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Old Apr 15th 2014, 02:59 AM   #1
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Post Solving oscillation-related problems using energy

I am very much interested in how exactly can one solve harmonic oscillation-problems using solely the KE - PE approach. I am a high-school student, so a little in-depth explanation will very much be appreciated. There seems to be great potential in solving such problems by finding the mass- and elasticity constant equivalent for a harmonic oscillator to find out the period of oscillations and such, but I frankly have no idea nor have I found any clear article on the web for that. So any links to good documentation or tips or some sort of explanations would be very much appreciated!

I have some sample problems that can be solved using this approach, in case they might help any of you make me understand better the process of solving them .

Sample problem #1:

The period oscillations of the "swinging" of the surface of a lake around a vertical axis. The dimensions are: L,l (bottom surface), H (equilibrium height), and d (<< H) (the "amplitude" of oscillations). I can sketch a drawing if any of you require.


Sample problem #2:

The period of oscillations of a marble (of mass M) that can slide along a string (of length 2L) fixed in two points distance 2d apart. The trajectory is rather obviously an ellipse. The system is set up vertically. (AO = OB = d; AC + BC = 2L - at any given moment)

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Old Apr 15th 2014, 05:59 AM   #2
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Usually for analysis of harmonic motion it's best to set up a differential equation and then solve for the frequency. For example a simple pendulum operates according to the DE:

mg sin(theta) - mL (d^2 theta/dt^2) = 0

But this approach is a bit advanced for high school - especially given the complicated geometry of the eliptical pendulum that you want to study. Using energy methods you can determine the peak velocity of the pendulum:

Max KE = Max PE

(1/2)mv^2 = mgh

where h= the max height that the pendulum attains in its arc. But this doesn't provide the period of oscillation.

As for your first problem - I don't understand what you mean by "swinging" of a lake about a vertical axis. I understand waves on a lake surface, but you are asking about some other phenomenon - please elaborate.
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Old Apr 15th 2014, 10:18 AM   #3
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I am not exactly interested in solving the problem by writing its equation of motion (which nonetheless is rather complicated), but by taking an energetic approach (so that the only variables I take into account are the kinetic and potential energy), which won't require accounting for all the forces and their components and such.

Here is a sketch I have drawn in a couple of minutes for problem #1.
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Old Apr 15th 2014, 11:22 AM   #4
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It appears that you are interested in the "sloshing" of water back and forth across the lake, not "swiveling about a vertical axis," which sounds more like a merry-go-round spinning on its axis.

As noted earlier I don't believe you can derive the period of oscillation purely from energy principals. The period is dependent on the nature of the restoring force acting on the system when out of equilibrium, and the equations of motion for that force acting on the system.

The speed of waves on water depends on whether they are "deep water" or "shallow water" waves. For deep water, where the depth is greater than half the wavelength of the wave, speed depends on wavelength (the longer the wavelength the faster the wave). For shallow water waves, where depth is less than half wavelength, the speed of the wave depends only on the water depth, and is equal to sqrt(gh). Note that your factor 'd' doesn't play in this at all.
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Old Apr 15th 2014, 11:33 AM   #5
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I saw a so-to-say proof of this problem by finding the coordinates of the com at a certain position (x) and somehow relating the speed of the com with the variation of the height (x dot), if that makes any sense. And then writing the kinetic energy and the potential energy to somehow derive a mass equivalent and an elasticity constant equivalent.

The answer was like pi * L / sqrt (3gH)

Edit:
We know that PE = k_e * x^2 / 2 and KE = m_e * (x dot)^2 / 2

Last edited by lbicsi; Apr 15th 2014 at 11:36 AM.
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