Question

Which of the following functions of time represent $\left(a\right)$ simple harmonic, $\left(b\right)$ periodic but not simple harmonic, and $\left(c\right)$ non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):

A) $\sin \omega t-\cos \omega t$

B) ${\sin }^3\omega t$

C) $3\cos (\frac{\pi }{4}-2\omega t)$

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

A) It is an $SHM$.

We know that both $\sin \omega t\text{and}\cos \omega t$ are $SHM$. A linear combination of $\sin \omega t\text{and}\cos \omega t$ can be represented as a single sine function. Therefore, the given function is $SHM$.

The given function is,

$\sin \omega t-\cos \omega t=\sqrt{2}(\sin \omega t.\frac{1}{\sqrt{2}}-\cos \omega t.\frac{1}{\sqrt{2}})$

$=\sqrt{2}(\sin \omega t\cos \frac{\pi }{4}-\cos \omega t\sin \frac{\pi }{4})$

$=\sqrt{2}\sin (\omega t-\frac{\pi }{4})$

This function represents SHM as it can be written
in the form $A\sin (\omega t+\varphi )$. Its period is $2\pi /\omega$.


B) It is periodic, but not $SHM$.
The given function is:
${\sin }^3\omega t=\frac{1}{2}(3\sin \omega t-\sin 3\omega t)$

The terms $\sin \omega t\text{and}\sin 3\omega t$ individually represent simple harmonic motion $\left(SHM\right)$.

However, the superposition of two $SHM$ with different frequencies is periodic but not simple harmonic.

Time period of the period function is the $LCM$ of the
time periods of superposing $SHMs$.

Time period of $\sin \omega tis2\pi /\omega$.

Time period of $\sin 3\omega tis2\pi /3\omega$.

Therefore, the time period here is, $2\pi /\omega$.


C) It is an $SHM$.

The given function is,
$3\cos (\frac{\pi }{4}-2\omega t)=3\cos (2\omega t-\frac{\pi }{4})$

This function represents simple harmonic motion because it can be written in the form $A\sin (\omega t+\varphi )$.

Its period is: $2\pi /2\omega =\pi /\omega$.


D) This is Periodic, but not $SHM$.

The given function is,

$\cos \omega t+\cos 3\omega t+\cos 5\omega t$

Each individual cosine function represents $SHM$. However, the superposition of three simple harmonic motions with different frequencies is periodic, but not simple harmonic.

Time period of the periodic function is the $LCM$ of the time periods of superposing $SHMs$.

Time period of $\cos \omega tis,2\pi /\omega$

Time period of $\cos 3\omega tis,2\pi /3\omega$

Time period of $\cos 5\omega tis,2\pi /5\omega$

Therefore, the time period here is, $2\pi /\omega$.


E) It is non-periodic motion.
The given function is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

F) The given function $1+\omega t+{\omega }^2{t}^2$ is non-periodic.
The function is non-periodic (physically not acceptable as the function $\to \infty$ as t $\to \infty$).