Which of the following functions of time represented (a) periodic and (b) non-periodic motion ? Given the period for each case of periodic motion `( omega` is any positive constant(i)`sin omega t-cos omega t(ii) log ( 2 omega)`

Explanation

(i) `sin omega t-cos omega t` periodic function . It can be written as
`sqrt(2) (sinomegat-(1)/(sqrt(2))-cos omegat (1)/(sqrt(2)))`
`= sqrt(2) ( sin omegacos. (pi)/( 4)- cos omega t sin .(pi)/(4))`
`= sqrt(2) sin (omegat - (pi)/(4))`
`= sqrt(2) sin (omegat - (pi)/( 4)+2pi)`
`= sqrt(2) sin [ omega( t + (2pi)/(omega))-(pi)/(4)]`
The periodic time of the function is `(2pi)/(omega)`
(ii) The function `log ( 2 omega t )` increases monotonically with time t. It, therefore,never repeats its value and is a non-periodic function . As ` trarr oo, log ( 2 omegat)` diverge to `oo`. It therefore, cannot represent any type of physical displacement.