Simple and driven harmonic motion

In summary: They are very concise and to the point. The problem with these lectures is that they are for a much more advanced audience than what you need for this question. So if you want to understand the whole concept behind the SHO, you will need to do more reading. However, if you just want the solutions, these lectures will be helpful.
  • #1
Andrei0408
50
8
Homework Statement
SUPERPOSITION OF TWO PERPENDICULAR SIMPLE
HARMONIC OSCILLATIONS : find the trajectory equation for a mass
point simoultaneously subjected at 2 SHO with the same frequency

DRIVEN HARMONIC MOTION : find the amplitude and the initial
phase for the steady solution and the resonance frequency and amplitude

In caps I wrote the name of the lecture, also how can I demonstrate the equation in the green box? I've also attached my attempt so far but I'm unsure how I should continue. Thank you!!
Relevant Equations
I've attached the equations that I believed to be relevant in the thread, I hope I was right
I know you can't solve it and just give it to me, I just want to know what I'm supposed to do, if you need any more information or clarification please let me know. Thank you for taking the time to help me!
 

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  • #2
The statement of the problem tells you what to do. For the first part you need to "
find the trajectory equation for a mass point simultaneously subjected at 2 SHO with the same frequency". The first figure you posted has already started you along the way by giving you equations (1) and (2). You need to manipulate them so that you can get the trajectory that appears in the green box.

For the second part you need to write down the differential equation for the driven harmonic oscillator and find the amplitude assuming that the transients have died out. This is something that you can find in any intermediate level textbook on Classical Mechanics and on the web.

Your request is too general for me to be more specific. Start working, post your work and we will help you to the extent that we can.
 
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  • #3
kuruman said:
The statement of the problem tells you what to do. For the first part you need to "
find the trajectory equation for a mass point simultaneously subjected at 2 SHO with the same frequency". The first figure you posted has already started you along the way by giving you equations (1) and (2). You need to manipulate them so that you can get the trajectory that appears in the green box.

For the second part you need to write down the differential equation for the driven harmonic oscillator and find the amplitude assuming that the transients have died out. This is something that you can find in any intermediate level textbook on Classical Mechanics and on the web.

Your request is too general for me to be more specific. Start working, post your work and we will help you to the extent that we can.
Thank you! I've solved the first one but I'm still having some trouble regarding the second one, and I wrote the differential equation. I also know the solution x(t) but I don't fully understand how I can get to that solution. Could you help me a bit more?
 

Attachments

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  • #4
Andrei0408 said:
Thank you! I've solved the first one but I'm still having some trouble regarding the second one, and I wrote the differential equation. I also know the solution x(t) but I don't fully understand how I can get to that solution. Could you help me a bit more?
Like I said, you can find the derivation of the solution in textbooks or on the web. For example, http://farside.ph.utexas.edu/teaching/315/Waves/node13.html
 
  • #5
There is a really good set of lectures about SHOs here:


The recitation lectures are the good part IMO.
 

1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion in which the restoring force is directly proportional to the displacement from equilibrium. This means that the object will oscillate back and forth around a central point, with the same period and amplitude.

2. What is the equation for simple harmonic motion?

The equation for simple harmonic motion is x(t) = A*cos(ωt + φ), where x(t) is the displacement from equilibrium at time t, A is the amplitude, ω is the angular frequency, and φ is the phase angle.

3. What is the difference between simple and driven harmonic motion?

Simple harmonic motion occurs when a system is oscillating due to a restoring force, while driven harmonic motion occurs when an external force is applied to the system, causing it to oscillate with a different frequency and amplitude.

4. How is simple harmonic motion related to the concept of equilibrium?

In simple harmonic motion, the system will always oscillate around an equilibrium point, where the net force is equal to zero. This means that the object will always return to its original position after completing one full oscillation.

5. What are some real-life examples of simple harmonic motion?

Some common examples of simple harmonic motion include the swinging of a pendulum, the motion of a mass on a spring, and the vibration of a guitar string. Other examples include the motion of a diving board, the movement of a metronome, and the oscillation of a child on a swing.

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