Question

Distance of the centre of mass of a solid uniform cone from its vertex is $z_0$. If the radius of its base is $R$ and its height is $12cm$ then $z_0$ is equal to _______$cm$.

Answer 1

Answer: 9

$dm=\rho \pi r^{2}dz$
From figure,
$\tan \alpha =\frac{r}{z}=\frac{R}{h}$
$\therefore r=\frac{R}{h}Z$
Now, $z_{CM}=\frac{\int zdm}{\int dm}=\frac{\underset{0}{\overset{h}{\int }}\rho \pi r^{2}zdz}{\frac{1}{3}\pi R^{2}h\rho }$
$=\frac{3}{R^{2}h}\underset{0}{\overset{h}{\int }}{\left(\frac{R}{h}z\right)}^{2}zdz$
$=\frac{3}{h{R}^2}\left(\frac{R^2}{h^2}\right)\underset{0}{\overset{h}{\int }}{z}^{3}dz$
$=\frac{3}{h^3}{\left[\frac{z^4}{4}\right]}_0^h=\frac{3h}{4}$
$=\frac{3\times 12}{4}=9cm$