Distance of the centre of mass of a solid uniform cone from its vertex is z₀ . If the radius of its base is R and its height is h then z₀ is equal to :

Let density of cone = $\mathrm{\rho }$.

Centre of mass, $y_{\mathrm{cm}}=\frac{\int \mathrm{ydm}}{\int \mathrm{dm}}$
$=\frac{{\int }_0^{\mathrm{h}} {\mathrm{y\pi r}}^2\mathrm{dy\rho }}{\frac{1}{3}{\mathrm{\pi R}}^2\mathrm{h\rho }}=\frac{{\int }_0^{\mathrm{h}} {\mathrm{r}}^2\mathrm{ydy}}{\frac{1}{3}{\mathrm{R}}^2\mathrm{h}}$    .....(i) 

For a cone, we know that
$\frac{\mathrm{r}}{\mathrm{R}}=\frac{\mathrm{y}}{\mathrm{h}}\therefore \mathrm{r}=\frac{\mathrm{y}}{\mathrm{h}}R$

$y_{\mathrm{cm}}=\frac{{\int }_0^{\mathrm{h}} 3{\mathrm{y}}^3\mathrm{dy}}{{\mathrm{h}}^3}=\frac{3{\left[\frac{{\mathrm{y}}^4}{4}\right]}_0^{\mathrm{h}}}{{\mathrm{h}}^3}=\frac{3}{4}\mathrm{h}$