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  4. A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of move 0.1 kg moving in a straight line with velocity 80 m / s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad / s ) of the rod immediately after the collision will be .

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Q. A thin rod of mass $0.9\, kg$ and length $1\, m$ is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of move $0.1 \,kg$ moving in a straight line with velocity $80\, m / s$ hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in $rad / s$ ) of the rod immediately after the collision will be ______.Physics Question Image

JEE MainJEE Main 2020System of Particles and Rotational Motion

Solution:

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$\vec{ L }_{ i }=\vec{ L }_{ f }$
$mvL = I \omega$
$mvL =\left(\frac{ ML ^{2}}{3}+ mL ^{2}\right) \omega$
$0.1 \times 80 \times 1=\left(\frac{0.9 \times 1^{2}}{3}+0.1 \times 1^{2}\right) \omega$
$ 8=\left(\frac{3}{10}+\frac{1}{10}\right) \omega=\frac{4}{10} \omega$
$\omega=20 rad \frac{ rad }{ sec }$