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  4. A uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speeds 2v and v respectively, strike the bar and stick to the bar after collision. Then the angular velocity of the bar (about the centre of mass) is <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-koncm67ddcno.png />

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Q. A uniform bar of length $6a$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ moving in the same horizontal plane with speeds $2v$ and $v$ respectively, strike the bar and stick to the bar after collision. Then the angular velocity of the bar (about the centre of mass) is

Question

NTA AbhyasNTA Abhyas 2020System of Particles and Rotational Motion

Solution:

Solution
From conservation of angular momentum about point $C$
$L_{i n i t i a l \left(2 m\right)}+L_{i n i t i a l \left(m\right)}=L_{f i n a l \left(s y s t e m\right)}$
$m\left(2 v\right) \, \left(2 a\right)+2m\left(v\right)\left(a\right)=\left[\frac{1}{12} \left(8 m\right) \left(6 a\right)^{2} + m \left(2 a\right)^{2} + 2 m \left(a\right)^{2}\right]\omega $
$6 \, mva=30 \, ma^{2}\omega $
$\omega =\frac{v}{5 a}$