Calculating Center of Mass in Tank with Muddy Suspension | Homework Help

AI Thread Summary
To calculate the center of mass of a tank shaped like a lower hemisphere with radius R and a muddy suspension density defined as ρ(x,y,z) = e^-h(x,y,z), the height function h(x,y,z) needs to be determined. The user suggests that h(x,y,z) could be z or z + R, seeking clarification on this point. Additionally, there is a request for graphical illustrations to help establish the limits of integration for the problem. The discussion also touches on the need for guidance on the phi limits of integration in the context of spherical coordinates. Understanding these elements is crucial for accurately solving the problem.
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Homework Statement


Let Ω be a tank whose shape is that of the lower hemisphere of radius R. The tank with a muddy suspension whose density ρ is ρ(x,y,z):=e^-h(x,y,z), where h(x,y,z) is the height of (x,y,z) above the lowest point of the tank. Find the center of mass in the tank


Homework Equations





The Attempt at a Solution


First of all, how does one determine the height, h(x,y,z)? I guess it would be R but I am not able to give a reasoning to my guess. I would appreciate if someone could give me a graphical illustration on how to find the limits of integration for this problem as well (ignore this if it will cause too much hassle). Thanks
 
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If z is the vertical coordinate, and z =0 at the bottom of the tank, then h(x,y,z) = z
 
I got a feeling its z+R. Could anyone let me know what are the phi limits of integration?
 

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