The Paradox of Torque Along the Y-Axis for a Gyrocompass

[breez]

Hey, while reading Kleppner & Kolenkow's discourse on gyrocompasses, I ran into a statement I could not wrap my head around:

"The pivots at A and B allow the system to swing freely about the y axis, so there can be no torque along the y-axis."

AB is an axis along the y-axis, which is the axis along which the gyrocompass can freely swing. The spin axis is along the x-axis.

How can this be true? If the gyrocompass can swing along the y-axis from initially no angular velocity along the y-axis, it must have some torque to accelerate it, and hence torque along the y-axis. This seems extremely paradoxical to me. Can someone explain?

 

[pallidin]

Not sure if this helps, but I found this:

"A gyrocompass is essentially a gyroscope, a spinning wheel mounted on gimbals so that the wheel's axis is free to orient itself in any way. Suppose it is spin up with its axis pointing in some direction other than the celestial pole. Because of the law of conservation of angular momentum, such a wheel will maintain its original orientation. Since the Earth rotates, it appears to a stationary observer on Earth that a gyroscope's axis is rotating once every 24 hours. Such a rotating gyroscope cannot be used for navigation. The crucial additional ingredient needed for a gyrocompass is some mechanism that results in applied torque whenever the compass's axis is not pointing north..."

Source: http://en.wikipedia.org/wiki/Gyrocompass

 

[sir-pinski]

A gyrocompass works by utilizing the principles of angular momentum and precession. It consists of a spinning gyroscope that is mounted on a set of pivots, allowing it to rotate freely in any direction. The gyroscope is initially aligned with the Earth's rotational axis, or the North-South axis, and as it spins, it maintains this alignment due to its angular momentum. This means that the gyroscope will resist any changes in its orientation, including changes caused by the rotation of the Earth.

Now, when the gyrocompass is placed on a moving platform, such as a ship or an airplane, the Earth's rotation causes a torque on the gyroscope, trying to tilt it along the y-axis. However, because of its angular momentum, the gyroscope resists this torque and instead precesses, or rotates, along the x-axis. This precession continues until the gyroscope aligns itself with the new rotational axis, which is now the North-South axis of the moving platform. This allows the gyrocompass to always point towards true North, regardless of the motion of the platform it is on.

Going back to the statement you mentioned, the pivots at A and B are designed in a way that they allow the gyroscope to freely rotate along the y-axis, without any external torque acting on it. This is crucial for the gyrocompass to function properly, as any torque along the y-axis would interfere with the precession and accuracy of the compass. So, the statement is not paradoxical, as the gyrocompass is still able to rotate along the y-axis, but without any external torque affecting it.