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 case of periodic motion (omega is any positive constant): 1)sin omega t - cos omega t2 ) sin^3 omega t3) 3 cos (pi//4 - 2omega t) 4)cos omegat+ cos 3 omega t + cos 5 omega t5)e^(-omega^2 t^2)6) 1+ omega t + 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 case of periodic motion (omega is any positive constant): 1)sin omega t - cos omega t2 ) sin^3 omega t3) 3 cos (pi//4 - 2omega t) 4)cos omegat+ cos 3 omega t + cos 5 omega t5)e^(-omega^2 t^2)6) 1+ omega t + omega^2 t^2
Answer» Solution :`SIN omega t - cos omegat - sqrt2 [ sin omega t cos PI/4 - cos omega t sin (pi)/(4)] = sqrt2 sin (omega t- pi/4)` It is simple harmonic with a time period T =` (2pi)/(omega)` 2) `sin^3 omega t ` is a periodic function but not simple harmonic because `a ALPHA - y `condition is not satisfied. It is time period is `T= (2pi)/(omega)` 3) `3 cos (pi//4 - 2omega t) = 3 cos (2 omega t - pi//4) ` it is simple harmonic with a time period `T= (2pi)/(2omega) = pi/omega ` 4) `cos omega t + cos 3 omega t + cos 5 omega t ` is a periodic function 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 the time period ` (2pi)/(omega` 5) `e^(-omega^2 t^2)` decreases and tends tozero as `t to oo` 6) `1 + omega t + omega^@ t^2 ` is not periodic as function with increase in t with out repetition.