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sporkstorms
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I'm given some initial conditions for a 2-d isotropic oscillator:
At t=0: x=A, y=4A, dx/dt = 0, dy/dt = 3wA
Solving the differential equations of motion and using those conditions, I get the following:
[tex]let\ \gamma = tan^{-1}(-3/4)[/tex]
[tex]x(t) = A cos(\omega t)[/tex]
[tex]y(t) = 5A cos(\omega t + \gamma)[/tex]
The problem then asks to show that the motion is confined to a box of dimensions 2A and 10A. To me this seems inherent just by looking at the amplitudes of x and y, but maybe I'm missing something?
The book (Fowles & Cassiday, 7th ed) goes into this big long spiel to show the confinement of motion. It rewrites y in terms of x, skips a million trig substitutions, and ends up with an equation of the form:
[tex]ax^2 + bxy + cy^2 + dx +ey = f[/tex]
And it says this can tell you, based on the discriminant, whether it's an ellipse, a parabola, or a hyperbola, and what it's bounds are.
So I took my x and y (listed above), put y in terms of x, did some trig substitutions, rearranged, squared both sides, and ended up with:
[tex]x^2 - 8xy + y^2 = 9[/tex]
Now, how does this help me describe the motion any more than my original equations for x and y? And how does this help me to show that the motion is confined to a box of dimensions 2A and 10A any more than the amplitudes of the original equations do?
Or should I ignore that whole part of the book? (probably not, but you never know)
My position equations seem ok since they agree with the IC's, and http://sporkstorms.org/tmp/2Doscillator.png" seems sane, and is clearly between -A,A and -5A,5A (which is what the problem text suggested).
At t=0: x=A, y=4A, dx/dt = 0, dy/dt = 3wA
Solving the differential equations of motion and using those conditions, I get the following:
[tex]let\ \gamma = tan^{-1}(-3/4)[/tex]
[tex]x(t) = A cos(\omega t)[/tex]
[tex]y(t) = 5A cos(\omega t + \gamma)[/tex]
The problem then asks to show that the motion is confined to a box of dimensions 2A and 10A. To me this seems inherent just by looking at the amplitudes of x and y, but maybe I'm missing something?
The book (Fowles & Cassiday, 7th ed) goes into this big long spiel to show the confinement of motion. It rewrites y in terms of x, skips a million trig substitutions, and ends up with an equation of the form:
[tex]ax^2 + bxy + cy^2 + dx +ey = f[/tex]
And it says this can tell you, based on the discriminant, whether it's an ellipse, a parabola, or a hyperbola, and what it's bounds are.
So I took my x and y (listed above), put y in terms of x, did some trig substitutions, rearranged, squared both sides, and ended up with:
[tex]x^2 - 8xy + y^2 = 9[/tex]
Now, how does this help me describe the motion any more than my original equations for x and y? And how does this help me to show that the motion is confined to a box of dimensions 2A and 10A any more than the amplitudes of the original equations do?
Or should I ignore that whole part of the book? (probably not, but you never know)
My position equations seem ok since they agree with the IC's, and http://sporkstorms.org/tmp/2Doscillator.png" seems sane, and is clearly between -A,A and -5A,5A (which is what the problem text suggested).
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