Which of the following functions of time represent (a) simple harmonic b. periodic but not simple harmonic. And c. non periodic motion? Give period for each of case of periodic motion (omega is ay positive constant) , 1. sinometa t-cos omegat 2. sin^(3)omega t 3. 3 cos (pi//4-2omegat) 4. cos omegat+cos 3 omega t+cos 5 omega t 5. e^(-omega^(2)t^(2) 6.1+omegat+omega^(2)t^(2)

Which of the following functions of time represent (a) simple harmonic

1.	Which of the following functions of time represent (a) simple harmonic b. periodic but not simple harmonic. And c. non periodic motion? Give period for each of case of periodic motion (omega is ay positive constant) , 1. sinometa t-cos omegat 2. sin^(3)omega t 3. 3 cos (pi//4-2omegat) 4. cos omegat+cos 3 omega t+cos 5 omega t 5. e^(-omega^(2)t^(2) 6.1+omegat+omega^(2)t^(2)
Answer» Solution :1. `sin OMEGA t-cos omega t=sqrt(2)[sin omega t "cos "(pi)/4-cos omega t "sin"(pi)/4]` `=sqrt(2)(omegat-(pi)/4)` It is simple harmonic with a time period `T=(2pi)/(omega)` 2. `sin^(3) omegat` is a periodic FUNCTION but not simple harmonic because `a alpha -y` condition ios not satisfied. Its time period is `T=(2pi)/(omega)` 3. `3cos (pi//4-2omegat)=3cos (2omegat-pi//4)` it is simple harmonic with a time period `T=(2pi)/(2omega0=(pi)/(omega)` 4. `cos omega t+cos 3 omegat+ cos 5 omega t` is a periodic funtion but not simple harmonic. The time periods of each periodic function are `(2pi)/(omega),(2pi)/(3omega)` and `(2pi)/(5omega)`. Since `(2pi)/(omega)` is the multiple of the other two periods, the given sum is periodic with time period `(2pi)/(omega)` 5. `e^(-omega^(2)r^(2))` is not periodic as t increases the function `e^(-omega^(2)t^(2))` DECREASES and tends to zero as `to to oo` 6. `1+omegat+omega^(2)t^(2)` is not periodic as function increases with increase in t with out repetition.