# Velocity And Angular Velocity Of A Sphere Rolling Down A Ramp

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 $$\omega$$.

• Did you mean "and $r$ stays constant shouldn't increasing $v$ also increase $\omega$?"
– user191954
Dec 3, 2018 at 12:03
• Your problem lies in the last sentence. Which $v$ do you mean?
– user197851
Dec 3, 2018 at 12:04
• I mean the $v$ with which the sphere rolls down the plane. Dec 3, 2018 at 12:14
• Yes, it was supposed to be " shouldn't increasing v also increase ω?" Dec 3, 2018 at 12:15
• 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.)
– user197851
Dec 3, 2018 at 12:29

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.