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Which of the following funchtions of time represent (a) periodic and (b) non-periodic motion? Give the period for each of period motion. `(omega` is any positive constant ) (i) `sinomegat+cosomegat` (ii) `sinomegat+cos2omegat+sin4omegat` (iii) `e^(-omegat)` (iv) `logomegat` |
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Answer» (i) `sinomegat+cosomegat` `=sqrt(2)[(1)/(sqrt(2))sinomegat+(1)/(sqrt(2))cosomegat]` `=sqrt(2)sin(omegat+pi//4)` As , `sqrt(2)sin(omegat+(pi)/(4))=sqrt(2)sin(omegat+(pi)/(4)+2pi)` so the given function is a periodic onoe and its period is `2pi//omega.` (ii)` sinomegat+cos2omegat+sin4omegat, ` it represents the periodic function with different angular frequency. Since, the period is the least time interval after which a function is repeated in its value. Here, `sinomegat,` has a period, `T=(2pi)/(omega),cos2omegat` has a period`(2pi)/(2omega)=(pi)/(omega)=(T)/(2)` and `sin4omegat` has a period ` (2pi)/(4omega)=(T)/(4)`. The last two terms repeat after any integral multiple of their period. Therefore, each term in the function repeats itself after time interval T. That is why, the given function is a periodic function iwth a period `T=2pi//omega` . (iii) The value of funtion `e^(-omegat)` , decreases with increasing time t and as `tprop oo`, it tends to zero. Therefore, the function is non-periodic. (iv) The value of function `logomegat` increases with time t. As `tprop oo`, `logomegat` approaches to `oo`. Therefore, the value of thhis function never repeates. Hence, it represents non-periodi function. |
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