Rolling without Slipping: A mistaken formula Ok, so I was playing around with some formulas and am a little confused now. I was trying to find the acceleration of a car on a normal road. For simplicity, I just use a single tire. I start with off with the rotational Newton's second law and the no slipping condition: $$ \alpha I = rf_{\rm friction}, \quad a = \alpha r$$ I let the moment of inertia be $I=\beta mr^2$, $\beta$ being the coefficient. Then,
$$\alpha \beta mr^2 = rF_f\\ a \beta mr= rF_f\\F_f = mg\mu \\ a = \frac{g\mu}{\beta} $$
So far I don't see any errors in my work so I accept this formula.
However, a cylinder rolling around an axis through its center has a coefficient where $\beta = 1/2$ which quite erroneously implies:
$$a=2g\mu\\ma = 2mg\mu\\\Sigma F = 2f_{\rm friction}$$
So in rolling without slipping, if a car is pushed forward by friction, how come the formula I got has a forward force of double the friction? I'd love any help in understanding my mistake here. Thanks in advance!
 A: The force of friction is not necessarily $mg\mu$. The formula $f=mg\mu$ refers to the max friction -- the $\mu$ here is the maximum static friction coefficient.
So when your wheel is rolling, you cannot immediately conclude that you're dealing with max static friction -- you must find out what your friction force is first.
Also, do note that if the wheel (or the car) is not accelerating, then in idealized conditions, there is no friction acting on the wheel -- the wheel will keep rolling and translating at a constant angular speed and constant velocity, and no torque or net force will be present. See here for more.
In summary, the friction force is between $0\leq f\leq f_{\rm max}=\mu N$, and you can't just assume it's $f_{\rm max}$.
A: In your equation: $\alpha I = rF_{friction}$, you are equating the wheel's angular acceleration to the torque placed on it from friction.
But for a car, friction is not the only torque.  Instead, the axle is being driven from the transmission or from the brakes.  You can't assume friction spontaneously arises.
