A horizontal pipe has a cross sectional area of 40.0 cm^2 at the wider portions and 10.0 cm^2 at the constriction. Water is flowing in the pipe, and the discharge from the pipe is 6.00*10^-3 m^3/s(6.00 L/s). Find (a) the flow speeds at the wide and the narrow portions; (b) the pressure difference between those portions; (c) the difference in height between the mercury columns and the U-shaped tube.
Marqious, you err. Water is incompressible, so the flow speed has to change with area.
Pls ignore, Marquious has been banned because of spamming.
ok for part a I got v at the wide portion as .015 m/s and at the narrow portion .06 m/s but I have no idea if that is right
it should be 1.5 m/s and 6.0 m/s
Thank you
To solve this problem, we can use the principles of fluid dynamics. There are a few important equations we need to consider:
1. Continuity equation: A1v1 = A2v2
2. Bernoulli's equation: P1 + 1/2 ρv1^2 + ρgh1 = P2 + 1/2 ρv2^2 + ρgh2
(a) To find the flow speeds at the wide and narrow portions of the pipe:
We are given the discharge Q = 6.00 * 10^-3 m^3/s. We can use the continuity equation to relate the flow speeds at the wide and narrow portions:
A1v1 = A2v2
The cross-sectional areas are given as A1 = 40.0 cm^2 = 4.00 * 10^-3 m^2 and A2 = 10.0 cm^2 = 1.00 * 10^-3 m^2.
Plugging in the values, we have:
(4.00 * 10^-3 m^2)v1 = (1.00 * 10^-3 m^2)v2
Simplifying, we find:
v1 = (1.00 * 10^-3 m^2/4.00 * 10^-3 m^2)v2
v1 = 0.25v2
Therefore, the flow speed at the wide portion is 0.25 times the flow speed at the narrow portion.
(b) To find the pressure difference between the wide and narrow portions:
To find the pressure difference, we can use Bernoulli's equation:
P1 + 1/2 ρv1^2 + ρgh1 = P2 + 1/2 ρv2^2 + ρgh2
Since the pipe is horizontal, we can neglect the term ρgh1 and ρgh2. Also, the velocities v1 and v2 are given as v1 = 0.25v2 according to the continuity equation.
The equation becomes:
P1 + 1/2 ρ(0.25v2)^2 = P2 + 1/2 ρv2^2
Simplifying:
P1 + 0.0625ρv2^2 = P2 + 0.5ρv2^2
Subtracting P2 and rearranging:
P1 - P2 = 0.4375ρv2^2
(c) To find the difference in height between the mercury columns in the U-shaped tube:
Unfortunately, the problem does not provide the necessary information to calculate the difference in height. If there were additional details about the system, such as the length and diameter of the U-shaped tube or the specific properties of the mercury, we could use the principles of hydrostatic pressure to determine the difference in height.
In summary:
(a) The flow speed at the wide portion is 0.25 times the flow speed at the narrow portion.
(b) The pressure difference between the wide and narrow portions is 0.4375ρv2^2, where ρ is the density of the fluid.
(c) The problem does not provide enough information to calculate the difference in height between the mercury columns in the U-shaped tube.
a) use the equation of continuity
b) Use bernoullis equation, I can check your work.
c) Change pressure in b to mmHg.