$A$ spherical rigid ball is released from rest and starts rolling down an inclined plane from height $h=7 \, m$,as shown in the figure. It hits a block at rest on the horizontal plane (assume elastic collision). If the mass of both the ball and the block is $m$ and the ball is rolling without sliding,then the speed of the block after collision is close to ............. $m/s$.

$A$ uniform rod of mass $8m$ and length $6a$ is placed on a horizontal table. Two point masses $m$ and $2m$ are moving with speeds $2v$ and $v$ respectively,as shown in the figure. They strike the rod and stick to it after the collision. Calculate the speed of the centre of mass of the system after the collision.

$A$ uniform disc is rolling on a horizontal surface. At a certain instant,$B$ is the point of contact and $A$ is the topmost point of the disc,where $R$ is the radius of the disc.

$A$ metal rod of length $L$ and mass $m$ is pivoted at one end. $A$ thin disk of mass $M$ and radius $R$ $(R < L)$ is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case $A$) The disc is not free to rotate about its center and (case $B$) the disc is free to rotate about its center. The rod-disc system performs $SHM$ in a vertical plane after being released from the same displaced position. Which of the following statement$(s)$ is (are) true?

$A$ wagon of $200\, kg$ is moving on a smooth track with a velocity of $2\, m/s$. $A$ man of $80\, kg$ runs in the wagon with a velocity such that the speed of the centre of mass of the system is zero. Find the relative velocity of the man with respect to the wagon in $m/s$.

$A$ particle is moving in a uniform circular motion with angular momentum $L$. If the frequency of motion is doubled and its kinetic energy is halved,the new angular momentum will be ...