Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):
(a) sin ωt – cos ωt
(b) sin3 ωt
(c) 3 cos (π/4 – 2ωt)
(d) cos ωt + cos 3ωt + cos 5ωt
(e) exp (–ω2t2)
(f) 1 + ωt + ω2t2
The condition for a function to be periodic is that it must identically repeat itself after a fixed interval of time. For a function to represent an SHM, it must have the form of cos (
or sin (
with a time period T.
(a) sin ωt – cos ωt = √2(
-
)..(Multiply & divide by √2 )
= (sin ωt.cosπ/4 - cos ωt.sinπ/4)
= √2.sin (ωt - π/4)
Hence, it is an SHM with a period 2π/ω.
(b) sin3 ωt = 1/3(3sin ωt - sin3ωt)
Each term here, sin ωt and 3ωt represent SHM. But B. is the result of superposition of two SHMs, is only periodic not SHM. Its time period is 2π/ω.
(c) It can be seen that it represents an SHM with a time period of 2π/2ω.
(d) It represents periodic motion but not SHM. Its time period is 2π/ω.
(e) An exponential function never repeats itself. Hence, it is a non periodic motion.
(f) It clearly represents a non-periodic motion.