Question

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) $\sin \omega t - \cos \omega t$
(b) $\sin 3\omega t$
(c) $3\cos (\dfrac{\pi }{4} - 2\omega t)$
(d) $\cos \omega t + \cos 3\omega t + \cos 5\omega t$
(e) $\exp ( - {\omega ^2}{t^2})$
(f) $1 + \omega t + {\omega ^2}{t^2}$

Answer 1

Hint A simple harmonic motion (SHM )is defined as the oscillatory motion where the restoring force of the system is directly proportional to its displacement from equilibrium position. Periodic motion is defined as a motion which repeats itself after a fixed interval of time(regular), whereas a non periodic motion also repeats itself but will not be in a fixed interval of time (irregular).The regular interval is called the time period.

Complete step by step answer:
For a motion to be simple harmonic as said earlier should have a restoring force which is proportional to displacement. This can be expressed as $a = - k \times x$. A simple harmonic motion always satisfies the equation $\dfrac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = 0$. Some of the common examples of simple harmonic motion are sine functions , cosine functions and any of their linear combinations. Generally every SHM can be written in the form of $Asin(\omega \times t + \phi )$. Here $\omega $ is the angular frequency, $\phi $ is the phase difference and $A$ is the amplitude. Also note that sine and cosine are periodic functions too.
$\sin \omega t - \cos \omega t$
= $\sqrt 2 (\dfrac{1}{{\sqrt 2 }}\sin \omega t - \dfrac{1}{{\sqrt 2 }}\cos \omega t)$ (using $\sin (a + b) = \sin a\sin b - \cos a\cos b$)
= $\sqrt 2 \sin (\omega t - \dfrac{\pi }{4})$
This represents an SHM and is a periodic motion with a time period of $\dfrac{{2\pi }}{\omega }$.
$\sin 3\omega t$
$\dfrac{1}{4}(3\sin \omega t - \sin 3\omega t)$
Even the both the individual terms are SHM ,their superposition is not a SHM. Their time period will be the least common multiple of both the functions(LCM) The period is $\dfrac{{2\pi }}{\omega }$.
(c) $3\cos (\dfrac{\pi }{4} - 2\omega t)$
$ - 3\cos (2\omega t - \dfrac{\pi }{4})$
This is a SHM and periodic motion with a period $\dfrac{{2\pi }}{{2\omega }}$=$\dfrac{\pi }{\omega }$.
(d) $\cos \omega t + \cos 3\omega t + \cos 5\omega t$
The terms are individually simple harmonic, but their superposition is not.
The function is a periodic function with time period $\dfrac{{2\pi }}{\omega }$.(LCM)
(e) $\exp ( - {\omega ^2}{t^2})$
This is an exponential function. This is neither periodic nor simple harmonic.
This is a non-periodic function.
(f) $1 + \omega t + {\omega ^2}{t^2}$
This too is an exponential function. Thus non periodic.
(a), (c) belongs to (A)Simple harmonic motion
(b), (d) belong to (B) Periodic motion
(e), (f) belongs to (C ) Non periodic motion.

Note: All simple harmonic motions are periodic. Simple pendulum is one of the examples. But all periodic motions are not simple harmonics. The rotation of earth about its axis is an example.