Work liquid pipe problem

In summary, the work required to pump the mysterious liquid out of the outlet of the hemisphere shaped tank with a vertical outlet pipe is 30π(4-z^2)(1-z)dz, where z ranges from -2 to 0. The density of the liquid is 300/98 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2. The tank has a radius of 2m and the vertical pipe is 1m. The volume of the liquid is 16π/3 m^3.
  • #1
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A hemisphere shaped tank with vertical outlet pipe is full of mysterious liquid. Find the work required to pump the mysterious liquid out of the outlet. The density of the mysterious liquid is 300/98 kg/m^3. [Radius is 2m and vertical pipe is 1m]

W= F*d
=m*a*d
=D*v*a*d
D=300/98 kg/m^3, V=16pi/3 m^3, a=9.8 m/s^2

Im having trouble finding the distance. Can you help me out and correct any of the information I stated above?
 
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  • #2
Think of the "hemisphere shaped tank" as the bottom of the sphere [itex]x^2+ y^2+ z^2= 4[/itex] or [tex]z= -\sqrt{4- x^2- y^2}[/tex]. At each z< 0, a "layer" of liquid, of thickness dz, is a circle of radius [tex]r= \sqrt{x^2+ y^2}= \sqrt{4- z^2}[/tex]. That has area [itex]\pi r^2= \pi(4- z^2)[/itex] and so volume [itex]\pi(4- z^2)dz[/itex]. It's mass is
[tex]m= \frac{300}{98}\pi (4- z^2)dz[/tex]
and so its weight is
[tex]mg= \frac{300}{98}(9.8)\pi (4- z^2)dz= 30\pi(4- z^2)dz[/itex].
(You see the reason for that "mysterious" density!)

That "layer" has to be lifted to the top of the hemi-sphere, a distance of -z m (z is negative, remember), plus the additional 1 m of pipe, for a total of -z+ 1 m so the work done is
[tex]mgh= 30\pi (4- z^2)(1-z)dz[/tex]

To find the total work, integrate that from z= -2 to z= 0.
 
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1. What is a work liquid pipe problem?

A work liquid pipe problem is a common issue in fluid mechanics where there is a blockage or restriction in a pipe that causes a decrease in the flow rate of a liquid through the pipe. It can be caused by various factors such as clogs, bends, or changes in the diameter of the pipe.

2. How do you identify a work liquid pipe problem?

A work liquid pipe problem can be identified by a decrease in the flow rate of the liquid, unusual noises coming from the pipe, or visible signs of blockages such as leaks or bulges in the pipe. It is important to regularly monitor the flow rate and condition of pipes to catch any potential problems early on.

3. What are the consequences of a work liquid pipe problem?

A work liquid pipe problem can lead to a variety of consequences, including reduced efficiency and productivity, increased energy consumption, and potential damage to the pipe system or equipment connected to it. It can also create safety hazards if the liquid is hazardous or corrosive.

4. How can a work liquid pipe problem be solved?

The solution to a work liquid pipe problem depends on the specific cause. In some cases, it may require unclogging the pipe or repairing any damages. In other cases, it may involve replacing the pipe or making modifications to the system. It is best to consult with a professional to determine the most effective solution.

5. How can work liquid pipe problems be prevented?

Regular maintenance and inspection of pipes can help prevent work liquid pipe problems. It is also important to properly design and install pipes to reduce the likelihood of blockages or restrictions. Monitoring the flow rate and pressure of the liquid can also help identify any potential issues before they become major problems.

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