water flows steadily through a horizontal pipe of non uniform cross section. if the pressure of water is 4*10^4 N/m^2 at a point where the velocity of flow is 2ms^-1 and cross section is 0.02 m^2 what is the pressure at a point where cross section reduces to 0.01m^2?
To find the pressure at a point where the cross section of the pipe reduces, we can use Bernoulli's equation, which states that the total pressure at any point in a flowing fluid system is the sum of the hydrostatic pressure, dynamic pressure, and potential energy.
The equation is as follows:
P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at points 1 and 2 (initial and final points, respectively),
ρ is the density of the fluid,
v1 and v2 are the velocities at points 1 and 2,
g is the acceleration due to gravity, and
h1 and h2 are the heights at points 1 and 2.
In this case, we can assume the elevation difference (h) is constant, and therefore, can be ignored in our calculations. Additionally, since the pipe is horizontal, the height difference can be considered zero.
Therefore, Bernoulli's equation simplifies to:
P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2
Now let's plug in the given values into the equation:
P1 = 4 * 10^4 N/m^2 (given)
v1 = 2 m/s (given)
A1 = 0.02 m^2 (given)
A2 = 0.01 m^2 (given)
To solve for P2, we need to find v2. We can use the principle of conservation of mass:
A1v1 = A2v2
Substituting the given values:
(0.02 m^2)(2 m/s) = (0.01 m^2)v2
Solving for v2:
0.04 m^2/s = 0.01 m^2v2
v2 = 0.04 m/s
Now, substituting the values into Bernoulli's equation:
P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2
Plugging in the values:
4 * 10^4 N/m^2 + 1/2ρ(2 m/s)^2 = P2 + 1/2ρ(0.04 m/s)^2
Since the density (ρ) of water is approximately constant, we can cancel it out:
4 * 10^4 N/m^2 + 1/2(2 m/s)^2 = P2 + 1/2(0.04 m/s)^2
Simplifying:
4 * 10^4 N/m^2 + 2 N/m^2 = P2 + 0.0008 N/m^2
P2 = (4 * 10^4 N/m^2 + 2 N/m^2) - 0.0008 N/m^2
P2 = 4 * 10^4 N/m^2 + 1.9992 N/m^2
Therefore, the pressure at the point where the cross section reduces to 0.01 m^2 is approximately 4 * 10^4 N/m^2 + 1.9992 N/m^2, or 4.01992 * 10^4 N/m^2.