Angular velocity of a rod rotating around a vertical axis

[alaix]

Homework Statement

An homogeneous rod is fixed to an extremity and is rotating around a vertical axis, doing an angle of theta with the vertical. (in the scheme I made I wrote alpha but it's theta! (θ))

If the length of the rod is L, show that the angular velocity needed to make it turn is

\omega = \sqrt{3g/2L cos(θ)}

Homework Equations

\tau = r x F
I = 1/2 m L²

\tau = I\alpha

The Attempt at a Solution

Here is what I tried

I considered that all the exterior forces (ie. gravity) was acting on the center of mass of the rod, which is situated in the middle, at L/2.

Therefore

Torque = r x F = 1/2 L mg sin(θ)

Torque = I\alpha

Where I = 1/3 mL²

Therefore

\alpha = \frac{3mgL sin(θ)}{2mL^{2}} = \frac{3g sin(θ)}{2L}

Since I'm looking for the angular VELOCITY, and since angular acceleration = d\omega/dθ

\alpha dθ = d\omega

By integrating both sides I find

\omega = -\frac{3g cos(θ)}{2L}


Which is ALMOST the answer I'm looking for... what am I missing?

Thanks!

 

[cepheid]

Homework Statement

An homogeneous rod is fixed to an extremity and is rotating around a vertical axis, doing an angle of theta with the vertical. (in the scheme I made I wrote alpha but it's theta! (θ))

If the length of the rod is L, show that the angular velocity needed to make it turn is

\omega = \sqrt{3g/2L cos(θ)}

There is a lot of ambiguity here as to what the final answer is supposed to be. You should include some extra parentheses under the square root sign to make it clear what's in the numerator and what's in the denominator.

Homework Equations

Since I'm looking for the angular VELOCITY, and since angular acceleration = d\omega/dθ

\alpha dθ = d\omega

No, actually that's wrong: \alpha \neq d\omega/d\theta. Rather, \alpha = d\omega / dt.