A uniform cylindrical body of radius r has a conical nose. The length of the cylindrical and conical parts are 4r and 3r respectively. Mass of the conical part is M. The body rests on a horizontal surface as shown A ring of radius $\frac{r}{2}$ is to be tightly fitted on the nose of the body the maximum permissible mass of the ring so that the body does not topple is $M = (\frac{29}{k}) m .$ Find k

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Detailed Solution

In critical case the COM of the system will lie at the centre of the circular base of the conical part (Just above A; at O)
Volume of cone $= \frac{1}{3} π r^{2} (3 r) = π r^{3}$
Volume of cylinder $= π r^{2} (4 r) = 4 π r^{3}$
Mass of cylinder is 4M
Question Image
COM of the cone is at a distance of $\frac{h}{4} = \frac{3 r}{4}$ from o.
Let mass of the ring =m.
$m \times 1.5 r + M \times 0.75 r = 4 M \times 2 r$
$∴ m = \frac{29}{6} M$