Minimum angular velocity of a point mass on a rotating rod

[Rob2024]

Homework Statement

A point mass $m$ is moving on a circular horizontal path attached to a massless rod length $l=50$ cm. The other
end of the rod is attached to the pivot point $O$ on a infinitely long vertical rod. There is no friction
along the vertical rod.

When $O$ is held motionless, the rod makes an angle 20$^\circ$ with the vertical direction.

If the pivot point $O$ is released and it falls on the rod without friction. What's the minimum angular speed of the mass during the subsequent motion?

Relevant Equations

Coonservation of angular momentum
Conservation of energy

Because the system has no friction, when the point mass falls, I think both angular momentum and energy are conserved. Further because the rod is massless, there cannot be a vertical force from mass on the rod. The difficulty is to determine the property of tension. It determines the subsequent motion of the point mass. I am not too sure where to go from here, It seems odd the tension in the rod is not along the rod. Is this a bad question?

Thanks for your help.

 

[haruspex]

What can the net torque on the angled rod be? What does that tell you about the forces between the two rods?

 

[Rob2024]

hmm there cannot be any net torque either, this implies the point mass cannot exert horizontal force on the rod. This means the point mass travels at constant horizontal velocity until the rod becomes horizontal. I think I see what's happening.

 

[haruspex]

hmm there cannot be any net torque either, this implies the point mass cannot exert horizontal force on the rod. This means the point mass travels at constant horizontal velocity until the rod becomes horizontal. I think I see what's happening.

What about gravity? Or did you just mean its horizontal component is constant?

 

[Rob2024]

What about gravity? Or did you just mean its horizontal component is constant?

The component.

 

[haruspex]

The component.

Ok.
So what will be the trajectory of the mass, and up to what point?
The question asks for the "minimum angular speed". It’s not clear to me how they are defining that. I assume they mean "the minimum magnitude of the angular velocity about the vertical rod", whereas taking it literally you'd have to consider the curve of the path in 3D.