1.

Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the period for each case of periodic motion [omega is any positive constant]. (i) sinomegat+cosomegat (ii) sinomegat+cos2omegat+sin4omegat (iii) e^(-omegat) (iv) log(omegat)

Answer» <html><body><p></p>Solution :(i) sin `omegat + <a href="https://interviewquestions.tuteehub.com/tag/cos-935872" style="font-weight:bold;" target="_blank" title="Click to know more about COS">COS</a> omegat` is a periodic function, it can <a href="https://interviewquestions.tuteehub.com/tag/also-373387" style="font-weight:bold;" target="_blank" title="Click to know more about ALSO">ALSO</a> be written as `sqrt2 sin (omegat + pi/4)`. Now `sqrt2 sin (omegat + pi//4)= 2 sin (omegat + pi//4+2pi)` <br/> `=sqrt2sin[omega(t+2pi//omega)+pi//4]` <br/> The periodic time of the function is `2pi//omega`. <br/> (ii) This is an <a href="https://interviewquestions.tuteehub.com/tag/example-978283" style="font-weight:bold;" target="_blank" title="Click to know more about EXAMPLE">EXAMPLE</a> of a periodic motion. It can be noted that each term represents a periodic function with a different angular <a href="https://interviewquestions.tuteehub.com/tag/frequency-465761" style="font-weight:bold;" target="_blank" title="Click to know more about FREQUENCY">FREQUENCY</a>. Since period is the least interval of time after which a function repeats its value, `sinomegat` has a period `T_(0)= 2pi//omega , cos 2 omegat` has a period `pi//omega =T_(0)//2`, and `sin 4omegat` has a period `2pi//4omega = T_(0)//4`. The period of the first term is a <a href="https://interviewquestions.tuteehub.com/tag/multiple-1105557" style="font-weight:bold;" target="_blank" title="Click to know more about MULTIPLE">MULTIPLE</a> of the periods of the last two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is `T_(0)`, and thus, the sum is a periodic function with a period `2pi//omega`. <br/> (iii) The function e–wt is not periodic, it decreases monotonically with increasing time and tends to zero as `t to oo` and thus, never repeats its value. <br/> (iv) The function `log(omegat)` increases monotonically with time t. It, therefore, never repeats its value and is a nonperiodic function. It may be noted that as `t to oo, log(omegat)` diverges to `oo`. It, therefore, cannot represent any kind of physical displacement.</body></html>


Discussion

No Comment Found

Related InterviewSolutions