- #1
Xsnac
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Homework Statement
I have 2 perpendicular oscilations and I have to find the trajectory equation.
$$x=A\cos\omega t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)$$
$$y=B\cos(\omega t+\Delta\phi)$$
Homework Equations
$$\cos (x+y) =\cos x\cos y -\sin x\sin y$$
$$\cos^{2} x+\sin^{2} x =1$$
and from (1)
$$\cos\omega t =\frac {x}{A}$$
The Attempt at a Solution
I basicaly spent 2 hours trying to algebraicaly manipulate the equationThe closes I could get was:
$$\frac {y^{2}} {B^{2}} -2\frac {xy} {AB}\cos\Delta\phi + \frac {x^{2}} {A^{2}}\cos^{2}\Delta\phi = (1 - \frac{x^{2}} {A^{2}} )sin^{2}\Delta\phi\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2)$$
The end result should be:
$$\frac {x^{2}} {A^{2}} +\frac {y^{2}} {B^{2}}-2\frac {xy} {AB}\cos\Delta\phi=sin^{2}\Delta\phi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (3)$$
I have no ideea how to get from (2) to (3).
Thank you in advance for reading this wall of text.
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