Reasoning behind predicting a gyroscopes motion

Lets say we have a gyroscope that has a flywheel spinning with constant angular velocity. The gyroscope is held in place in a position where the massless metal rod connecting to the flywheel is parallel to the ground. We now let go and are asked to predict the gyroscoped motion.

When researching gyroscopes the predicted motion of precession at constant vertical height was explained by saying, in essence, that the precession causes the angular momentum to change in the correct way, the derivative of angular momentum being equal to torque, based on the gravitational torque acting on the system. This question is not so much about gyroscopes specifically but getting a better grasp of the way one reasons in physics.

To me the above explanation seems incomplete. My reasoning would be> if the normal force is not equal to the weight, the body will move vertically causing a change in angular momentum in a direction in which there is no torque => contradiction. Now the only possible motion that remains to ensure that torque is equal to the derivative of angular momentum is precession. Does my reasoning make sense? I am right in my feeling that the first argument is not satisfactory or am I misunderstanding something?

• The pure precession model is an approximation in the limit of unbounded angular momentum. A more complete model is given in, for instance, arxiv.org/abs/1007.5288 which references Feynman's explanation in the The Feynman Lectures where he addresses exactly how you should be reasoning about such a system. Jun 18 '17 at 20:28