A hollow hemispherical bowl of mass m, radius R is placed on a smooth horizontal surface. A particle (also of mass m) strikes the hemi spherical bowl at point P and sticks to it. The velocity of particle just after the collision in vertically upwards direction is $\frac{ku}{25}$ ;where k is an integer. Find k.

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Detailed Solution

$V_{y} = ωxcos θ = \frac{ωR}{2}$
Using conservation of angular momentum about centre of mass.
$\begin{array}{l} mu \frac{R}{4} = \{\frac{2}{3} {mR}^{2} - \frac{{mR}^{2}}{4} + \frac{5 {mR}^{2}}{16}\} ω + \frac{5 {mR}^{2}}{16} ω \\ = \frac{25}{24} {mR}^{2} ω \\ \Rightarrow ω = \frac{6 u}{25 R} \\ \Rightarrow V_{y} = \frac{3 u}{25} \end{array}$
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