$A$ 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.

$A$ wheel of radius $0.5 \ m$ and a moment of inertia of $10 \ kg \cdot m^2$ is rotating freely at an angular speed of $70 \ rev/min$. The wheel can be stopped in $5.0 \ s$ by pressing a wet cloth against the rim and exerting a radially inward force of $88 \ N$. The coefficient of kinetic friction between the wheel and wet cloth is:

$A$ solid sphere is in purely rotational motion about its diameter. The ratio of its angular momentum $(L)$ and kinetic energy $(K)$ is $\frac{\pi}{22}$. Find the angular velocity $(\omega)$ of the sphere. (Take $\pi = \frac{22}{7}$) (in $rad/s$)

$A$ motor of an engine rotates at an angular speed of $100 \ rpm$ about its axis. When the switch is turned off,it comes to rest in $15 \ s$. How many rotations does it complete before coming to rest?

$A$ solid sphere and a disc of the same radius are released from the same height on an inclined plane and they reach the bottom of the plane at different times. This is due to their:

$A$ thin uniform bar lies on a frictionless horizontal surface and is free to move in any way on the surface. Its mass is $0.300 \, kg$ and length is $2 \, m$. Two particles,each of mass $0.100 \, kg$,are moving on the same surface towards the two ends of the bar in a direction perpendicular to the bar. One particle moves with a velocity of $10 \, m/s$ towards one end,and the other moves with a velocity of $5 \, m/s$ towards the other end. If the collision between the particles and the bar is completely elastic and both particles strike the bar simultaneously,find the velocity of the centre of mass of the bar after the collision in $m/s$.