- #1

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- Homework Statement:
- To find the centroid of a hollow and solid hemisphere.

- Relevant Equations:
- Pappus's centroid theorem

I recently learned how to calculate the centroid of a semi-circular ring of radius ##r## using Pappus's centroid theorem as

##\begin{align}

&4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\

&d=\frac {2r}{\pi}\nonumber

\end{align}##

Where ##d## is the distance of center of mass of the ring from its base.

Similarly its second theorem can also be used to calculate C.O.M of semi-circular disc of radius ##r##,

##\begin{align}

&\frac{4}{3} \pi r^3=(2 \pi d)(\dfrac {\pi r^2}{2})\nonumber\\

&d=\frac {4r}{3 \pi}\nonumber

\end{align}##

Now this is great because now I don't have to do the lengthy calculations to find the C.O.M using ##d=\dfrac{\int ydm}{M}##

So I wanted to extend this to find the centroid of hemispheres and cones but then how do I find the area or volume traced by rotating a hemisphere?

How do I extend this to 3D object or can I extend this to 3D objects?

##\begin{align}

&4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\

&d=\frac {2r}{\pi}\nonumber

\end{align}##

Where ##d## is the distance of center of mass of the ring from its base.

Similarly its second theorem can also be used to calculate C.O.M of semi-circular disc of radius ##r##,

##\begin{align}

&\frac{4}{3} \pi r^3=(2 \pi d)(\dfrac {\pi r^2}{2})\nonumber\\

&d=\frac {4r}{3 \pi}\nonumber

\end{align}##

Now this is great because now I don't have to do the lengthy calculations to find the C.O.M using ##d=\dfrac{\int ydm}{M}##

So I wanted to extend this to find the centroid of hemispheres and cones but then how do I find the area or volume traced by rotating a hemisphere?

How do I extend this to 3D object or can I extend this to 3D objects?