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 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

Answer: (C)

Let density of cone is $\rho$, then density of sphere is $12 \rho$. Mass of cone:
$m_{1}=\frac{1}{3} \pi(2 R)^{2}(4 R) \rho=16 \pi \rho R^{3} / 3$
mass of sphere: $m_{2}=\frac{4}{3} \pi R^{3}(12 \rho)=48 \pi \rho R^{3} / 3$
$y_{ cm }= \frac{m_{1} O Q_{1}+m_{2} O Q_{2}}{m_{1}+m_{2}} $
$ = \frac{\frac{16 \pi \rho R^{3}}{3}\left(\frac{4 R}{4}\right)+\frac{48}{3} \pi \rho R^{3}(5 R)}{\frac{16 \pi \rho R^{3}}{3}+\frac{48 \pi \rho R^{3}}{3}}=4 R$