A carpenter has constructed a toy as shown in | vedclass

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(C) Let the density of the cone be $\rho$. Then its mass $m_1$ is given by:$m_1 = \frac{1}{3} \pi (2R)^2 (4R) \rho = \frac{16}{3} \pi R^3 \rho$The cen

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at a distance of $4R$ from $O$

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(C) Let the density of the cone be $\rho$. Then its mass $m_1$ is given by:$m_1 = \frac{1}{3} \pi (2R)^2 (4R) \rho = \frac{16}{3} \pi R^3 \rho$The centre of mass of the cone is at a height $h/4$ from its base,where $h = 4R$. Thus,$y_1 = \frac{4R}{4} = R$ from $O$.Similarly,the mass of the sphere $m_2$ with radius $R$ and density $12\rho$ is:$m_2 = \frac{4}{3} \pi R^3 (12\rho) = 16 \pi R^3 \rho = 3 m_1$The centre of mass of the sphere is at its centre $O_2$. The distance of $O_2$ from $O$ is $y_2 = 4R + R = 5R$.Taking the line of symmetry as the $y$-axis with the origin at $O$,the centre of mass $Y_{CM}$ of the toy is:$Y_{CM} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} = \frac{m_1(R) + (3m_1)(5R)}{m_1 + 3m_1} = \frac{16 m_1 R}{4 m_1} = 4R$Thus,the centre of mass of the toy is at a distance of $4R$ from $O$.

$A$ 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 the cone,the position of the centre of mass of the toy is given by:

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