I want to analyze the following system, where the mass of the axle is negligible and the Wheel is spinning.
In particular I need to calculate the amount of torque about the point where the string meets the axle.
Unfortunately I get two different answers depending on the procedure:
First (and correct) way
Taking into account the whole system, wheel and axle, there are two forces. The weight, with point of application the center of mass of the wheel; and a force with equal magnitude but opposite orientation, with point of application the place where the string meets the axle. Therefore there is a net torque.
Second (and wrong) way
Taking into account only the wheel, there are the same two forces, but both are applied to the center of mass of the Wheel. Therefore there is no torque.
There must be some amount of torque because otherwise the wheel wouldn't experience precession, and it does. So why is my second approach wrong?