The displacement y of a particle executing periodic motion is given by $y = 4{\cos ^2}(t/2)\sin (1000t)$. This expression may be considered to be a result of the superposition of ........... independent harmonic motions
IIT 1992, Diffcult
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(b)$y = 4{\cos ^2}\left( {\frac{t}{2}} \right)\sin 1000\;t$
$ \Rightarrow y = 2(1 + \cos t)\sin 1000\;t$
$ \Rightarrow y = 2\sin 1000\;t + 2\cos t\;\sin 1000t$
$ \Rightarrow y = 2\sin 1000\;t + \sin 999\;t + \sin 1001\;t$
It is a sum of three $S.H.M.$
$ \Rightarrow y = 2(1 + \cos t)\sin 1000\;t$
$ \Rightarrow y = 2\sin 1000\;t + 2\cos t\;\sin 1000t$
$ \Rightarrow y = 2\sin 1000\;t + \sin 999\;t + \sin 1001\;t$
It is a sum of three $S.H.M.$
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