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

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" /> (A) zero (B) (2 v/3 a) (C) (v/5 a) (D) (3 v/5 a)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-koncm67dmuyd.png" /> From conservation of angular momentum about point C Li n i t i a l (2 m)+Li n i t i a l (m)=Lf i n a l (s y s t e m) m(2 v) (2 a)+2m(v)(a)=[(1/12) (8 m) (6 a)2 + m (2 a)2 + 2 m (a)2]ω 6 mva=30 ma2ω ω =(v/5 a)

zero

$\frac{2 v}{3 a}$

$\frac{v}{5 a}$

$\frac{3 v}{5 a}$

From conservation of angular momentum about point $C$
$L_{ini t ia l (2 m)} + L_{ini t ia l (m)} = L_{f ina l (sys t e m)}$
$m (2 v) (2 a) + 2 m (v) (a) = [\frac{1}{12} (8 m) (6 a)^{2} + m (2 a)^{2} + 2 m (a)^{2}] ω$
$6 m v a = 30 m a^{2} ω$
$ω = \frac{v}{5 a}$

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

zero

3a2v​

5av​

5a3v​

From conservation of angular momentum about point C Linitial(2m)​+Linitial(m)​=Lfinal(system)​ m(2v)(2a)+2m(v)(a)=[121​(8m)(6a)2+m(2a)2+2m(a)2]ω 6mva=30ma2ω ω=5av​

$\frac{2 v}{3 a}$

$\frac{v}{5 a}$

$\frac{3 v}{5 a}$

From conservation of angular momentum about point $C$
$L_{ini t ia l (2 m)} + L_{ini t ia l (m)} = L_{f ina l (sys t e m)}$
$m (2 v) (2 a) + 2 m (v) (a) = [\frac{1}{12} (8 m) (6 a)^{2} + m (2 a)^{2} + 2 m (a)^{2}] ω$
$6 m v a = 30 m a^{2} ω$
$ω = \frac{v}{5 a}$