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Q.

A small block of mass M is moving with a velocity $v_{1} = v_{0}$ on a frictionless surface as shown in the figure. It passes over a cylinder of radius R and mass M, which has a smooth fixed axis and is initially at rest. When the block first makes contact with the cylinder, it slips on the cylinder, but the friction is large enough so that slipping ceases before it losses contact with the cylinder. Finally, it goes to the dotted position with velocity $V_{2} = \frac{2}{β} v_{0} .$ Find the value of $β$

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answer is 3.

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Detailed Solution

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from conservation of angular momentum
$M v_{1} R = I ω + M v_{2} R$
For no slipping $ω = \frac{v_{2}}{R}$
$M v_{1} R = \frac{M R^{2}}{2} . \frac{v_{2}}{R} + M v_{2} R$
$v_{1} = \frac{1}{2} v_{2} + v_{2} \Rightarrow v_{2} = \frac{2}{3} v_{1} = \frac{2}{3} v_{0}$
$∴ β = 3$

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A small block of mass M is moving with a velocity v1=v0 on a frictionless surface as shown in the figure. It passes over a cylinder of radius R and mass M, which has a smooth fixed axis and is initially at rest. When the block first makes contact with the cylinder, it slips on the cylinder, but the friction is large enough so that slipping ceases before it losses contact with the cylinder. Finally, it goes to the dotted position with velocity V2=2βv0. Find the value of β