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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 motoin [omega is any positive constant]. (i) sin omega t + cos omega t (ii) sin omega t +cos 2 omega t + sin 4 omega t (iii) e^(-omegat) (iv) log (omegat) |
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Answer» Solution :(i) `sin omega t+cos omega t` is a periodic function. It can also be written as `sqrt(2)sin(omegat+pi//4)` Now `sqrt(2) sin (omegat+pi//4)=sqrt(2)sin(omega t+pi//4)+2pi)` `=sqrt(2)sin [omega (t+2pi//omega)+pi//4]` The periodic time of the function isd `2pi//omega` (ii) This is an example of a periodic motion. It can be noted that each term represents a periodic functionwith a different angular frequency. Sinc eperiod is the least interval of time after which a function repeats its value, `sin omega t ` has a period `T_(0)=2pi//omega:cos 2 omega` has a period `pi//omega=T_(0)//2` and `sin4 omega t` has a period `2pi/4omega=T_(0)//4`. The period of the first term is a multiple of the periods of thelast two terms. Therefore, the SMALLEST interval of time after whcih the sum of the three terms repeats is `T_(0)`,an THUS the sum is a periodic functionwilth 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 it value. (iv) The function`log(omegat)` increase monotonicaly with time t. It therfore, never repeats it value and is a non periodic function. it may be noted that as t tens to `prop , log (omegat)` diverges to `oo`, it therfore, canot represent any kind of PHYSICAL displacement. |
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