Saved Bookmarks
| 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 ofperiodic motion (w is any positive constant): (a) sin omega t - cos omega t (b) sin^(3) omega t (c) 3 cos (pi//4 - 2 omega t) (d) cos omega t + cos 3 omega t + cos 5 omega t (e) "exp" (-omega^(2)t^(2)) (f) 1+ omega t + omega^(2)t^(2). |
|
Answer» Solution :The function will represent a periodic motion,if it is identically repeated after a fixed interval of time and will represent S.H.M. if it can be written uniquely in the form of a cos `((2pi t)/(T)+phi)` or ` sin ((2pi t)/(T)+phi)`, where T is the time period. (a) `sin omega t - cos omega t = sqrt(2) [(1)/(sqrt(2)) sin omega t -(1)/(sqrt(2)) cos omega t]` `=sqrt(2) [sin omega t cos.(pi)/(4)-cos omega t sin.(pi)/(4)]` `= sqrt(2) sin (omega t-(pi)/(4))` It is a S.H.M and its period is `2pi//omega` (B) `sin^(3) omega t = (1)/(3) [3 sin omega t - sin omega t]` Here each TERMS `sin omega t` and `sin 3 omega t` individually represents S.H.M But (ii) which is the outcome of the superposition of two SHMs will only be periodic but not SHMs. Its time period is `2pi//omega`. (c) `3 cos ((pi)/(4)-2 omega t) = 3 cos (2 omega t -(pi)/(4)) [:' cos (-theta) = cos theta]` Clearly it represents SHM and its time period is `2pi//2omega`. (d) `cos omega t + cos 3 omega t +cos 5 omega t`. It is represents the periodic but no S.H.M Its time period is `2pi//omega` (E) `e^(-w2t2)`. It is an exponential function which never repeats itself. Therefore it represents non-periodic motion. (f) `1 + wt +w^(2)t^(2)` also represents non periodic motion. |
|