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A rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius R is placed horizontally at rest with its length parallel to the edge such that the axis of the cylinder and the horizontal edge are in same vertical plane shown in the figure. There is sufficient friction present at the edge, so that a very small displacement causes the cylinder to roll off the edge without slipping. Then the ratio of the translational to rotational kinetic energies of the cylinder when its center of mass is in horizontal line with the edge is......

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answer is 6.

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Detailed Solution

$m g \cos θ = m r ω^{2}$

$m g r (1 - \cos θ) = \frac{1}{2} \frac{3}{2} m r^{2} ω^{2}$

Rotation K.E = $\frac{1}{2} \frac{m r^{2}}{2} ω^{2} = \frac{1}{7} m g r \cos θ$

Translational K.E = $\frac{6}{7} m g r \cos θ$

$∴$ Ratio = 6

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