A figure shows a section of a long tube that narrows near its open end to a diameter of 1.0 mm. Water at 20 degrees celsius flows out of the open end at 0.020 L/s. What is the gauge pressure at point P, where the diameter is 4.0 mm?
To find the gauge pressure at point P, we can use Bernoulli's principle, which relates the pressure, velocity, and height of a fluid at different points in a flow.
Bernoulli's principle states that the total energy per unit volume of a flowing fluid is constant along a streamline. This means that the sum of the pressure energy, kinetic energy, and gravitational potential energy per unit volume is the same at any two points in the flow.
In this problem, we can assume the water flow is steady and incompressible, so the Bernoulli equation can be simplified to:
P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at points 1 (near the open end) and 2 (at point P), respectively.
ρ is the density of water (1000 kg/m^3).
v1 and v2 are the velocities of water at points 1 and 2, respectively.
g is the acceleration due to gravity (9.8 m/s^2).
h1 and h2 are the heights of points 1 and 2, respectively (assuming the height difference is negligible).
We need to solve for P2, the pressure at point P. We are given the diameter at point 1 (1.0 mm) and point 2 (4.0 mm), and the flow rate at point 1 (0.020 L/s).
First, let's find the velocity at point 1:
v1 = Q/A1
Where Q is the flow rate (0.020 L/s) and A1 is the cross-sectional area at point 1.
A1 = πr1^2
Where r1 is the radius at point 1 (0.5 mm).
Next, let's find the velocity at point 2:
v2 = Q/A2
Where A2 is the cross-sectional area at point 2.
A2 = πr2^2
Where r2 is the radius at point 2 (2.0 mm).
Now, substituting these values into the Bernoulli equation and solving for P2, we can find the gauge pressure at point P.