Question

A uniform solid right circular cone of base radius R is joined to a uniform solid hemisphere of radius R and of the same density, as shown. The centre of mass of the composite solid lies at the centre of base the cone. The height of the cone is

• A. 1.5 R
• B. $\sqrt{3}$ R
• C. 3 R
• D. $2\sqrt{3}$ R

Answer 1

The correct option is C 3 R

We know com of cone is at a height of $\frac{h}{4}$ from its base
and com of hemisphere is at a height of $\frac{3R}{8}$ from its base.
frac{2}{3}\,\pi R{\,^3} \times \,d$
Let the base be origin
Mass of hemisphere = $\frac{1}{2}\left( {\frac{4}{3}\pi {R^3}} \right)\, \times \,d\, = \,\
So Xcom = 0 = $Mc\times \frac{h}{4}-Mh\times \frac{3R}{8}$
$Mc\times \frac{h}{4}=Mh\times \frac{3R}{8}$
$\frac{1}{3}\pi R^{2}h\times d\times \frac{h}{4}=\frac{2}{3}\pi R^2\times d\times \frac{3R}{8}$
$\frac{h^2}{12}=\frac{2}{3}\times \frac{3}{8}{R}^2$
$h=3R$