A uniform rigid sphere having radius \(R\) moving with velocity \({ v }_{ o }\) and
angular velocity \({ \omega }_{ o } =\cfrac { { v }_{ o } }{ R } \)
strikes a horizontal rough surface having coefficient of friction \(\mu \) .
If coefficient of restitution for collision is \(e=1/2\).

Let angular velocity and Velocity of centre of sphere after**immediately** after the collision is in 'x' and 'y' direction will be expressed as :

###### This is part of my set Deepanshu's Mechanics Blasts

Let angular velocity and Velocity of centre of sphere after

\[ \begin{align} { v }_{ x } &= \cfrac { a }{ b } \times \mu { v }_{ o }\\ { v }_{ y }&= \cfrac { c }{ d } \times { v }_{ o }\\ { { \omega } }^{ " }&= \cfrac { (\alpha -\beta \times \mu ) }{ \gamma R } \times { v }_{ o } \end{align} \]

Then find the value of :

\[a + b + c + d + \alpha +\beta +\gamma. \]

**Details and assumptions**

\(\bullet \) gcd (a , b ) = 1

\(\bullet \) gcd (c,d ) = 1

\(\bullet \) gcd (\(\alpha \) , \(\beta \) ) = 1

\(\bullet \) gcd ( \(\gamma \) , \(\beta \) ) = 1

\(\bullet \) All are positive integers.

\(\bullet \) Take clockwise direction as positive.

Source : My Friend give me as challange , Unfortunately He is not on Brilliant .

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