Harmonic oscillations of the electromechanical system (normal modes)

[sergiokapone]

Homework Statement

http://imagizer.imageshack.us/v2/275x215q90/661/kIVMcC.png
Mathematical pendulum is the part of the oscillating circuit.
The system is in a constant uniform magnetic field. Oscillations is small. Find the normal modes of oscilations.

Homework Equations

$\begin{cases} ml^2\ddot \phi+ mgl\sin\phi=1/2\dot q Bl^2 \\ \frac{q}{C}+L\ddot q = 1/2Bl^2\dot \phi \end{cases}$
+
$\sin\phi \approx \phi$

The Attempt at a Solution

[/B]
Submit a solution in the form $q=q_0e^{i\omega t}$ and $\phi=\phi_0e^{i\omega t}$.
It is more clear to me how to find for a solution. But I'm not sure for a signs near right sides of equations, e. g. near the $1/2\dot q Bl^2$ and $1/2 Bl^2\dot \phi$. How I can determine right signs.

I get right sides in first equation as mean Ampere force: $F=1/2IlB=1/2\dot q l B$,
and in second equation as EMF: $\epsilon=-B\frac{dS}{dt}$, where $dS=1/2lv$, $v=-l\dot \phi$. Thus, $\epsilon=\frac{Bl^2}{2}\dot\phi$

Another words, how can I determine right signs for the Ampere force and EMF?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[Dr.D]

You have not indicated the direction you are taking for positive current; seems to me like this is a first step.

 

[sergiokapone]

You have not indicated the direction you are taking for positive current; seems to me like this is a first step.

Ok, let it be clockwise.

 

[paranormal]

I would first commend the student for their attempt to find the normal modes of oscillations in this electromechanical system. The equations and diagrams provided show a clear understanding of the physical setup and the mathematical concepts involved.

To answer the question about determining the right signs for the Ampere force and EMF, I would suggest looking at the direction of the forces and currents involved. In the first equation, the Ampere force is acting in the direction of the velocity of the charge, which is determined by the sign of the dot product between the magnetic field and the velocity. Similarly, in the second equation, the EMF is acting in the direction opposite to the change in magnetic flux, which is determined by the right-hand rule.

In general, it is important to pay attention to the direction of forces and currents in electromechanical systems, as they can greatly affect the behavior of the system. I would encourage the student to continue exploring and learning about these types of systems, as they have many real-world applications and can lead to interesting and valuable research opportunities.