Bouncing Rotation

Two blocks of masses $\SI{1}{kg}$ and $\SI{2}{kg}$ are connected by an ideal spring having spring constant $k=\SI{600}{N/m}$ . They are at rest on a smooth horizontal ground. A sphere rotating with initial angular velocity $\omega=\SI{30}{rad/s}$ , having radius $\SI{50}{cm}$ and mass $\SI{1}{kg}$ is falling from a height $\SI{5}{m}$ (from the bottom of the sphere to the top of the block as shown.

The coefficient of restitution and coefficient of friction are given by $\dfrac{1}{2}$ and $\dfrac{1}{3}$ respectively. Assume The rotation of sphere doesn't cease before it rebounds from the block .

Take $g=\SI{10}{m/s^2}$ .

Then

• Find the velocity of the 1-kg block (that is $m$ ) just after the collision.

Answer is $\SI{v}{m/s}$ .

• Find the maximum elongation in the spring.

Answer is $\dfrac{x}{y}$ meters , $\gcd(x,y)=1$ and $x,y$ are positive numbers

Find the value of $v+x+y$ .