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I was given a question on a recent test that asked me the following:

A sphere is allowed to roll down a smooth inclined plane (No Friction), as it rolls down does its velocity remain constant, and does its angular velocity remain constant?

Now obviously I answered that both the velocity and angular velocity increase as the sphere rolls down the ramp, but I was marked wrong, my teacher said that the angular velocity doesn't increase and remains constant, because there isn't a torque acting on the sphere, but the velocity does increase.

Now here's my question, even if there wasn't a torque acting on the sphere, the velocity was still increasing, and since $v=r\omega$, and $r$ stays constant shouldn't increasing $v$ also increase $\omega$.

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  • $\begingroup$ Did you mean "and $r$ stays constant shouldn't increasing $v$ also increase $\omega$?" $\endgroup$ – user191954 Dec 3 '18 at 12:03
  • $\begingroup$ Your problem lies in the last sentence. Which $v$ do you mean? $\endgroup$ – user197851 Dec 3 '18 at 12:04
  • $\begingroup$ I mean the $v$ with which the sphere rolls down the plane. $\endgroup$ – Kosh Rai Dec 3 '18 at 12:14
  • $\begingroup$ Yes, it was supposed to be " shouldn't increasing v also increase ω?" $\endgroup$ – Kosh Rai Dec 3 '18 at 12:15
  • $\begingroup$ But the formula $v=r\omega$ does not involve the speed with which the sphere is moving. It refers to a different $v$. Remember, in your case, there is no friction. (Also perhaps the word "rolls" is misleading, and the word "slides" would be more accurate.) $\endgroup$ – user197851 Dec 3 '18 at 12:29
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A sphere is allowed to roll down a smooth inclined plane (No Friction), as it rolls down does its velocity remain constant, and does its angular velocity remain constant?

We wouldn't normally refer to the sphere as "rolling" when there's no friction acting on it. "Sliding" (or "slipping") is the preferred term in that situation.

Now here's my question, even if there wasn't a torque acting on the sphere, the velocity was still increasing, and since $v=rω$, and $r$ stays constant shouldn't increasing $v$ also increase $ω$.

As the sphere slides down the incline, its velocity $v$ increases due to the Earth's gravitational acceleration.

However, without friction (and thus without torque) $\omega$ remains constant. $v=rω$ only applies for rolling without sliding, where there is enough friction to prevent sliding altogether.

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