Water (density = 1.0 × 103 kg/m3) flows through a horizontal tapered pipe. At the wide end its speed is 4.0m/s and at the narrow end it is 5.0 m/s. The difference is pressure between the two ends is:
P + (1/2)(density)V^2
is the same for both ends
P1 - P2 = (1/2)(density)(V2^2 - V1^2))
Well, water sure knows how to go with the flow! Now, let's tackle your question. To find the difference in pressure, we can use Bernoulli's principle. According to this principle, the difference in pressure between two points in a fluid flow is equal to the difference in kinetic energy per unit volume.
Using the equation for kinetic energy per unit volume:
P1 - P2 = (1/2) * (ρ * v2^2 - ρ * v1^2)
Where:
P1 is the pressure at the wide end,
P2 is the pressure at the narrow end,
ρ is the density of water,
v1 is the speed at the wide end, and
v2 is the speed at the narrow end.
Plugging in the given values:
P1 - P2 = (1/2) * (1.0 × 10^3 kg/m^3 * (5.0 m/s)^2 - 1.0 × 10^3 kg/m^3 * (4.0 m/s)^2)
Now it's just some math:
P1 - P2 = (1/2) * (1.0 × 10^3 kg/m^3 * 25 m^2/s^2 - 1.0 × 10^3 kg/m^3 * 16 m^2/s^2)
P1 - P2 = (1/2) * (1.0 × 10^3 kg/m^3 * 9 m^2/s^2)
P1 - P2 = 4,500 Pa
So, the difference in pressure between the two ends of the pipe is 4,500 Pa. Now, that's some high-pressure comedy!
To find the difference in pressure between the two ends of the tapered pipe, we can use Bernoulli's equation. Bernoulli's equation states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume at any two points in a steady, incompressible fluid flow system must remain constant.
The equation can be written as:
P1 + 1/2 * ρ * V1^2 + ρ * g * h1 = P2 + 1/2 * ρ * V2^2 + ρ * g * h2
Where:
P1 and P2 are the pressures at the wide end and narrow end of the pipe, respectively.
ρ is the density of the fluid (in this case, water) = 1.0 × 10^3 kg/m^3.
V1 and V2 are the velocities at the wide end and narrow end of the pipe, respectively.
g is the acceleration due to gravity = 9.8 m/s^2.
h1 and h2 are the heights at the wide end and narrow end of the pipe, respectively.
In this case, since the pipe is horizontal, the heights h1 and h2 can be assumed to be the same, so we can cancel out the ρ * g * h terms.
The equation simplifies to:
P1 + 1/2 * ρ * V1^2 = P2 + 1/2 * ρ * V2^2
Now we can substitute the given values:
P1 + 1/2 * 1.0 × 10^3 kg/m^3 * (4.0 m/s)^2 = P2 + 1/2 * 1.0 × 10^3 kg/m^3 * (5.0 m/s)^2
Simplifying further:
P1 + 1/2 * 1.0 × 10^3 kg/m^3 * 16.0 m^2/s^2 = P2 + 1/2 * 1.0 × 10^3 kg/m^3 * 25.0 m^2/s^2
P1 + 8.0 × 10^3 N/m^2 = P2 + 12.5 × 10^3 N/m^2
Rearranging the equation:
P2 - P1 = 12.5 × 10^3 N/m^2 - 8.0 × 10^3 N/m^2
P2 - P1 = 4.5 × 10^3 N/m^2
Therefore, the difference in pressure between the two ends of the tapered pipe is 4.5 × 10^3 N/m^2.
To find the difference in pressure between the two ends of the tapered pipe, we can use Bernoulli's equation, which relates the pressure, velocity, and density of a flowing fluid.
Bernoulli's equation is given by:
P + (1/2)ρv^2 + ρgh = 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.
In this case, the fluid is water. Given that the density of water is 1.0 × 10^3 kg/m^3, we can substitute the known values into Bernoulli's equation.
At the wide end of the pipe, the velocity is 4.0 m/s, and at the narrow end, the velocity is 5.0 m/s. The height (h) is not given, but since the pipe is horizontal, we can assume it is constant throughout.
We can set up Bernoulli's equation for both ends of the pipe and then find the difference in pressure between the two ends.
For the wide end:
P1 + (1/2)ρv1^2 + ρgh1 = constant
For the narrow end:
P2 + (1/2)ρv2^2 + ρgh2 = constant
Since the heights are the same at both ends and the fluid is horizontal, we can ignore the height term.
Therefore, the equation for the wide end becomes:
P1 + (1/2)ρv1^2 = constant
And the equation for the narrow end becomes:
P2 + (1/2)ρv2^2 = constant
We want to find the difference in pressure, which can be obtained by subtracting the two equations:
(P1 - P2) + (1/2)ρ(v1^2 - v2^2) = 0
Now we can substitute the known values into the equation:
(P1 - P2) + (1/2)(1.0 × 10^3 kg/m^3)((4.0 m/s)^2 - (5.0 m/s)^2) = 0
Simplifying:
(P1 - P2) + (1/2)(1.0 × 10^3 kg/m^3)(16 m^2/s^2 - 25 m^2/s^2) = 0
(P1 - P2) + (1/2)(1.0 × 10^3 kg/m^3)(-9 m^2/s^2) = 0
(P1 - P2) - (4.5 × 10^3 kg/m^3)(9 m^2/s^2) = 0
(P1 - P2) - 40.5 × 10^3 N/m^2 = 0
(P1 - P2) = 40.5 × 10^3 N/m^2
Therefore, the difference in pressure between the two ends of the tapered pipe is 40.5 × 10^3 N/m^2.