * π)
r = 0.13 m
To solve this problem, we can use the principle of conservation of mass. The volume flow rate, which represents the amount of water passing through a given point in a given amount of time, is constant at different points along the flow.
The formula for the volume flow rate (Q) is given by:
Q = A * v
Where:
Q = volume flow rate
A = cross-sectional area of the pipe
v = velocity of the water
We can use this formula to find the cross-sectional area of the pipe at both points and then equate the two volume flow rates to find the radius of the smaller pipe.
Step 1: Calculate the cross-sectional area of the first pipe.
Given:
Radius of the first pipe (r1) = 0.40 m
Using the formula for the area of a circle (A = π * r^2), we can calculate the cross-sectional area (A1) of the first pipe:
A1 = π * r1^2
A1 = π * (0.40)^2
A1 = 0.16π
Step 2: Calculate the volume flow rate of the first pipe.
Given:
Velocity of water in the first pipe (v1) = 15 m/s
Using the formula for volume flow rate (Q = A * v), we can calculate the volume flow rate (Q1) of the first pipe:
Q1 = A1 * v1
Q1 = 0.16π * 15
Q1 = 4.8π
Step 3: Calculate the cross-sectional area of the smaller pipe.
Given:
Velocity of water in the smaller pipe (v2) = 45 m/s
To find the cross-sectional area (A2) of the smaller pipe, we rearrange the formula for the volume flow rate:
A2 = Q2 / v2
Using the volume flow rate from the first pipe (Q1 = 4.8π) and the velocity of the water in the smaller pipe (v2 = 45), we can calculate A2:
A2 = (4.8π) / 45
Step 4: Calculate the radius of the smaller pipe.
To calculate the radius (r2) of the smaller pipe, we rearrange the formula for the area of a circle:
A2 = π * r2^2
r2 = √(A2 / π)
Using the calculated cross-sectional area of the smaller pipe (A2), we can find the radius (r2):
r2 = √((4.8π) / 45π)
r2 = √(4.8 / 45)
r2 ≈ 0.171 m
Therefore, the radius of the smaller pipe is approximately 0.171 m.
To find the radius of the smaller pipe, we can use the principle of the conservation of mass, which states that the mass flow rate must be the same at any point in the pipe. The mass flow rate is given by the product of the density (ρ) of the fluid, the cross-sectional area (A) of the pipe, and the velocity (v) of the fluid:
Mass flow rate = ρ * A * v
Since the two pipes are connected and the water flows continuously, the mass flow rate must be the same for both pipes. Therefore, we can equate the mass flow rate for the two pipes:
ρ1 * A1 * v1 = ρ2 * A2 * v2
Given that the velocity (v1) of the water in the first pipe is 15 m/s, the radius (r1) of the first pipe is 0.40 m, and the velocity (v2) of the water in the second pipe is 45 m/s, we need to find the radius (r2) of the second pipe.
To solve for r2, we rearrange the equation:
ρ1 * A1 * v1 = ρ2 * A2 * v2
We know that the cross-sectional area (A) of a pipe is given by the formula:
A = π * r^2
Substituting the formulas for the cross-sectional area into the equation, we have:
ρ1 * (π * r1^2) * v1 = ρ2 * (π * r2^2) * v2
Now, we can cancel out the common terms (π) and rearrange the equation to solve for r2:
(r2^2) = (ρ1 * r1^2 * v1) / (ρ2 * v2)
Finally, taking the square root of both sides of the equation, we can find the radius (r2) of the smaller pipe:
r2 = √[(ρ1 * r1^2 * v1) / (ρ2 * v2)]
Note: To find the radius, we need to know the densities (ρ) of the fluid in both pipes.