period for each case of periodic motion [ ω is any positive constant]. (i) sin ω t + cos ω t (ii) sin ω t + cos ω t + sin ω t (iii) e – ω t (iv) log ( ω t ) Answer (i) sin ω t + cos ω t is a periodic function, it can also be written as sin ( ω t + π / ). Now sin ( ω t + π / )= sin ( ω t + π / + π ) sin [ ω (t + π / ω ) + π / ] The periodic time of the function is π / ω . (ii) This is an example of a periodic motion.
It can be noted that each term represents a periodic function with a different angular frequency. Since period is the least interval of time after which a function repeats its value, sin ω t has a period T = π / ω ; cos ω t has a period π / ω = T / ; and sin ω t has a period π / ω = T / . The period of the first term is a multiple of the periods of the last two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is T , and thus, the sum is a periodic function with a period π / ω .
(iii) The function e – ω t is not periodic, it decreases monotonically with increasing time and tends to zero as t → ∞ and thus, never repeats its value. (iv) The function log( ω t ) increases monotonically with time t . It, therefore, never repeats its value and is a non- periodic function. It may be noted that as t → ∞ , log( ω t ) diverges to ∞ .
It, therefore, cannot represent any kind of physical displacement. ⊳ . SIMPLE HARMONIC MOTION Consider a particle oscillating back and forth