The resultant of two rectangular simple harmonic $$motionsx=asin$$ω$$t+$$ϕ and $$y=bsin$$ωt of the same frequency and unequal amplitudes but differing in phase by π$$2$$ is

Question

The resultant of two rectangular simple harmonic $$motionsx=asin$$ω$$t+$$ϕ and $$y=bsin$$ωt of the same frequency   and unequal amplitudes but differing in phase by π$$2$$  is

Infinity Learn · Accepted Answer

Let us consider two simple harmonic motion $$y=bsin$$ωt ……………………..(1)x=asin$$ω$$t+$$ϕ ………………………$$(2)x=asin$$ωtcosϕ$$+acos$$ωtsinϕ  From $$(1) $$yb=$$$$sin$$ωt  and  $$1$$−$$y2b2=$$$$cos$$   ωtSubstituting these is equation (2) we get $$xa=ybcos$$ϕ$$+1$$−$$y2b2sin$$ϕ Taking square on both sides we get xa−ybcosϕ$$2=1$$−$$y2b2sin$$ϕ$$2$$ Thus $$xa2+yb2cos2$$ϕ−$$2xyabcos$$ϕ$$=1$$−$$y2b2sin2$$ϕSimplifying  this we $$getxa2+yb2$$−$$2xyabcos$$ϕ$$=sin2$$ϕIn this case   ϕ$$=$$π$$2Making$$ the equation an ellipse

The resultant of two rectangular simple harmonic motions $x = a \sin (ω t + ϕ)$ and $y = b sin ω t$ of the same frequency and unequal amplitudes but differing in phase by $\frac{π}{2}$ is

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The resultant of two rectangular simple harmonic motionsx=asinωt+ϕ and y=bsinωt of the same frequency and unequal amplitudes but differing in phase by π2 is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

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The resultant of two rectangular simple harmonic motions $x = a \sin (ω t + ϕ)$ and $y = b sin ω t$ of the same frequency and unequal amplitudes but differing in phase by $\frac{π}{2}$ is