Lagrange Equations: Rectilinear Rod Rotating with Constant Angular Velocity

[Fabio010]

Consider a rectilinear rod (A-B) with negligible mass that is attached, without friction, to a vertical OZ axis. The rod rotates about that axis with a constant angular velocity ω and it maintains a angle α with OZ.

A particle of mass m moves about the rod and it is attracted by the gravity.a) Indicate, and justify the degrees of freedom of the material point.


To write the degrees of freedom i have to know the equations of x, y and z.

My problem is the z component equation.

For x and y we have the following equation:
x^2+y^2 = r^2

r is the radius (A-B)

Now i need the equation for z..

any types?

 

[TSny]

If I'm understanding the setup, wouldn't r and z make up two legs of a right triangle with $\alpha$ as an angle in the triangle? So, z can be expressed in terms of r and $\alpha$.