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

[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 α to the horizon and a man of mass M′ 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 √2M′a(M+M′)gsinα, 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 θ to the vertical of the radius of the disc through O is cos−1gaw2, unless w2<ga and then θ is zero.

[2002, 30M]