Water is flowing in a pipe as depicted in the figure. As shown, p = 117 kPa, d = 4.67 cm, d' = 3.06 cm, v = 4.36 m/s, and h = 1.11 m. What pressure is indicated on the upper pressure gauge? Take y = 0 at the bottom of the left pipe so that the center of the left pipe is y1 = d/2.
See my response to "Physics please help!!" below
To determine the pressure indicated on the upper pressure gauge, we can apply Bernoulli's equation, which relates the pressure, velocity, and height of a fluid flowing in a pipe.
Bernoulli's equation is given as:
P1 + (1/2)ρv1^2 + ρgy1 = P2 + (1/2)ρv2^2 + ρgy2
Where:
P1 and P2 are the pressures at two different points in the fluid (in this case, at the bottom of the left pipe and at the location of the upper pressure gauge),
ρ is the density of the fluid (water, in this case),
v1 and v2 are the velocities at two different points in the fluid,
g is the acceleration due to gravity,
y1 and y2 are the heights of the two points relative to a reference level (in this case, y1 is the center of the left pipe).
We are given the following values:
P1 = 117 kPa (pressure at the bottom of the left pipe),
d = 4.67 cm (diameter of the left pipe),
d' = 3.06 cm (diameter of the right pipe),
v = 4.36 m/s (velocity),
h = 1.11 m (height difference),
y1 = d/2 (center of the left pipe).
First, let's convert the diameter and velocity to the appropriate units:
d = 4.67 cm = 0.0467 m
d' = 3.06 cm = 0.0306 m
v = 4.36 m/s
Using these values, we can now apply Bernoulli's equation:
P1 + (1/2)ρv1^2 + ρgy1 = P2 + (1/2)ρv2^2 + ρgy2
Since we are solving for the pressure indicated on the upper pressure gauge (P2), we need to rearrange the equation:
P2 = P1 + (1/2)ρ(v1^2 - v2^2) + ρg(y1 - y2)
Now, let's substitute the given values into the equation and calculate:
P2 = 117 kPa + (1/2)(1000 kg/m^3)(4.36 m/s)^2 - (1000 kg/m^3)(9.8 m/s^2)(0.0467 m/2 - 0.0306 m)
P2 = 117 kPa + (1/2)(1000 kg/m^3)(19.0096 m^2/s^2) - (1000 kg/m^3)(9.8 m/s^2)(0.00735 m)
P2 = 117 kPa + 9504.8 Pa - 68.235 Pa
P2 = 126.4 kPa
Therefore, the pressure indicated on the upper pressure gauge is approximately 126.4 kPa.