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I am facing trouble with the following question

A rod $AC$ of length $l$ and mass $m$ is kept on a horizontal smooth surface. It is free to rotate and move. A particle of same mass $m$ moving on the plane with velocity $v$ strikes rod at $B$ elastically as shown in the figure below.

enter image description here

The question is to find the angular velocity of the rod and its final velocity.

Attempt

I tried basically by applying conservation of energy , linear momentum and angular momentum.

Let the final situation be something like this

enter image description here

Now conservation of energy gives $$1/2mv^2=1/2m(v_5^2+v_6^2)+1/2m(v_3^2+v_4^2)+1/2(1/12ml^2w^2)$$

and conservation of angular momentum $$m(4v/5)(3l/4)=(1/12ml^2w^ 2)w-mv_4(3l/4)$$

and by linear momentum in two directions $$ 3v/4=v_3+v_6$$ and $$4v/5=v_5-v_4$$

But I have 5 unknowns and I could manage only 4 equations. Moreover solving these equations is lengthy and I think there must be some quicker way to solve this being a entrance exam question with time restrictions. Any help would be highly appreciated. Thanks.

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  • $\begingroup$ What happens to the incoming mass after collision? Is the collision elastic or inelastic? $\endgroup$ – Farcher Apr 19 '17 at 6:29
  • $\begingroup$ @Farcher sorry I forgot to mention that collision is elastic. Nothing is known about the velocity of incoming mass after collision. $\endgroup$ – Pink Apr 19 '17 at 7:06
  • $\begingroup$ If the collision is elastic then the incoming mass can impart no linear momentum to the bar along the direction of the long axis of the bar. $\endgroup$ – Farcher Apr 19 '17 at 7:10
  • $\begingroup$ @Farcher how can we say that? The collision is oblique in nature. $\endgroup$ – Pink Apr 19 '17 at 7:13
  • $\begingroup$ Which forces causes a change in momentum along the axis of the rod? $\endgroup$ – Farcher Apr 19 '17 at 7:16
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Im not solving the whole question but u can also write the missing equation for cofficient of resistution

$e=\frac{Velocity of sepration}{Velocity of approach}$

(e=1 for elastic collision)

And I think this question can also solved by conserving angular momentum about point of collision as Net torque about all the forces are 0 about this point.If you like you can also use Impulse momentum theorem though it will produce same result as yours.

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  • $\begingroup$ Thank you. I wanted to know if velocity of saparartion depends in w. $\endgroup$ – Pink Apr 19 '17 at 7:57
  • $\begingroup$ Use conventional translatory motion approach. e has no relation with $\omega$. $\endgroup$ – Abhash Jha Apr 19 '17 at 8:01

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