Question

A hemisphere and a solid cone have a common base. The center of mass of the common structure coincides with the centre of the common base. If $R$ is the radius of hemisphere and $h$ is height of the cone then:

• A. $\frac{h}{R}=\sqrt{3}$
• B. $\frac{h}{R}=\frac{1}{\sqrt{3}}$
• C. $\frac{h}{R}=3$
• D. $\frac{h}{R}=\frac{2}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{h}{R}=\sqrt{3}$

Let $S=$ density

B and C be the center of mass of cone and hemisphere. Now for common com at base.

$\Rightarrow \frac{3r}{8}\times \left(\frac{2}{3}\pi r^3\right)ȷ-\frac{h}{4}\left(\frac{1}{3}\pi r^{2}h\right)\rho =0$

$\Rightarrow \frac{3}{8}\times \frac{2}{3}{r}^2-\frac{h^2}{12}=0$

$\Rightarrow r^2-\frac{h^2}{3}=0$

$\Rightarrow \frac{h^2}{3}=r^2$

$\Rightarrow {\left(\frac{h}{r}\right)}^2=3$

$\Rightarrow \frac{h}{r}=\sqrt{3}.$

Hence, the answer is $\frac{h}{r}=\sqrt{3}.$