- #1

akrill

- 3

- 2

- Homework Statement
- Given a hemisphere of radius r and uniform density, find the centre of mass of the hemisphere

- Relevant Equations
- None given.

- Place hemisphere in xyz coordinates so that the centre of the corresponding sphere is at the origin.
- Then notice that the centre of mass must be at some point on the z axis ( because the 4 sphere segments when cutting along the the xz and xy planes are of equal volume)
- y
^{2}+ x^{2}= r^{2} - We want two volumes, V
_{1}and V_{2}which are equal, only cutting parallel to the flat side of the hemisphere at some distance h. - Recall volume of revolution formula, V = π∫y
^{2}dx - V
_{1}= ∫_{0}^{h}r^{2}- x^{2}dx - Similarly, V
_{2}= ∫_{h}^{r}r^{2}- x^{2}dx - Then, by equating the two integrals and doing some rearrangement, I got to: h
^{3}-3r^{2}h +r^{3}= 0 - Also, 0 < h < r obviously.