Question

Show that the motion of a particle represented by$y=\sin \omega t-\cos \omega t$ is a simple harmonic motion with a time period of $\dfrac{2\pi }{\omega }$.

Answer 1

A simple harmonic motion of an object is an oscillatory motion under a restoring force which is proportional to the displacement of that object from an equilibrium position. We can prove that the given equation represents a S.H.M by comparing the given equation with the standard equation of the S.H.M.

Complete step by step answer:
$y(t)=Acos(\omega t+\Phi )$ is a standard equation for the simple harmonic motion.
Mathematically the standard equation for SHM is:
$y(t)=Acos(\omega t+\Phi )$,
where $y(t)$ is the displacement of the object from its mean position as a function of time, $A$ is the amplitude or the maximum displacement from the mean position, $\omega$ is the angular frequency which is equal to $2\pi f$and $f$ is the number of oscillations per second, $t$ is time in seconds and $\Phi$ is the phase of the motion or the initial angular displacement of the object from its mean position.

Also $f=\dfrac{1}{T}$,
where $T$ is the time period of the motion.
The given equation is $y=\sin \omega t-\cos \omega t$, taking out $\sqrt{2}$ from RHS, the equation becomes,
$y=\sqrt{2}(\dfrac{1}{\sqrt{2}}\sin \omega t-\dfrac{1}{\sqrt{2}}\cos \omega t)$
$\Rightarrow y=\sqrt{2}(\cos \dfrac{\pi }{4}\sin \omega t-\sin \dfrac{\pi }{4}\cos \omega t)$
$\because \cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}=\sin \dfrac{\pi }{4}$

Applying the following trigonometric identity to the above equation,
$\sin (A-B)=\sin A\cos B-\sin B\cos A$
we get:
$y=\sqrt{2}(\sin (\omega t-\dfrac{\pi }{4}))$
Now comparing this equation with the standard equation of SHM,
$y(t)=Acos(\omega t+\Phi )$
we can say that the given equation, $y=\sin \omega t-\cos \omega t$ represents a simple harmonic equation of angular frequency $\omega$. Since $\omega =2\pi f$ and $f=\dfrac{1}{T}$, where$T$ is the time period. We can say that the time period of the given equation is $2\pi /\omega$.

Note: The direction of the restoring force is always towards the equilibrium position, which means that $F=-kx$ where $k$ is a constant of proportion and the negative sign represents the direction of force. Real life examples of SHM include pendulum, swing, spring-mass system, musical instrument etc. If we look around us we can see many other SHMs also.