In the figure below it is stated that the friction force should always be greater than the tangential force in order to prevent slipping between two frictional wheels. My question is that if we apply newton's second law to that contact point won't the tangential force resultant points upwards with the direction of the friction? And thus the wheels have an opposite rotating direction? 
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1$\begingroup$ The diagram does not make clear which body is force $P$ applied on. Also consider, that Newton's law only applies on the center of mass of objects, and not any arbitrary point (at least in the calculation of acceleration). $\endgroup$– John AlexiouCommented Feb 6, 2022 at 20:46
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1 Answer
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You need to be careful in your analysis. When the wheels spin without slipping they naturally spin in opposite directions. The left one clockwise and the right one counterclockwise, as drawn. So the concern is them spinning in the same direction, not opposite directions.
By Newton’s 3rd law they will have an equal and opposite force. And those opposite directed forces will produce same directed torques about their respective axes. But the motion those opposite directed torques will act to produce depends on other factors.
First, their initial tangential velocities. If the initial tangential velocities do not match, then those torques will act to make the fast one slow down and the slow one speed up until they do match.
Second, any external torques. The torque between the two wheels can serve to transport external torque from one wheel to the other. If they are initially co-rotating then an external torque which would increase the speed of one wheel will be reduced by the friction torque, and the other wheel’s speed will be increased by the friction torque. This will keep them spinning together if the force is less than the slipping friction.
Finally, if they already have matching tangential velocities and if there are no external torques then there will be no friction (in the ideal case). They will spin together freely with no friction or frictional torque.
