Gyrocompass motion: Zero torque in the free axis

[RicardoMP]

Hi!
At the moment I'm studying rigid body motion, more specifically, the gyrocompass. As you can see in the attached picture (Introduction to Mechanics -Kleppner-Kolenkow-Chap.7), the gyrocompass rotates about the z axis and the spin angular momentum is reoriented towards the z axis, creating a angular momentum along the AB axis. As I've read, the pivots at A and B allow the system to swing freely about the y-axis (which is along AB), so there can be no torque along the y axis. Why is that? Why is the torque along any free axis zero? I know that there are a couple of contributions to torque along the y axis, for example, the change of direction of the spin angular momentum. Why must the sum of all these contributions be zero? My objective is to understand intuitively this concept so any time I'm faced with such an axis, I immediately assume that the torque along it is zero.
Sorry for the long post and thank you in advance!

 

[A.T.]

As I've read, the pivots at A and B allow the system to swing freely about the y-axis (which is along AB), so there can be no torque along the y axis. Why is that?

Look at the definition of torque, if an applied force passes through an axis, then it produces no torque around that axis. If the forks A & B are frictionless, can they apply any force to the cylindrical axle, that doesn't pass right through it's axis?

 

[RicardoMP]

Look at the definition of torque, if an applied force passes through an axis, then it produces no torque around that axis. If the forks A & B are frictionless, can they apply any force to the cylindrical axle, that doesn't pass right through it's axis?

Oh! Indeed! It is that simple. And so, the sum of all contributions to the torque along that axis equals the cross product R x F, in this case is zero, since the angle between R and F is zero.
Thank you!.

 

[A.T.]

the cross product R x F, in this case is zero, since the angle between R and F is zero.

R x F is zero because R is zero, in this case.

 

[RicardoMP]

Ah yes, of course! My bad!
Thank you! :)

 

[A.T.]

Ah yes, of course! My bad!
Thank you! :)

Well depending on what R is meant to be, you can also say: R x F is zero because the angle between R and F is 180° .