Why is only one component of spin angular momentum considered in the precession of a top?

Question

This was how the orbital angular velocity/ angular velocity of precession was shown to be calculated in my textbook ($\phi$ is angle made by top with vertical)

$$dL=L_\text {spin}\sin\phi d\theta$$

But , $$\omega_ \text{precs} =\frac{d\theta}{dt}$$

$$\omega_ \text {precs}= \left|\frac{d\vec L}{dt}\right| \left[ \frac{1}{L_ \text {spin}\sin\phi} \right]$$

Now $\frac{d\vec L}{dt}$ is equal to torque from gravity $=mgr \sin\phi$

So, $$\omega_\text {precs}= \frac{mgr}{L_\text {spin}}$$

My doubt is : Why is the torque changing only the horizontal component of the angular momentum vector, if its also perpendicular to the vertical component (since its direction is out of the plane)?

What I am thinking is that the torque will pull down on the vertical component and slowly align it with the direction of the torque itself. While this is happening it will of course also be shifting around the horizontal component but remain always perpendicular to it. As the previously vertical component will also now be in the same direction as torque this means that the top is no longer vertical at all? And is simply going around circles on the ground? I don't know, I am really confused.

Answer 1

The torque is always horizontal (in the direction of precession). Since $\vec{\tau}=d\vec{L}/dt$, the vertical component of $\vec{L}$ does not change.

However you are right in your intuition that the torque doesn't only affect the horizontal component of the "spin (figure) axis" of the top. For a fast top, this axis is very closely, but not perfectly, aligned with the angular momentum. The torque causes the the spin axis to oscillate up and down as well. This is motion is called nutation. A spinning top precesses and nutates simultaneously. This is illustrated by the following figure from Goldstein's Classical Mechanics:

Here, the direction of the spin (figure) axis of the top is visualized for three different initial conditions. The nutation amplitude ($\theta_1 - \theta_2$) is approximately inversely proportional to the square of the spin angular velocity: the faster the top spins, the less it nutates. The nutation frequency is approximately proportional to the spin angular velocity. For a very fast top, nutation can be imperceptible. Since the figure axis closely tracks the angular momentum, your equation for the precession angular velocity is still valid.