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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+cos 2omegat+sin 4 omegat iii) e^(-2omegat) (iv) log(omegat) |
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Answer» Solution :i) ` sinomegat+cosomegat` is a periodic function. It can also be written as `sqrt(2)sin(omegat+pi//4)` The periodic time of the function is `2pi//omega` ii) This is an example of a periodic motion. It can be noted that each term represents a periodic function with a different ANGULAR frequency. Since PERIOD is the least interval of time after which a function repeats its VALUE, `sin omegat` has a period `T_(0) = 2pi//omega: cos 2omegat`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 multiple 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`. iii) The function `E^(-omegat)`is not periodic, it decreases monotonically with increasing time and tends to zero as t tens to `prop` and thus, never repeats its value. IV) The function `log(omegat)` increase monotonically with time t. It, therefore, never repeats its value and is a non periodic function. It may be noted that as t tens to `prop log(omegat)` diverges to `prop`, it, therefore, cannot represent any kind of physical displacement. |
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