capacitor, changing with time in an AC circuit, is also a displacement variable. In the same way, pressure variations in time in the propagation of sound wave, the changing electric and magnetic fields in a light wave are examples of displacement in different contexts. The displacement variable may take both positive and negative values. In experiments on oscillations, the displacement is measured for different times.
The displacement can be represented by a mathematical function of time. In case of periodic motion, this function is periodic in time. One of the simplest periodic functions is given by f ( t ) = A cos ω t ( .3a) If the argument of this function, ω t , is increased by an integral multiple of π radians, the value of the function remains the same. The function f ( t ) is then periodic and its period, T , is given by ( .3b) Thus, the function f ( t ) is periodic with period T , f ( t ) = f ( t+T ) The same result is obviously correct if we consider a sine function, f ( t ) = A sin ω t .
Further, a linear combination of sine and cosine functions like, f ( t ) = A sin ω t + B cos ω t ( .3c) is also a periodic function with the same period T. Taking, A = D cos φ and B = D sin φ Eq. ( .3c) can be written as, f ( t ) = D sin ( ω t + φ ) , ( .3d) Here D and φ are constant given by and tan φ = – D = A + B B A The great importance of periodic sine and cosine functions is due to a remarkable result proved by the French mathematician, Jean Baptiste Joseph Fourier ( – ): Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients . u Example .
Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the