Slipping Sphere on Steep Incline

[dreamnoir]

I'm trying to wrap my head around how a rolling and slipping sphere would be behave on an incline (with an angle of θ) that is too steep for pure rolling. I believe I understand the behaviour up to that point but once we reach the position where the amount of friction required to maintain rolling, F > μN, is surpassed my understand falls apart. It's at this point we'll no longer see pure rolling occur I'm having some trouble figuring out what happens exactly.

When we have a ball moving a flat surface and it's rotating too quickly (front spin), vR < ω, then we have a positive translational acceleration and negative rotational acceleration due to friction, or a > 0 and [itex]\alpha < 0 [/itex]. When it's rotating too slowly (back spin), vR > ω, then we have a negative translation acceleration and positive rotational acceleration due to friction, a < 0 and [itex]\alpha > 0[/itex]. On a surface that the ball can roll on eventually vr = ω will become true and the ball will be rolling.

When we have a incline with too steep an angle and F > μN then we never get into pure rolling. However, I'm not sure what happens here. Since the rotational acceleration, [itex]\alpha = \frac{\mu NR}{I}[/itex], isn't linked to the translation acceleration, a = sinθg - μcosθg, we can end up up in the situation where [itex]\alpha > aR[/itex] so we end up with front spin after some time interval. This switches the condition of slipping and the friction direction reverses until we no longer have front spin and now it's back spin. This behaviour seems to oscillate back and forth forever.

I don't think this happens and something else is going on. It seems like friction would always oppose the relative direction of motion at the point of contact but if that's true I don't understand entirely how rolling with slipping down a steep incline works then. Some explanation would very helpful.

 

[jbriggs444]

When we have a incline with too steep an angle and F > μN then we never get into pure rolling. However, I'm not sure what happens here. Since the rotational acceleration, α=μNR I , isn't linked to the translation acceleration, a = sinθg - μcosθg, we can end up up in the situation where α>aR so we end up with front spin after some time interval. This switches the condition of slipping and the friction direction reverses until we no longer have front spin and now it's back spin. This behaviour seems to oscillate back and forth forever.

If α>aR then you are not into the slipping regime. The frictional force is able to supply enough angular acceleration to match the translational acceleration and you roll without slipping. Or, if you're already slipping, the roll rate is changing to match the translation rate.