of time - - - - Unit Oscillations Solution Using equation ( . ) v = ω x - ⇒ v = ω ( A − x ) Therefore, at position x , x ω ( ) ( ) Similarly, at position x , x ω ( ) ( ) Subtrating ( ) from ( ), we get x x ω ω ( ) ( ) x x ω ( ) ω π ⇒ x x x x ( ) Dividing ( ) and ( ), we get x x v x v x ( ) ( ) ⇒ ω ω ( ) Dividing equation ( ) and equation ( ), we have x x v x v x π . . Time period, frequency, phase, phase difference and epoch in SHM.
(i) Time period The time period is defined as the time taken by a particle to complete one oscillation. It is usually denoted by T . For one complete revolution, the time taken is t = T, therefore ω T = 2π ⇒ T = π ω ( . ) Then, the displacement of a particle executing simple harmonic motion can be written either as sine function or cosine function.
y ( t )= A sin π T t or y ( t ) = A cos π T t where T represents the time period. Suppose the time t is replaced by t + T , then the function y ( t + T ) = A sin π T ( t + T ) = A sin( π T t + 2π) = A sin π T t = y ( t ) y ( t + T ) = y ( t ) Thus, the function repeats after one time period. This y ( t ) is an example of periodic function. (ii) Frequency and angular frequency The number of oscillations produced by the particle per second is called frequency.
It is denoted by f . SI unit for frequency is s − or hertz (In symbol,