Does this solution exhibits resonance

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In summary, the conversation is about finding the general solution for a given equation and determining if it exhibits resonance. The solution is x = A sin( \omega_{0}t) + B cos (\omega_{0}t) + \frac {1} {\omega_0^2 - \omega^2} cos (\omega t). The denominator of the last term is important in determining resonance and a video is provided to demonstrate.
  • #1
ElDavidas
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In a question, I've been asked to find the general solution of the equation:

[tex] \ddot{x} + \omega^2{}_{0} = cos( \omega t) [/tex]

[tex] (\omega , \omega_{0} > 0, \omega \neq \omega_{0})[/tex]

I've found the solution to this.

It then asks me if this solution exhibits resonance. What does this mean and how do you determine this?
 
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  • #2
If you found the solution then it should be apparent whether it exhibits resonance. It might be helpful if you displayed your solution here.

Incidentally, if you have excluded the possibility of [itex]\omega = \omega_0[/itex] then you have excluded the possibility of [exact] resonance! :)
 
  • #3
Tide said:
It might be helpful if you displayed your solution here.

Ok, this is my answer:

[tex] x = A sin( \omega_{0}t) + B cos (\omega_{0}t) + \frac {1} {\omega_0^2 - \omega^2} cos (\omega t) [/tex]
 
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  • #4
The denominator of the last term is what you're after. What happens to the solution as [tex]\omega[/tex] -> [tex]\omega_0[/tex]? I'll try to find you a legendary video of resonant behaviour.edit: here's a small clip http://www.camerashoptacoma.com/mpegs/TacomaNarrowsBridge.mpg
 
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FAQ: Does this solution exhibits resonance

What is resonance in a solution?

Resonance in a solution is the phenomenon in which the electrons within a molecule or ion can freely move between different atomic orbitals, resulting in multiple possible structures for the molecule or ion.

How can I determine if a solution exhibits resonance?

To determine if a solution exhibits resonance, you can analyze the structure of the molecule or ion and look for the presence of multiple resonance structures. This can be done through techniques such as molecular orbital theory or Lewis structures.

What are the benefits of resonance in a solution?

Resonance in a solution can result in increased stability and delocalization of charge, making the molecule or ion less reactive. It can also contribute to the unique properties and behaviors of the solution, such as color and magnetism.

Can all molecules exhibit resonance?

No, not all molecules can exhibit resonance. Molecules with single bonds and no lone pairs of electrons cannot exhibit resonance because there are no multiple possible structures for the molecule.

How does resonance affect the properties of a solution?

Resonance can affect the properties of a solution by making it more stable, less reactive, and potentially changing its color or magnetic properties. It can also impact the acidity or basicity of a solution, as resonance can influence the distribution of electrons within a molecule or ion.

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