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If a purely rolling body (say, a sphere) is first rolling on a rough surface and suddenly after a distance the surface is found to be smooth, then what effects does it have on the state of rolling of the body? Will it now slip?

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  • $\begingroup$ Are we talking about a level surface, or an inclined plane? $\endgroup$ Nov 8 '21 at 19:32
  • $\begingroup$ A level surface, my bad $\endgroup$ Nov 9 '21 at 4:33
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Suppose when the sphere reaches the smooth surface, it is rotating with angular speed $\omega$ and the centre of mass is moving with speed $u=r\omega$.

Since the surface is now smooth, there is no longer any torque on the body so it's angular acceleration becomes zero. Therefore it will continue to rotate with constant angular speed $\omega$.

However the centre of mass will accelerate with magnitude $g\sin\theta$, so that at time $t>0$ the speed of the centre of mass will be $v\ge u$. Therefore $v \ge r\omega$.

The sphere will therefore be both rolling and slipping.

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  • $\begingroup$ Ah yes thank you, but I forgot to mention in my question that I asked about a level surface, not inclined, I have edited my question. $\endgroup$ Nov 9 '21 at 4:36
  • $\begingroup$ This answer still applies. Just set $\theta=0$. Please do not edit your question in a way that invalidates existing answers. I have reverted the edit. $\endgroup$
    – Dale
    Nov 9 '21 at 4:45

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