Simple harmonic motion and the properties of waves.

Discussion in 'Physics & Math' started by Elucinatus, Aug 25, 2008.

  1. Elucinatus Registered Member

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    36
    I am seeking some help about the properties of waves and simple harmonic motion.



    - What is it that retards the motion of a spring? Why does a spring not oscillate back and forth for all eternity in simple harmonic motion?

    - Aren't periodic waves just tons of pulse waves grouped together? What is the actual difference between pulse waves and periodic waves?

    - When particles from the medium of a sine wave undergo cyclical motion and are for a time displaced, oscillating in simple harmonic motion, what is it that actually causes the particles to oscillate. As in up....AND DOWN. As far as I am sure of in my studies mediums for waves such as water and air do not have the same elastic properties to follow simple harmonic motion that a spring has.

    - I believe in simple harmonic motion theory, the theory is "changing the amplitude does not affect the period, or duration of oscillation". How is this possible when it is the amplitude (displacment of the spring) which determines the acceleration of the spring?


    I appreciate any and all help or criticism. It will help me greatly in my studies of physics. I am following the model of a spring for simple harmonic motion if that clears up any confusion.
     
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  3. Reiku Banned Banned

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    The necessary conditions accepted in a Harmonic Motion, is that a restoring force exists, so that \(F_{R}=-kx\), where k is a proportionality constant, and x is the displacement.

    For k, to explain what the proportionality constant means, is that for two quantities, such as x and y, are only proportional if there exists a functional relationship with a constant and nonzero number, so that y=k(x).

    So if you travelled at a constant speed, then the distance and time are proportional, and the proportional constant is the speed.

    From \(F_{R}=F_{RH}\), a restoring for causing the motion, is exactly the same as the horizontal componant called the centripetal force.

    So,

    \(-kx=-m (4\pi^{2}x/T^{2}\) is substituting definitions of both, so

    \(k=m4\pi^{2}/T^{2}\) is valid, because -x is the same in both cases, and so is k.

    If you worked the algebra through, you will finally get \(T=2\pi \sqrt{m/k}\) which is the equation that shows that the period of simple harmonic motion [T], is calculatable using the mass of the object, and the constant of proportionality for the displacement of the object.

    So if the mass is larger, the longer the period of vibration, also, the stiffer the spring, the less vibration. This should answer your first question.
     
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  5. Elucinatus Registered Member

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    What is it that causes the vibration and oscillation to eventually give way and become motionless?
     
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  7. Reiku Banned Banned

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    Energy. There will be a given force on a spring, that gives way by pushing against the earth, and the energy will weaken.
     
  8. Trippy ALEA IACTA EST Staff Member

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    A spring has stiffness, as Reiku pointed out, but the reason why a mass on a spring doesn't just keep going and going and going is because it takes work to overcome that stiffness. Essentially what happens, is every time the spring bends away from the centerline/rest position, the spring actually experiences a little bit of elastic deformation part of the metal that makes up the spring itself becomes compressed, and part of it becomes stretched. Ultimately, which part becomes stretched, and which part becomes compressed depends on which direction the mass is moving in (compression in the direction of motion, extension in the opposite direction). This alternating compression and extension transforms the kinetic energy of the motion of the mass into heat energy, which is radiated away, and also transferred through the body of the spring.

    There's also a small amount of kinetic energy lost as work done to overcome the friction of the air.


    First you need to clarify what you mean by periodic waves. Are you referring to standing waves?

    It's a little more complicated then that - in essence, a medium has some form of restoring force. When talking about water, liquids have a tendencey to completely fill the container, but when you look at an ocean wave (for example) over a full wave length, you've effectively 'dug a hole' in the water, and then 'piled it up' beside the hole. That difference in potential provides the restoring force. Also bear in mind that the water that was piled up, and the water that was pushed aside will tend to continue travelling in the direction that they're going in once they reach the 'zero' level (IE the 'falling' water will continue moving downwards, and the 'rising' water continues moving upwards) however, moving in either direction requires doing work to overcome a force, so it only moves a certain distance - real ocean waves ar emore complicated then this, because any particle in an ocean wave tends to move in a circle, rather then simply up and down. Something similar happens in air. You create the initial compression (or expansion), and the restoring force is provided, essentially, by a combination of gravity, and a fluids tendency to fill part or all of the volume it's contained within, and the impetus to form the next part is provided by the momentum of the particles (as it is with a mass on a spring).

    Believe it or not, you've answered you're own question. Because the displacement determines acceleration, displacing it further makes it move faster, so it ends up covering a greater distance, faster, but in the same amount of time.

    First we have F=kx, and F=ma.

    So, if we double the displacement, we double the force, and we double the acceleration.

    But, if you think back to your linear kinematics, and to the equation:

    x=(at²)/2 + v₀t + x₀

    If we take the special case where v₀= x₀=0 then we get:

    x=(at²)/2

    Which, because it's time we're interested in, rearranges to:

    t=√(2x/a)

    So, bearing in mind that earlier we showed that doubling the displacement doubles the acceleration, and we've just shown that the time taken to travel through one quarter of the cycle is proportional to the displacement divided by the acceleration, then we can see that doubling the acceleration and doubling the displacement has no effect on the time taken to transverse that distance.

    Hope that helps (and makes sense).
     
  9. Trippy ALEA IACTA EST Staff Member

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    Pipped at the post while typing.

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  10. Reiku Banned Banned

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    Oh gosh!

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  11. przyk squishy Valued Senior Member

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    Friction, both internal (within the spring as it deforms) and some air resistance.
    Something like that. Actually after you learn Fourier analysis, you'll probably be more likely to do it the other way around: you can express a single pulse as a superposition of periodic waves (a "wave packet").
    The interatomic forces between the constituent particles of the medium: try to push them too close or pull them too far apart and they'll be pulled back - it's a negative feedback mechanism that'll keep the particles oscillating around their equilibrium positions (until the energy you give them dissipates). If all the particles in a medium oscillate 'in sync' you'll have a macroscopic oscillation of the entire system.
    Water molecules are held at fixed distances around one another but not in any overall macroscopic structure, so if you pull two molecules apart, a third will just slide in between them. This even happens with metals: spring coils only work by allowing a lot of small local stretches to accumulate into one large one. Stretch a spring too far and it'll be permanently deformed.
    The amplitude is also (half) the distance the spring has to oscillate through (which would increase the period).
     
  12. Elucinatus Registered Member

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    36
    Thanks for the help, gentlemen.

    This book right here which I'm studying out of makes a mention of Pulse and Periodic Waves....
    The model in the book shows how "a single flip o the wrist creates a pulse wave on a taut rope." It is basically a rope attached to the wall and someone holding the other end with their hand producing the disturbance in the rope.

    The book goes on to say.
    "A wave that consists of a single traveling pulse is called a pulse wave. Now imagine that you continue to generate pulses at one end of the rope. Together, these pulses form what is called a 'periodic wave'."

    So my question then was isn't a "periodc wave(s)" just a bunch of pulse waves? Is it the same thing?

    The book goes onto say that "a wave whose source vibrates with simple harmonic motion is called a 'sine wave'.
     
  13. fleem Registered Member

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    The term "pulse" does not imply any exact shape, although typically one would not describe a very complex pattern as a "pulse". That might more likely be called a "pulse train" or something along those lines.

    I take exception to the book calling a single pulse a "pulse wave". Its just a pulse, not a "pulse wave".

    A periodic wave is a bunch of identical pulses separated by the same amount of time. Again, the shape of the pulses is not specified.

    A sine wave, however, has a definite shape. Simple harmonic motion is described with a sine wave. Motion that follows some other periodic pattern is not called "simple harmonic motion"--you might just say "periodic motion" (which does include harmonic motion, but not necessarily), I suppose, or something along those lines.

    On a side note, the word "harmonic" means different things in different contexts. In one context "harmonic" means NOT the fundamental frequency, but some integral multiple of it. In the context of "harmonic motion" it means ONLY the fundamental frequency. Yes, its lame.
     

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