A sphere of uniform density has mass and radius . Find its moment of inertia about an axis through its centre.
A marble of uniform density is released from rest on a plane inclined at an angle to the horizontal. Let the time taken for the marble to travel a distance down the plane be: (i) if the plane is perfectly smooth; or (ii) if the plane is rough and the marble rolls without slipping.
Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that .
Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is for some constant . What is the new value for ?