Derivation of resonant frequency for SHM systems

In summary: 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.
  • #1
Bonnie
18
1

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.
 
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  • #2
Bonnie said:

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. :smile:
 
  • #3
berkeman said:
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. :smile:

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?
 
  • #4
Bonnie said:
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. :smile:

Bonnie said:
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?
 
  • #5
berkeman said:
Just use the Upload button in the lower right of the Edit window to Upload the PDF file. :smile: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?
 

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  • PHYS205 Assignment 6.pdf
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  • #6
Thanks. So can you show us your work on these two problems now? :smile:
 
  • #7
berkeman said:
Thanks. So can you show us your work on these two problems now? :smile:
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 :/
 
  • #8
Bonnie said:
I have to 'derive an expression for the resonant frequency, ω0'
Bonnie said:
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? :smile:
 
Last edited:

Related to Derivation of resonant frequency for SHM systems

1. What is the resonant frequency for SHM systems?

The resonant frequency for SHM (simple harmonic motion) systems is the frequency at which the system will oscillate with the maximum amplitude or displacement. It is also known as the natural frequency of the system.

2. How is the resonant frequency derived for SHM systems?

The resonant frequency for SHM systems can be derived using the equation: f0 = 1 / 2π√(k/m), where f0 is the resonant frequency, k is the spring constant, and m is the mass of the system. This equation is based on Hooke's Law and the equation for the angular frequency of SHM.

3. Why is it important to know the resonant frequency of a SHM system?

Knowing the resonant frequency of a SHM system is important because it allows us to predict the behavior of the system and avoid any potential resonance disasters. Resonance can cause excessive vibrations and damage to the system, so understanding the resonant frequency can help engineers design systems to avoid resonance or operate them at a safe frequency.

4. How does the mass and spring constant affect the resonant frequency of a SHM system?

The resonant frequency of a SHM system is directly proportional to the square root of the spring constant and inversely proportional to the square root of the mass. This means that as the spring constant increases, the resonant frequency will also increase, and as the mass increases, the resonant frequency will decrease.

5. Can the resonant frequency of a SHM system be changed?

Yes, the resonant frequency of a SHM system can be changed by altering the mass or the spring constant. By changing the mass, the resonant frequency will change proportionally, and by changing the spring constant, the resonant frequency will change inversely proportionally. Additionally, damping can be used to decrease the amplitude of the oscillations and shift the resonant frequency to a lower value.

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