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Consider two simple harmonic motion alon...

Consider two simple harmonic motion along x and y- axis having same frequencies but different amplitudes as `x=A sin (omega t+varphi)` (along x axis) and `y= B sin omega t`( along y axis).
then show that `(x^(2))/(A^(2))+(y^(2))/(B^(2))-(2xy)/(AB) cos varphi = sin^(2) varphi` and also discuss the special cases when
`varphi=(pi)/(2) and A=B`
Note : when a particle is subjected to two simple harmonic motion at right angle to each other the particle may move along different paths.

Text Solution

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a. `y=(B)/(A)x,` equation is a straight line passing through origin with positive slope.
b. `y=-(B)/(A)x` equation is a straight line passing through origin with negative slope.
c. `(x^(2))/(A^(2))+(y^(2))/(B^(2))=1,` equation is an ellipse whose centre is origin .
d. `x^(2)+y^(2)=A^(2),` equation is a circle whose center is origin.
e. `(x^(2))/(A^(2))+(y^(2))/(B^(2))-(2xy)/(AB)(1)/(sqrt(2))=1/2` equation is an ellipse (oblique ellipse which means tilted ellipse. )
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