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Use spherical polar coordinates r, 9, 4) to find the CM of
Chapter 3, Problem 3.22(choose chapter or problem)
Use spherical polar coordinates \(r, \theta, \phi\) to find the CM of a uniform solid hemisphere of radius \(R\), whose flat face lies in the \(xy\) plane with its center at the origin. Before you do this, you will need to convince yourself that the element of volume in spherical polars is \(dV=r^2dr\sin\theta\ d\theta\ d\phi\). (Spherical polar coordinates are defined in Section 4.8. If you are not already familiar with these coordinates, you should probably not try this problem yet.)
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QUESTION:
Use spherical polar coordinates \(r, \theta, \phi\) to find the CM of a uniform solid hemisphere of radius \(R\), whose flat face lies in the \(xy\) plane with its center at the origin. Before you do this, you will need to convince yourself that the element of volume in spherical polars is \(dV=r^2dr\sin\theta\ d\theta\ d\phi\). (Spherical polar coordinates are defined in Section 4.8. If you are not already familiar with these coordinates, you should probably not try this problem yet.)
ANSWER:Step 1 of 3
It is given that the hemisphere is placed on XY plane with its center at the origin, therefore, the CM of hemisphere lies on Z-axis only. Its X and Y coordinates are zero.
Let the mass of hemisphere be M and its radius be R.
The volume is given by,
\(\begin{aligned}
V & =\frac{1}{2}\left(\frac{4}{3} \pi R^{3}\right) \\
& =\frac{2}{3} \pi R^{3}
\end{aligned}\)
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