Which of the following functions represents a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) function represents simple harmonic motion $(S.H.M.)$ if it satisfies the differential equation $\frac{d^2x}{dt^2} + \omega^2 x = 0$.Option $A$: $

Vedclass · Accepted Answer

$\sin \omega t - \cos \omega t$

Vedclass · Answer

(A) function represents simple harmonic motion $(S.H.M.)$ if it satisfies the differential equation $\frac{d^2x}{dt^2} + \omega^2 x = 0$.Option $A$: $x = \sin \omega t - \cos \omega t$. This can be written as $x = \sqrt{2} [\frac{1}{\sqrt{2}} \sin \omega t - \frac{1}{\sqrt{2}} \cos \omega t] = \sqrt{2} \sin(\omega t - \frac{\pi}{4})$. This is a standard $S.H.M.$ equation.Option $B$: $x = \sin^2 \omega t = \frac{1 - \cos 2\omega t}{2}$. This represents motion with a constant shift and frequency $2\omega$,which is not simple harmonic.Options $C$ and $D$: These are superpositions of two different frequencies ($\omega$ and $2\omega$),which result in periodic motion but not simple harmonic motion.Therefore,the correct option is $A$.

Which of the following functions represents a simple harmonic oscillation?

Similar Questions

Define the amplitude of $SHM$ and draw two different $SHM$s in one figure having two different amplitudes.

Explain that simple harmonic motion is the projection of uniform circular motion on a diameter of the reference circle and obtain the velocity and acceleration.

$A$ ball is released from height $h$ which makes a perfectly elastic collision with the ground. The frequency of the periodic vibratory motion is ($g$ = acceleration due to gravity).

Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?
$(a)\; a=0.7 x$
$(b)\; a=-200 x^{2}$
$(c)\; a=-10 x$
$(d)\; a=100 x^{3}$

Equations $y = 2A \cos^2 \omega t$ and $y = A (\sin \omega t + \sqrt{3} \cos \omega t)$ represent the motion of two particles.

Vedclass Test Series

Exam Paper Generator

Online Exam Module