Water flows under pressure from one pipe into a second pipe having half the diameter. The volume flow rate in the narrow pipe is ________ that in the wide pipe, and the speed of the water in the narrow pipe is _______ that in the wide pipe
Q = u A = constant
The volume flow rate in the narrow pipe is ___ the same as_____ that in the wide pipe, and the speed of the water in the narrow pipe is ___4 times____ that in the wide pipe
pi R^2 = 4 pi(R/2)^2
flow rate in narrow tube is 16times less than in wide tube. speed in narrow tube will be 4time smaller than wide tube
Oh pipe, oh pipe, you make me laugh! Let's dive into your watery math. When water flows, it's quite the sight, from one pipe to another, day and night.
Now, the volume flow rate, my friend, behaves as a sneaky trend. In the narrow pipe, it's always true, that it's half the wide pipe's volume, it's you. So, if the wide pipe's flow makes you cheer, the narrow pipe will bring half the cheer.
Now let's talk about speed, oh what a treat! In the narrow pipe, it's swift and neat. The water there, it twirls and twines, moving at twice the wide pipe's times. So, if the wide pipe's speed you find just fine, the narrow pipe will be doubly divine.
So, to sum it up with a jolly rhyme, the narrow pipe sings a half-flow chime. And in terms of speed, it dances with grace, being twice as fast in that tight little space. Oh, how water pipes can bring us cheer, with their flow rates and speeds, crystal clear!
To determine the volume flow rate and speed of the water in the narrow pipe compared to the wide pipe, we can use the principle of continuity, which states that the mass flow rate of an incompressible fluid remains constant within a closed system.
The volume flow rate is the volume of fluid passing through a section of the pipe per unit of time. It is calculated using the formula:
Volume flow rate = Cross-sectional area × Velocity
Now, let's consider the two pipes in question. We'll call the wide pipe "Pipe 1" and the narrow pipe "Pipe 2."
Given that water flows under pressure from Pipe 1 to Pipe 2, we will assume that the volume flow rate remains constant between the two pipes.
1. Volume flow rate in the narrow pipe:
Since Pipe 2 has half the diameter of Pipe 1, its cross-sectional area will be one-fourth (πr2) of Pipe 1's cross-sectional area.
Therefore, the volume flow rate in the narrow pipe will be one-fourth of the volume flow rate in the wide pipe.
In other words, the volume flow rate in the narrow pipe is 1/4 of the volume flow rate in the wide pipe.
2. Speed of the water in the narrow pipe:
As we mentioned earlier, the volume flow rate remains constant between the two pipes. Since volume flow rate = Cross-sectional area × Velocity, we can rearrange the equation to solve for velocity:
Velocity = Volume flow rate / Cross-sectional area
Since the volume flow rate is the same for both pipes, and the cross-sectional area of the narrow pipe is one-fourth that of the wide pipe, we can conclude that the speed of water in the narrow pipe is four times higher than that in the wide pipe.
In other words, the speed of the water in the narrow pipe is four times the speed in the wide pipe.
To summarize:
- The volume flow rate in the narrow pipe is one-fourth of the volume flow rate in the wide pipe.
- The speed of the water in the narrow pipe is four times the speed in the wide pipe.