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 \(M^{\prime}\) 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 \(\sqrt{\dfrac{2 M^{\prime} a}{\left(M+M^{\prime}\right) g \sin \alpha}}\), 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 \(\cos^{-1} \dfrac{g}{aw^2}\), unless \(w^2 < \dfrac{g}{a}\) and then \(\theta\) is zero.

[2002, 30M]