Gyroscope (Coriolis Torque and Force)

[andyonassis]

Homework Statement

Using integration, show that the total coriolis torque acting on the gyroscope when the wheel has a mass M is:
$$ \vec{N} = -\frac{1}{2}Mr^2\omega\dot{\theta} \sin \alpha \hat{y} $$
This torque is the basis of the gyrocompass.
The Coriolis force can produce a torque on a spinning object.
The $\hat{x},\hat{y}$ and $\hat{z}$ components of $d\vec{N}$ about the origin due to the coriolis force in the $xyz$ coordinate system which acts on a point mass $m$ on the gyroscope's rim is given by:
$$ d\vec{N} = -2r^{2}\omega\dot{\theta}[(\sin(\alpha)cos(\theta)(\sin\theta)\hat{x}-\cos^{2}(\theta) \sin (\alpha) \hat{y})] \times [dm]$$
$\alpha$ is the angle that the gyro makes with the earth and $\theta$ is the angle of the gyroscope that rotates around its axis.

Relevant Equations

Equations for coriolis force and torque.

I know that to prove the total coriolis torque from the coriolis torque on a point mass is to express dm as a function of $d\theta$ and integrate from 0 to $2\pi$ and then the x component disappears due to orthogonality of sine and cosine. But i am stuck at other parts.

 

[chrisk]

This is a Symon problem. I found the given expression for the torque is incorrect. Attached is the solution and an explanation of why the solution does not agree with textbook. I know the policy is not to provide solutions but in this case it's necessary to show the error in the stated problem.

 

[berkeman]

I know the policy is not to provide solutions but in this case it's necessary to show the error in the stated problem.

In this case it's a 4 year old thread, so posting the solution is fine. Thanks for updating the thread with the correct solution.