An object has a moment of inertia of $3 \ kg \cdot m^2$. It is rotating with an angular velocity of $2 \ rad/s$. If a mass of $12 \ kg$ is moving with a velocity of $v \ m/s$,at what value of $v$ will their kinetic energies be equal?

$A$ wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. $A$ constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point $Q$. Which of the following options is/are correct?

$A$ uniform rod $AB$ of length $1 \ m$ and mass $4 \ kg$ is sliding along two mutually perpendicular frictionless walls $OX$ and $OY$. The velocity of the two ends of the rod $A$ and $B$ are $3 \ m/s$ and $4 \ m/s$ respectively,as shown in the figure. Which of the following statement$(s)$ is/are correct?

$A$ uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly,what happens to its speed of rotation?

$A$ uniform rod $AB$ of length $L$ and mass $M$ is lying on a smooth table. $A$ small particle of mass $m$ strikes the rod with a velocity $v_0$ at point $C$ at a distance $x$ from the centre $O$. The particle comes to rest after the collision. The value of $x$ such that point $A$ of the rod remains stationary just after the collision is:

$A$ projectile of mass $3m$ explodes at the highest point of its path. It breaks into three equal parts. One part retraces its path,the second one comes to rest. The distance of the third part from the point of projection when it finally lands on the ground is ........$m.$ (The range of the projectile was $100\,m$ if no explosion would have taken place.)