Consider the case of a common demonstration which you would find in videos. A bicycle wheel mounted on a shaft is set into rotation, axis is made horizontal and the shaft stays horizontal for sometime and eventually goes down as the wheel spin slows down due to friction.

My questions are

1) What torque balances the gravitational torque vertically and the precession torque horizontally (applying Newton's second law) ?

2) Let the torque resisting the vertical fall of shaft be R and torque due to gravity be T. Since the shaft finally tilts down is R more than T and why does this happen?

3) How is the resisting torque (against gravity) R dependent on
a)Rate of spin
b)Gravitational torque T ?

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In short, the angular momentum is what causes it to stay up.

Forces alter the linear momentum vector (p). Torques alter the angular momentum vector (L). Looking at a gyroscope, the angular momentum is pointed up in some direction. The radius vector is pointed in the same direction as the angular momentum towards the center of mass. The force due to gravity is pointing down from the center of mass. The torque is then:

$\tau$ = r$\times$F

Which, using the right hand rule, acts perpendicular to both the force and radius vectors which points the torque sideways. The torque "pulls" on L. $\tau$ is always pependicular to L; this is circular motion.

Obviously, this only works if the angular momentum is strong enough to counteract the gravitational force. Friction will eventually slow down the angular velocity.

1. There is no vertical gravitational torque; it points sideways. The gravitational torque is the precessional torque, and nothing balances it. So the gyroscope spins.

2. Remember, torques are orthogonal to forces. It can be difficult to grasp this concept at first, but that is the difference maker.

3. There is no resisting torque as I said earlier, but the larger the gravitational force, the faster the precession. The larger the angular momentum, the slower the precession.