Pure rolling of sphere having non uniform mass density ?

[gunparashar]

in case of rolling without slipping of a solid sphere having uniform mass density the condition is
Vcm (velocity of center of mass ) = Rω or [a][/cm] = Rα ,which comes from the fact that if an object that rolls without slipping the geometric center of the body travels 1 circumference along the ground for every for every full rotation it makes around the geometric center .
but in case of a sphere whose mass density is non uniform so the center of mass will not be at geometric center ,so for a non uniform sphere to undergo rolling without slipping what condition we should apply ?

 

[mfb]

The condition is the same if you apply it to the geometric center - it comes purely from geometry. For uniform mass you just have the nice feature that the center of mass is in the geometric center as well.

 

[andrewkirk]

The problem is simple if the sphere is initially at rest, because the sphere will roll 'towards' its centre of mass. By that I mean that the direction of rolling will be that of the projection on the floor plane of the vector from the sphere's point of contact with the floor centre to the centre of mass. The sphere will not complete a full revolution because the centre of mass cannot rise higher than it was at the beginning. But if it is helped along by a push in that direction, it can roll a full revolution or more, and the rule stated above in bold will apply.

If the sphere is initially rolling in a direction that is not aligned with the abovementioned vector, or if it is given a push that does not align with that vector, the motion will become more complex nd the bolded rule may not apply.