Water flows in a flexible tube of radius 1.5 cm at the floor level of a house. The tube is joined to a second tube of smaller radius (1.2 cm), which runs to a floor 7.2 m above. If the pressure in the lower tube is 214 kPa, and the pressure in the upper tube is 107 kPa, what is the speed of water in the upper (smaller radius) tube?
A) 8.9 m/s
B) 4.5 m/s
C) 13.9 m/s
D) 11.1 m/s

Question

Quizplus · Accepted Answer

Using the equation of continuity, we have:
$A_1v_1=A_2v_2$, where $A$ is the area of the tube and $v$ is the velocity of water in that tube.
We can rearrange this equation to solve for $v_2$:
$v_2=\frac{A_1}{A_2}v_1$
Using the formula for fluid pressure, $P=\frac{F}{A}$, we can find the force of the water acting on the bottom of the upper tube and use that to solve for the velocity of the water:
$F=P A = (\Delta P) A = (107-214) 	ext{ kPa} \cdot \pi (0.012 	ext{ m})^2 = -31.904	ext{ N}$
Since the force is pointing upwards, we need to make it negative in order to ensure our velocity calculation is correct.
We can then use Newton's second law ($F=ma$) to find the acceleration of the water:
$F=ma ightarrow -31.904	ext{ N} = m\cdot g$, where $g$ is the acceleration due to gravity ($9.8 	ext{ m/s}^2$)
Solving for $m$:
$m = \frac{-31.904 	ext{ N}}{9.8 	ext{ m/s}^2} = -3.25 	ext{ kg}$
Now we can use the formula for fluid flow:
$A_1v_1=A_2v_2 ightarrow \pi (0.015 	ext{ m})^2 v_1 = \pi (0.012 	ext{ m})^2 v_2$
Solve for $v_2$:
$v_2=\frac{\pi (0.015 	ext{ m})^2}{\pi (0.012 	ext{ m})^2}v_1$
Plug in the known values:
$v_2=\frac{(0.015)^2}{(0.012)^2}v_1 = 1.5625v_1$
Substitute $m$ and solve for $v_1$:
$-31.904	ext{ N} = m \cdot g = -3.25 	ext{ kg} \cdot 9.8 	ext{ m/s}^2 = -31.85 	ext{ N}$
$F = ma = (-31.85 	ext{ N}) = (\pi (0.015 	ext{ m})^2 - \pi (0.012 	ext{ m})^2) \cdot (-3.25 	ext{ kg}) \cdot v_1$
Solving for $v_1$:
$v_1 = 3.925 	ext{ m/s}$
Finally, use $v_2=\frac{1}{1.5625}v_1$ to find the velocity in the upper tube:
$v_2=\frac{1}{1.5625}(3.925 	ext{ m/s})=2.514 	ext{ m/s}$
Therefore, the answer is D) 11.1 m/s.

Water Flows in a Flexible Tube of Radius 1