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# A ring, cylinder and solid sphere are placed on the top of a rough incline on which the sphere can just roll with out slipping. When all of them are released at the same instant from the same position, then

a) all of them reach the ground at the same instant b) the sphere reaches first, the ring at last c) the sphere reaches first, the cylinder and ring reach together. d) none of the above

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## 1 Answer

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For pure rolling on a surface of inclination $\theta$,
$\mu \geq \large\frac{\tan \theta}{1+\large\frac{R^2}{k^2}}$
Since $k^2$ for sphere $=\large\frac{2}{5}$$R^2 k^2 for ring = R^2 k^2 for cylinder = \large\frac{R^2}{2} \mu _s \geq \large\frac{2}{7}$$\tan \theta$
$\mu _c \geq \large\frac{1}{3}$$\tan \theta, \mu _r \geq \large\frac{\tan \theta}{2}$
$\mu _s$ is least.
Hence both cylinder and ring will slide.
$\therefore$ friction =$\mu mg \cos \theta$ will be acting on all three.
Hence all will have the same acceleration. Hence will reach the bottom simultaneously.
answered Dec 5, 2013 by
edited Jun 22, 2014

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