Question

A man has constructed a toy as shown in figure. If density of the material of the sphere is $12$ times of the cone compute the position of the centre of mass. [Centre of mass of a cone of height $h$ is at height of $h/4$ from its base.]

Answer 1

Let the density of cone = $\rho$

And density of sphere = $12\rho$

mass of cone = $m_1=\frac{1}{3}\pi R^{2}H\rho =\frac{4}{3}\pi R^3\rho$

mass of sphere = $m_2=\frac{48}{3}\pi R^3\rho$

so

$Xcm=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$

since

$X_1=\frac{4R}{4}=R$

$X_2=5R$

putting all values

$X_{c}m=\frac{61R}{13}$ (from bottom)