A thin uniform rod of length $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega $. Its centre of mass rises to a maximum height of

Two discs of moment of inertia $I_1$ and $I_2$ and angular speeds ${\omega _1}\,{\rm{and }}{\omega _2}$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational $KE$ of system will be 

A cord is wound round the circumference of wheel of radius $r$. The axis of the wheel is horizontal and moment of inertia about it is $I$. A weight $mg$ is attached to the end of the cord and falls from rest. After falling through a distance $h$, the angular velocity of the wheel will be

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement($s$) is/are correct, when the rod makes an angle $60^{\circ}$ with vertical ? [ $g$ is the acceleration due to gravity]

$(1)$ The radial acceleration of the rod's center of mass will be $\frac{3 g }{4}$

$(2)$ The angular acceleration of the rod will be $\frac{2 g }{ L }$

$(3)$ The angular speed of the rod will be $\sqrt{\frac{3 g}{2 L}}$

$(4)$ The normal reaction force from the floor on the rod will be $\frac{ Mg }{16}$

A stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any ways. A ball of mass $m$ moving with speed $v$ collides elastically with the stick as shown in the figure. If after the collision the ball comes to rest, then what should be the mass of the ball ?