Derivation of resonant frequency for SHM systems

[Bonnie]

Homework Statement

My question here isn't a specific question that has been given for homework, but a more general one. For an assignment I have to 'derive an expression for the resonant frequency, ω₀' for two different systems, the first for 'a mass M connected to rigid walls via two springs' and the second 'a free oscillator composed of an LC circuit'.
Without specificity to either question, what formula/'recipe' should I follow in general, in order to derive these expressions?

Homework Equations

The Attempt at a Solution

I am pretty sure I will have to start by finding an equation of motion? The problem here is less the algebra and more the understanding behind it.

 

[berkeman]

Homework Statement

My question here isn't a specific question that has been given for homework, but a more general one. For an assignment I have to 'derive an expression for the resonant frequency, ω₀' for two different systems, the first for 'a mass M connected to rigid walls via two springs' and the second 'a free oscillator composed of an LC circuit'.
Without specificity to either question, what formula/'recipe' should I follow in general, in order to derive these expressions?

Homework Equations

The Attempt at a Solution

I am pretty sure I will have to start by finding an equation of motion? The problem here is less the algebra and more the understanding behind it.

For the springs+mass problem, use the equations of motion and energy for spring-type problems. Can you list those for us?

For the LC circuit, you will start with the differential equations for the voltage and current behaviors for inductors and capacitors. Can you list those for us?

Thanks.

 

[Bonnie]

For the springs+mass problem, use the equations of motion and energy for spring-type problems. Can you list those for us?

For the LC circuit, you will start with the differential equations for the voltage and current behaviors for inductors and capacitors. Can you list those for us?

Thanks.

file://file/UsersB$/bem60/Home/My%20Documents/2nd%20Year/Phys205/PHYS205%20Assignment%206.pdf
^I have attempted to attach the file, but I doubt it will work unfortunately
For the spring question:
Equation of motion: ma = -Kx
so a +(K/m)x = 0

Energy: Ek = 1/2mv²
And Ep = 1/2Kx²
Total energy is conserved, so 1/2Kx² + 1/2mv² = E

For the LC circuit:
We use Kirchoff's Loop law:
-L(dI/dt) - (Q₂/C) + (Q₁/C) = 0

Is this the information you meant?

 

[berkeman]

file://file/UsersB$/bem60/Home/My%20Documents/2nd%20Year/Phys205/PHYS205%20Assignment%206.pdf
^I have attempted to attach the file, but I doubt it will work unfortunately

Just use the Upload button in the lower right of the Edit window to Upload the PDF file.

Is this the information you meant?

Pretty close. For the spring equations, I'd add in the force equation. Are there diagrams of the problems that are included in the PDF you will Upload?

 

[Bonnie]

Just use the Upload button in the lower right of the Edit window to Upload the PDF file.


Pretty close. For the spring equations, I'd add in the force equation. Are there diagrams of the problems that are included in the PDF you will Upload?

 

[berkeman]

Thanks. So can you show us your work on these two problems now?

 

[Bonnie]

Thanks. So can you show us your work on these two problems now?

Unfortunately I'm not sure what to do once I have the general equations of motion, I'm not looking for someone else to answer my question, just some guidance as to where to go from here :/

 

[berkeman]

I have to 'derive an expression for the resonant frequency, ω0'

I'm not sure what to do once I have the general equations of motion

One way to do it is to write the equations for the displacement of the mass as a function of time, and then solve them using the initial condition of the mass displaced from its equilibrium position by some amount. In the solution you will get a sinusoidal term, and the frequency of that term will be ω0.

For the LC circuit you would do the same thing, and make the initial condition as either a starting voltage across the caps, or a starting current around the circuit.

Does that help? Can you show us the equation of motion for the mass/spring system? Then what happens when you solve them with the IC being a starting displacement of the mass?