Water flowing out of a horizontal pipe emerges through a nozzle. The radius of the pipe is 1.5 cm, and the radius of the nozzle is 0.42 cm. The speed of the water in the pipe is 0.68 m/s. Treat the water as an ideal fluid, and determine the absolute pressure of the water in the pipe.
To determine the absolute pressure of the water in the pipe, we can use Bernoulli's equation. Bernoulli's equation states that the total pressure of a fluid is constant along a streamline, and it can be expressed as:
P + 1/2 * ρ * v^2 + ρ * g * h = constant
Where:
P is the pressure of the fluid
ρ is the density of the fluid
v is the velocity of the fluid
g is the acceleration due to gravity
h is the height above a reference point
Since the water is flowing horizontally, we can assume that the height is constant and neglect the term involving the height in the equation.
Next, we need to find the pressure at two points in the pipe: at the pipe's radius (r1 = 1.5 cm) and at the nozzle's radius (r2 = 0.42 cm).
Let's calculate the pressures at both points:
At the pipe's radius (r1):
P1 + 1/2 * ρ * v1^2 = constant
At the nozzle's radius (r2):
P2 + 1/2 * ρ * v2^2 = constant
Now, let's calculate the velocities at both points:
The speed of the water in the pipe (v1) = 0.68 m/s
Since the water is flowing through a nozzle, the flow rate should be conserved. Therefore, the velocity at the nozzle (v2) can be calculated using the equation:
A1 * v1 = A2 * v2
Where A1 is the cross-sectional area of the pipe and A2 is the cross-sectional area of the nozzle. We can use the formula for the area of a circle: A = π * r^2.
Therefore, the cross-sectional areas are:
A1 = π * (r1)^2
A2 = π * (r2)^2
Substituting the given values into the equation, we get:
A1 = π * (1.5 cm)^2
A2 = π * (0.42 cm)^2
Finally, let's calculate the absolute pressure at the pipe's radius (P1) using the equation:
P1 = P2 + 1/2 * ρ * (v2^2 - v1^2)
Substituting all the values we have calculated, we can determine the absolute pressure of the water in the pipe.
To determine the absolute pressure of the water in the pipe, we can make use of Bernoulli's equation. Bernoulli's equation describes the conservation of energy for an ideal fluid flowing in a streamline manner.
The equation can be written as:
P + 1/2 * ρ * v^2 + ρ * g * h = constant
Where:
P is the pressure of the fluid
ρ is the density of the fluid
v is the velocity of the fluid
g is the acceleration due to gravity
h is the height of the fluid above a reference point
In this case, we can consider the water to be at the same height throughout the pipe and nozzle, so the term ρ * g * h is the same for both.
If we focus on the two terms related to pressure and velocity (P1, v1, for the pipe and P2, v2 for the nozzle), we can compare them using Bernoulli's equation:
P1 + 1/2 * ρ * v1^2 = P2 + 1/2 * ρ * v2^2
Since we want to determine the pressure in the pipe (P1), we rearrange the equation:
P1 = P2 + 1/2 * ρ * v2^2 - 1/2 * ρ * v1^2
Given the values in the problem statement, let's calculate the pressure:
P1 = P2 + 1/2 * ρ * v2^2 - 1/2 * ρ * v1^2
First, let's find the densities of the water in the pipe and nozzle. The density of water is approximately 1000 kg/m^3.
ρ = 1000 kg/m^3
Next, let's determine the velocities in the pipe and the nozzle. The speed of the water in the pipe is given as 0.68 m/s.
v1 = 0.68 m/s
To find v2, we can make use of the principle of conservation of mass. The volume flow rate of water is conserved, which means that the product of the area of the pipe and the velocity of the water in the pipe is equal to the product of the area of the nozzle and the velocity of the water at the nozzle.
A1 * v1 = A2 * v2
To find A1 and A2, we can use the formula for the area of a circle:
A = π * r^2
For the pipe:
r1 = 1.5 cm = 0.015 m
A1 = π * (0.015 m)^2
For the nozzle:
r2 = 0.42 cm = 0.0042 m
A2 = π * (0.0042 m)^2
Now, let's calculate v2:
A1 * v1 = A2 * v2
(π * (0.015 m)^2) * 0.68 m/s = (π * (0.0042 m)^2) * v2
(0.000707 m^2) * 0.68 m/s = (0.0000139 m^2) * v2
v2 = (0.000707 m^2 * 0.68 m/s) / (0.0000139 m^2)
v2 ≈ 34.36 m/s
Now we have all the values needed to calculate the pressure.
P1 = P2 + 1/2 * ρ * v2^2 - 1/2 * ρ * v1^2
P1 = P2 + 1/2 * 1000 kg/m^3 * (34.36 m/s)^2 - 1/2 * 1000 kg/m^3 * (0.68 m/s)^2
P1 = P2 + 1/2 * 1000 kg/m^3 * 1180.7696 m^2/s^2 - 1/2 * 1000 kg/m^3 * 0.4624 m^2/s^2
P1 = P2 + 590.3848 N/m^2 - 0.2312 N/m^2
P1 = P2 + 590.1536 N/m^2
Therefore, the absolute pressure of the water in the pipe is P1 = P2 + 590.1536 N/m^2.