Why I can't use $\mu N$ for solving the friction force of the rolling wheel? Suppose we have a wheel rolling without slipping down an inclined plane, and we have to find the tangential acceleration of the wheel. The standard approach is to use the equation
$$f\cdot R=(I\cdot a)/R$$
But I was curious as to why we can't use $f=\mu N$ to find the frictional force and use that to calculate the tangential acceleration?
Any help would be appreciated.
 A: You are making a common misconception. $ f = \mu N  $ is only true when the static friction force acting is at its maximum value. For example, if you pull a slab of wood by a string until it starts sliding. That will be the maximum static friction value right before it starts sliding. However, in rolling without slipping, this is not the case. We have to use further calculations using the equation that you mentioned in your question to find the frictional force itself.
A: As you have indicated in your torque equation, the friction must adjust itself so that the angular acceleration is consistent with the linear acceleration. That gives you one equation with two unknowns.  You also need a linear force equation.  ( The friction may be less than the maximum value of $μ_s$N.)
A: Another point in addition to the answers provided earlier for a rotating object experiencing a force of friction.
For rolling without slipping the force of friction does no work because there is no relative displacement of the instantaneous point of contact with respect to the force.  The force of friction does provide a torque, but does no work.  This results in a simple energy balance for the object as it rotates down the plane: kinetic energy of the center of mass plus rotational energy about the center of mass plus gravitational potential energy is constant.
(For sliding, the force of friction does work.  But for a rigid body there can be no change in the energy within the body- no "heating"- and the work done by friction only contributes to a change in the kinetic energy of the object, considering the energy of rotation about the center of mass as well as the translational energy of the center of mass.)
Most basic physics texts only address rolling friction for a rotating object. The text Analytical Mechanics by Fowles evaluates both cases: rolling and sliding friction for a rigid body.
A: As it was pointed in the other answers, the problem that you have in mind involves static friction, i.e.,
$$
f\leq \mu N.
$$
Note that this friction force is actually involved in deriving the rolling equation cited in the question.
An issue that goes a bit beyond the standard textbooks problems (but just a bit) is that friction on realistic rolling objects is not reduced to a static friction - one need to account, at least, for the rolling resistance.
A: Actually while pure rolling the point touching the ground does not slip so static friction acts which is due to bonds formed between surface and the contact point. now these bonds formed can provide force to prevent the tendency of sliding and thus it can be variable.
Now as while slipping the contact point can require lesser value than your $\mu N$.
Static friction just arises to fulfil the needs to prevent slipping and not provide extra force.
Thus it is always not $\mu N$.
