A horizontal pipe (Venturi Tube) 10.0 cm in diameter has a smooth reduction to a pipe 5.00 cm in diameter. If the pressure of the water in the larger pipe is 2.0 x 105 Pa and the pressure in the smaller pipe is 5.0 x 104 Pa, at what rate does water flow through the pipes. Hint: Use the continuity equation to express one speed in terms of the another.
To find the rate at which water flows through the pipes, we can use the continuity equation, which states that the mass flow rate through a pipe is constant. In other words, the product of the area of the pipe and the velocity of the water is constant.
Let's denote the velocity in the larger pipe as v1 and the velocity in the smaller pipe as v2. According to the continuity equation, we have:
A1 * v1 = A2 * v2
Where A1 and A2 are the cross-sectional areas of the larger and smaller pipes, respectively.
Given that the diameter of the larger pipe is 10.0 cm, we can calculate its radius (R1):
R1 = 10.0 cm / 2 = 5.0 cm = 0.05 m
The cross-sectional area of the larger pipe (A1) can be calculated using the formula for the area of a circle:
A1 = π * (R1)^2
Given that the diameter of the smaller pipe is 5.00 cm, we can calculate its radius (R2):
R2 = 5.00 cm / 2 = 2.50 cm = 0.025 m
The cross-sectional area of the smaller pipe (A2) can be calculated using the same formula:
A2 = π * (R2)^2
Now that we have the values for A1, A2, v1, and v2 in the equation A1 * v1 = A2 * v2, we can rearrange the equation to solve for the rate at which water flows:
v2 = (A1 * v1) / A2
Substituting the given values, we have:
v2 = [(π * (0.05 m)^2) * v1] / [π * (0.025 m)^2]
Simplifying the equation further, we can cancel out the π and simplifying the expression within the square brackets:
v2 = [(0.05 m)^2 * v1] / [(0.025 m)^2]
Finally, the value of v2 can be calculated using the given velocities:
v2 = [(0.05 m)^2 * v1] / [(0.025 m)^2]
= 4 * v1
So the rate at which water flows through the smaller pipe (v2) is four times the rate at which water flows through the larger pipe (v1).
Therefore, the rate at which water flows through the smaller pipe is four times the velocity of water in the larger pipe.