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For what max of vertical force acting tangentially upward ,will the cylinder undergo pure rolling in a rough horizontal surface?

In this case ,why can't we put torque equation about the axis touching the ground to find angular acceleration ?my approach

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You can find the angular acceleration about an axes passing through the point of contact of the cylinder and the ground , in fact you have done it correctly .

I think what you are missing is that the frictional force causes acceleration .

Hence
$f= m(Acm) $

Where the acceleration of the center of mass is given by :

$Acm = r\alpha$

Putting this in your equation :

$F= \frac 12 mr\alpha + uN $

You get :

$F = \frac 32 mr\alpha $

As you have already calculated.

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