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Translational, Rotational or Rolling?

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Question-
A $hollow\;sphere{\kern 1pt} A$ of mass $M$ and radius $R$, rolls without slipping on a smooth horizontal surface. It collides elastically and head on with a stationary and smooth $solid\;sphere{\kern 1pt} B$ of equal mass $M$ and radius $R$. Then, the ratio of Kinetic Energy of $B$ to that of $A$, immediately after collision is -

I just want to know what happens to $hollow\;sphere{\kern 1pt} A$ immediately after collision. Whether it executes -

  • $Pure\;Translational$ or
  • $Pure\;Rotational$ or
  • $Rolling\;Motion?$

Attempt-
I definitely know that $hollow\;sphere{\kern 1pt} A$ cannot execute $Pure\;Translational\;Motion$ because in this situation there is no possibility of a torque to act on it to produce angular acceleration. The surface and $solid\;sphere{\kern 1pt} B$ both are found to be smooth.
As of $Pure\;Rotational$ or $Rolling\;Motion$, I do not have any reason why either of them cannot happen.

asked Mar 23, 2017 in Physics Problems by DoubtExpert (340 points)
edited Mar 24, 2017 by DoubtExpert
What do YOU think happens, and why? The rules of the site require you to show some effort.

1 Answer

1 vote
 
Best answer

So in a elastic collision over here there would be interchanging of Linear velocities according to conservation of linear momentum.

Since no tangential forces are presents, and all surface are smooth , the angular velocity of sphere A is not affected.
hence body A has pure rotational with angular velocity $w $and body B has pure translation with linear velocity $v $.

so ratio would be simply $ \frac {1/2 M v^2} {1/2I w^2} $
so, on substituting $I = 2/3 MR^2$ and $w = v/R $
which gives $3/2$

answered Mar 24, 2017 by physicsapproval (2,320 points)
selected Mar 24, 2017 by DoubtExpert
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