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For a rigid smooth rolling ball rolling down a ramp (as seen above), the acceleration of the center of mass is given by:

$$a_{\text{com}, x} = - \frac{g \sin θ}{1+\frac{I_\text{com}}{MR^2}}.$$

However, if $θ=0$, the acceleration would be zero, meaning that the ball would travel at constant velocity. Why wouldn’t there be acceleration in the backwards direction caused by the static frictional force?

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Real rolling objects on a horizontal plane (i.e., for $θ=0$) do not slow down due to static or dynamic friction but aerodynamic drag and rolling friction (caused by deformation of the rolling object). The former two occur when two surfaces move against each other, which does not happen during pure rolling motion, i.e., as long as the rolling condition $ωR = v$ is met. As rolling friction is excluded (your ball is rigid) and air resistance is not mentioned, the ball will just continue rolling when on a plane.

Static friction plays a role on an inclined plane since it causes some of the downhill force to accelerate the ball rotationally instead of linearly. (This is as long as the downhill force is smaller than the static friction.) If you place a resting, non-rotating ball on an inclined plane without static or dynamic friction, it would simply slide down the plane without rotating.

Dynamic friction becomes relevant if the rolling condition is not met. We then have a rolling-with-slipping scenario, and the sliding friction linearly decelerates and rotationally accelerates the ball or vice versa until the rolling condition is met.

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    $\begingroup$ @ruakh: I mixed up the signs there: It has to be smaller instead of bigger. If the downhill force becomes larger than static friction, the ball will slip, and it will be dynamic friction making the ball turn (but never achieving rolling conditions). Let me edit that … $\endgroup$
    – Wrzlprmft
    Apr 22 at 21:00
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In the derivation of $a$ as a function of $\theta$ there is an implicit assumption that the ball does not slide down the slope at all, so its angular acceleration is $\dot \omega = \frac a R$, and so $f_sR = I \dot \omega = \frac {Ia} R$. But if $a=0$ then this condition implies that $f_s=0$ too, so on level ground the ball can travel at constant linear velocity $v$. To avoid sliding the ball must also have constant angular velocity $\omega = \frac v R$.

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There is not static friction. Not under ideal circumstances. If you spin a wheel in outer space, then it will just keep spinning. No need for static friction to keep up the rotation, since nothing is stopping the rotation.

Same for a wheel rolling on horizontal ground under ideal circumstances. It just keeps rolling, since nothing stops the rotation - no need for a static friction to be present. Thus there is no force present that could do negative work and thus reduce the rotational kinetic energy.

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  • $\begingroup$ If the ball is rolling, not sliding, then there must be static friction between the ball and the ramp. $\endgroup$
    – Solomon Slow
    Apr 22 at 12:58
  • $\begingroup$ @SolomonSlow there must be static friction if the ramp isn't horizontal. If it is horizontal, which is what the OP is asking about, it's not really a ramp but the static friction would go to zero. $\endgroup$
    – M. Enns
    Apr 22 at 13:42
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    $\begingroup$ @M.Enns, For a ball rolling at constant velocity on a level surface, I would say that the magnitude of static friction is zero, which, to my mind, is slightly different from saying that static friction "does not exist." Steeven said, "There is not static friction." Maybe I was mistaken, but it looked to me as if he was saying it would make no sense to even talk about static friction in the experiment regardless of whether its magnitude was zero or not. $\endgroup$
    – Solomon Slow
    Apr 22 at 13:53
  • $\begingroup$ @SolomonSlow I would not in physics distinguish between a zero vector and a non-existing vector. $\endgroup$
    – Steeven
    Apr 22 at 16:39
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Why wouldn't there be acceleration in the backwards direction caused by the static frictional force?

Static friction only exists to prevent sliding when there is an applied torque or horizontal force acting on the ball. On a horizontal surface, once the ball is rolling, if there are no applied or opposing horizontal forces (air resistance, rolling resistance, etc.) the ball will continue rolling due to its inertia. There would be no need for static friction. The surface might just as well be frictionless.

Hope this helps.

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    $\begingroup$ And here I was the demiurge had invented static friction existed to keep my trousers from falling off. (Joke aside, I find it confusing to assign a purpose to static friction and the resulting argument difficult to follow.) $\endgroup$
    – Wrzlprmft
    Apr 22 at 14:03
  • $\begingroup$ @Wrzlprmft suppose there is a box sitting on a horizontal rough floor. Do you think there is a static friction force acting on it? $\endgroup$
    – Bob D
    Apr 22 at 15:09
  • $\begingroup$ No, but I don’t see what this has to do with my criticism. $\endgroup$
    – Wrzlprmft
    Apr 22 at 15:15
  • $\begingroup$ @Wrzlprmft I’m getting there. Please be patient. $\endgroup$
    – Bob D
    Apr 22 at 15:18
  • $\begingroup$ @Wrzlprmft now let’s say I push on the box with a force of 5N and the box doesn’t move. Is there a static friction force and if so what is it and in what direction? $\endgroup$
    – Bob D
    Apr 22 at 15:24

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