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Explain periodic function. |
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Answer» Solution :The displacement can be represented by a mathematical functions of TIME. Periodic function are those functions which are used to represent periodic motion. One of the simplest periodic function is `f(t) =" A cos " omega t` The periodic time of this function is `T= (2PI)/(omega)` because, `omega t` is increased by an integral MULTIPLE of `2pi` radians, the value of the function remains the same. Thus, the function f(t) is periodic with period T. `therefore f(t) = f(t+T)` If we consider a sine function, `f(t) = Asin omega t` is a periodic function with the same period T. If we consider a sine and COSINE function with a linear combination, then `f(t) = A sin omega t+ B cos omega t` is also a periodic function with the same period T. If `A= D cos phi """......"(1)` `B= D sin phi """........"(2)` then `f(t) = D sin omega t cos phi + D cos omega t sin phi` `= D[sin omega t cos phi + cos omega t sin phi]` `=D[sin (omega t +phi)]` `f(t)= D sin (omega t+phi)` Where D and `phi` are constants. D is a resultant amplitude. Adding and squaring equation (1) and (2) `A^(2)+B^(2)= D^(2) cos^(2) phi + D^(2) sin^(2) phi` `= D^(2) [cos^(2) phi + sin^(2) phi]` `= D^(2)` `therefore = sqrt(A^(2)+B^(2))` and TAKING ratio of equation (2) and (1) `(B)/(A)= (D sin phi )/(D cos phi)` `(B)/(A)= tan phi` `therefore phi = tan^(-1) (B/A)`. |
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