1) A uniform rod $OA$, of length $2a$, free to turn about its end $O$, revolves with angular velocity $\omega$ about the vertical $OZ$ through $O$, and is inclined at a constant angle $\alpha$ to $OZ$; find the value of $\alpha$.

[2019, 10M]

2) A circular board is placed on a smooth horizontal plane and a boy runs round the edge of it at a uniform rate. What is the motion of the centre of the board? Explain. What happens if the mass of the board and boy are equal?

[2008, 12M]

3) A plank of mass $M$ is initially at rest along a line of greatest slope of a smooth plane inclined at an angle $\alpha$ to the horizon and a man of mass $M^{\mathrm{\prime }}$ starting from the upper end walks down the plank so that it does not move. Show that he gets to the other end in time $\sqrt{{\displaystyle \frac{2M^{\mathrm{\prime }}a}{(M+M^{\mathrm{\prime }})g\sin \alpha }}}$, where $a$ is the length of the plank.

[2005, 30M]

4) A thin circular disc of mass $M$ and radius $a$ can turn freely about a thin axis $OA$, which is perpendicular to its plane and passes through a point $O$ of its circumference. The axis $OA$ is compelled to move in a horizontal plane with angular velocity $w$ about its end $A$.

Show that the inclination $\theta$ to the vertical of the radius of the disc through $O$ is ${\cos }^{-1}{\displaystyle \frac{g}{a{w}^2}}$, unless $w^2<{\displaystyle \frac{g}{a}}$ and then $\theta$ is zero.

[2002, 30M]