Time period of a conical pendulum by D'Alembert's principle

[justwild]

Homework Statement

Finding the time period of a conical pendulum by D'Alembert's principle. The string is of a constant length and all dissipations are to be ignored.

Homework Equations

The time period of a conical pendulum is $2\pi \sqrt{\frac{r}{g\tan\theta}}$. I need to arrive at this result starting from the D'Alembert's principle.

The Attempt at a Solution

I assume a cylindrical coordinates system whose origin coincides with the fulcrum of the pendulum. Thus, for the conical pendulum with constant string length, the acceleration of the particle would be
$\ddot{\vec{r}}=\ddot{\rho}\hat{\rho}+\rho\ddot{\phi}\hat{\phi}$.
Now the virtual displacement can be given as
$\delta\vec{r}=\rho\delta\phi\hat{\phi}$.
And the force acting on the particle as $\vec{F}=-mg\hat{z}$.

Now if I substitute all these in the D'Alembert's principle, I won't be able to calculate the time period from there. This is obvious because the virtual displacement does not contain the $\hat{z}$ term and due to the dot product, the expression won't include any $g$ term which is necessary because the result does contain the $g$ term.
I am out of ideas for now and I would appreciate anyone from the PF helping me out.

 

[TSny]

Hello, justwild.

What about virtual displacements in the $\hat{\theta}$ direction. For acceleration in spherical coordinates see here.

 

[justwild]

What about virtual displacements in the $\hat{\theta}$ direction.

I don't understand. Even if I assume a spherical coordinates system why would there be a component of acceleration in the $\hat{\theta}$ direction?
Isn't $\theta$ supposed to remain constant during the motion?

 

[TSny]

I don't understand. Even if I assume a spherical coordinates system why would there be a component of acceleration in the $\hat{\theta}$ direction?
Isn't $\theta$ supposed to remain constant during the motion?

Yes, for the particular solution that you are considering θ will remain constant. However, in http://en.wikipedia.org/wiki/D'Alembert's_principle]d'Alembert's[/PLAIN] principle you can consider any virtual displacement that doesn't violate the physical constraint of the system.

 

[ion santra]

Use Cartesian co-ordinates. Take the length of the string=constant to be the constraint eq. Find delta(z).
Substitute values in d'alemberts', put delta (z).
You'll be able to find the time period