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.
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