Question

Find the centre of mass of a uniform solid cone.

Answer 1

Let us consider a uniform solid cone of mass $M$, radius $R$ and heightt $h$
$X_{cm}=0$ (by symmetry)
Let us consider a small element (disc) of $dm$, radius $r$ and thickness $dy$ at a distance $y$ the from base as shown
Then, $\rho =\frac{3M}{\pi R^{2}h}=\frac{dm}{\pi r^{2}dy}\Rightarrow dm=\frac{3M{r}^2}{R^{2}h}dy$
$Y_{CM}=\frac{1}{M}{\int }^{}ydm=\frac{1}{M}{\int }^{}y\frac{3M{r}^2}{R^{2}h}dy=\frac{3}{R^{2}h}\int y{r}^{2}dy=\frac{3}{h}{}^{}{\int }_0^{h}y{(1-\frac{y}{h})}^{2}dy[\because \frac{h-y}{r}=\frac{h}{R}\Rightarrow r=(1-\frac{y}{h})R]$
On solving we get $Y_{CM}=\frac{h}{4}$
$\therefore$ CM of a uniform solid cone is $(0,\cfrac{h}{4}) from the centre of base.