Paper 4, Section II, A

Consider a rigid body, whose shape and density distribution are rotationally symmetric about a horizontal axis. The body has mass $M$, radius $a$ and moment of inertia $I$ about its axis of rotational symmetry and is rolling down a non-slip slope inclined at an angle $\alpha$ to the horizontal. By considering its energy, calculate the acceleration of the disc down the slope in terms of the quantities introduced above and $g$, the acceleration due to gravity.

(a) A sphere with density proportional to $r^{c}$ (where $r$ is distance to the sphere's centre and $c$ is a positive constant) is launched up a non-slip slope of constant incline at the same time, level and speed as a vertical disc of constant density. Find $c$ such that the sphere and the disc return to their launch points at the same time.

(b) Two spherical glass marbles of equal radius are released from rest at time $t=0$ on an inclined non-slip slope of constant incline from the same level. The glass in each marble is of constant and equal density, but the second marble has two spherical air bubbles in it whose radii are half the radius of the marbles, initially centred directly above and below the marble's centre, respectively. Each bubble is centred half-way between the centre of the marble and its surface. At a later time $t$, find the ratio of the distance travelled by the first marble and the second. [ You may state without proof any theorems that you use and neglect the mass of air in the bubbles. ]

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