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Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic. and (c) non-periodic motion ? Give period for each case of periodic motion (omega is any positive constant). (a) sin omega t - cos omega t(b) sin^3 omega t (c) 3 cos (pi/4 -2 omega t) (d) cos omegat + cos3 omega t + cos 5 omega t (e) exp (-omega^2 t^2) (f) 1 + omega t + omega^2 t^2 |
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Answer» Solution :The function will represent a periodic motion, if it is identically repeated after a fixed INTERVAL of time and will represent S.H.M. if it can be written uniquely in the form of a `cos ((2pit)/(T) + PHI) ` or a `sin ((2pi t)/(T) + phi)` , where T is the time period (a) `sin omegat - cos omega t = SQRT2 [1/sqrt2sin omega t - 1/sqrt2cos omegat] = sqrt2 [ sin omega t cos pi/4 - cos omega t sin (pi)/4]` `=sqrt2 sin (omega t - pi/4 ) ` it is a S.H.M. and its period is `2pi//omega` (b) `sin^3 omega t = 1/4 [3 sin omega t - sin 3 omega t ]` Here each term sin `omega t ` and `sin 3 omega t ` individually represents S.H.M.But(ii) Which is the outcome of the superposition of two S.H.Ms will only be periodic but not S.H.M. Its time period is `2pi//omega` (c) `3cos (pi/4 - 2omega t) = 3cos (2omega t - pi/4) . [because cos(-theta) = cos theta]` Clearly it represents S.H.M. and its time period is `2pi//2omega` (d) `cos omegat + cos 3omega t + cos 5 omega t ` .It represents the periodic but not S.H.M. its time period is `2pi//omega` (e) `e^(-omega^2 t^2) `. It is an exponential function which never repeats itself. Therefore it represented non-periodic motion. (f) `1+ omega t + omega^2 t^2` also represents non periodic motion . |
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