question_answer 4)

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; \[\log \left[ \frac{\left( x+\upsilon t \right)}{{{x}_{0}}} \right]\]co is any positive constant). (a) \[\log \left[ \frac{\left( x+\upsilon t \right)}{{{x}_{0}}} \right]\] (b) \[\text{m}{{\text{s}}^{\text{-1}}}\] (c) \[\text{m}{{\text{s}}^{\text{-1}}}\] (d) \[{{10}^{5}}\] (e) \[{{\text{ }\!\!\upsilon\!\!\text{ }}_{\text{a}}}\text{=340m}{{\text{s}}^{\text{-1}}}\text{,}\] (f) \[{{\upsilon }_{w}}=1486m{{s}^{-1}}\]

Answer:

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 \[\cos \left( \frac{2\pi t}{T}+\phi \right)\] or \[\sin \left( \frac{2\pi t}{T}+\phi \right)\] where, T is the time period. (a) \[\sin \,\,\omega t-\cos \,\omega t=\sqrt{2}\left[ \frac{1}{\sqrt{2}}\sin \omega t-\frac{1}{\sqrt{2}}\cos \,\omega t \right]\] \[\text{cos }\!\!\omega\!\!\text{ t + cos 3 }\!\!\omega\!\!\text{ t + cos 5 }\!\!\omega\!\!\text{ t}\]\[\text{exp}\left( \text{-}{{\text{ }\!\!\omega\!\!\text{ }}^{\text{2}}}{{\text{t}}^{\text{2}}} \right)\] It is a simple harmonic function with period \[=-\frac{2\pi }{\omega }\] (b) \[\left( \frac{2\pi t}{T}+\phi \right)\] Here each term sin (01 and sin 3 (01 individually represents simple harmonic function. But (b) which is the outcome of the superposition of two simple harmonic functions will only be periodic but not simple harmonic. Its time period is \[\left( \frac{2\pi t}{T}+\phi \right)\] (c)\[\text{ }\!\!\omega\!\!\text{ t - cos }\!\!\omega\!\!\text{ t = }\sqrt{\text{2}}\] \[\left[ \frac{\text{1}}{\sqrt{\text{2}}}\text{sin }\!\!\omega\!\!\text{ t-}\frac{\text{1}}{\sqrt{\text{2}}}\text{cos }\!\!\omega\!\!\text{ t} \right]\] Clearly it represents simple harmonic function and its time period is \[\text{=}\sqrt{\text{2}}\left[ \text{sin }\!\!\omega\!\!\text{ t cos }\frac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}\text{ - cos }\!\!\omega\!\!\text{ t sin }\frac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}} \right]\] (d) \[\text{=}\sqrt{\text{2}}\text{sin}\left( \text{ }\!\!\omega\!\!\text{ t-}\frac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}} \right)\]It represents the periodic but not simple harmonic function. Its time period is \[\text{=}\frac{\text{2 }\!\!\pi\!\!\text{ }}{\text{ }\!\!\omega\!\!\text{ }}\]. (e) \[\text{si}{{\text{n}}^{\text{3}}}\text{ }\!\!\omega\!\!\text{ t =}\frac{\text{1}}{\text{4}}\left[ \text{3 sin }\!\!\omega\!\!\text{ t - sin 3 }\!\!\omega\!\!\text{ t} \right]\].It is an exponential function which never repeats itself. Therefore it represents non-periodic function. (vi)\[\text{2 }\!\!\pi\!\!\text{ / }\!\!\omega\!\!\text{ }\text{.}\] also represents non periodic function.