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1. Yes1. Wheel with radius ##R## on a horizontal surface: No friction

If the wheel sees an external torque it rotates, but does not translate. It slips.

2. Wheel on a horizontal surface: With Friction

If the wheel sees an external torque, it rotates and translates. If it sees a torque greater than ## R \mu_s N## it slips. slips means ##R \omega \neq v_{CM}##

3. Wheel on a horizontal surface with friction being held by an external force that stops translation. For some range of torque less than ## \mu_s N## the wheel remains static and does not slip. If the torque exceeds ## \mu_s N ## the wheel slips, but does not translate.

Are all these scenarios consistent with some understanding, or do I still not get something?

2. Not quite. The radius of inertia matters. If you consider moments and forces you get ##\tau\leq R\mu_sN(1+k^2)##, where the radius of inertia is

*kR*.

3. This one is ## R \mu_s N##. ## \mu_s N ## is not a torque.

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