Question

A carpenter has constructed a toy as shown in the adjoining figure. If the density of the material of the sphere is $12$ times that of cone, the position of the centre of mass of the toy is given by

• A. At a distance of $2R$ from O
• B. At a distance of $3R$ from O
• C. At a distance of $4R$ from O
• D. At a distance of $5R$ from O

Answer 1

The correct option is C At a distance of $4R$ from O

If density of cone is $\rho ,$ then mass of cone = density $\times$ volume
$\Rightarrow \rho \times \frac{1}{3}\pi (2R{)}^2\times 4R\Rightarrow \rho \times \frac{1}{3}\pi 16R^3=m$
Given that the density of sphere is 12 times the density of the cone.
Hence the mass of the sphere can be given as
$\Rightarrow 12\times \frac{4}{3}\pi R^3\times \rho \Rightarrow \frac{48}{3}\pi R^3\rho =3m$
Let us take the point O as origin. We know that the centre of mass of a uniform cone lies at a height $\frac{h}{4}$ where the total height is $h$.
Hence we can write the C.O.M of the combined system as
$Y_{cm}=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}\Rightarrow \frac{mR+3m\left(5R\right)}{4m}\Rightarrow 4R$