Question

A carpenter has constructed a toy as shown in the figure If the density of the material of the sphere is $12$times that of cone , the position of the centre of mass 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 the density of cone is p, then mass of cone = $density\times volume=\frac{p1}{3\pi \left(2R\right)²}$

$=\frac{16}{3\pi \left(R\right)³p}$

This mass is concentrated at a height from O= h/4 =4R/h = R

Mass of the sphere= m₂= density x volume=$\frac{12px4}{3\pi \left(R\right)³}=16\pi R³p$= m3

This mass is concentrated at O2, i.e. at a distance of 4R+R=5Rfrom O.Hence centre of mass of whole system will be

$Y_{cm}=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}$

=$\frac{mR+3m\left(5R\right)}{m+3m}$

=$\frac{16mR}{4m}$

=4R