Question

Which of the following expression does not represent SHM?

• A. $A\cos \omega t$
• B. $A\sin 2\omega t$
• C. $A\sin \omega t$ + $B\cos \omega t$
• D. $A{\sin }^2\omega t$

Answer 1

Answer: (B)

$A\sin 2\omega t$

A) $y=Acos\omega t$. . . . . . . .. (1)

$\frac{dy}{dt}=-A\omega sin\omega t$

$\frac{d^{2}y}{d{t}^2}=-A{\omega }^{2}cos\omega t$

$\frac{d^{2}y}{d{t}^2}=-{\omega }^{2}y$ (from equation 1)

This is the equation of S.H.M

B) $y=Asin2\omega t$. . . . . . . . . .(2)

$\frac{dy}{dt}=2A\omega cos2\omega t$

$\frac{d^{2}y}{d{t}^2}=-4A{\omega }^{2}sin2\omega t$

$\frac{d^{2}y}{d{t}^2}=-4{\omega }^{2}y$ (from equation 2)

This is not a equation of S.H.M

C) $y=Asin\omega t+Bcos\omega t$. . . . . . . .(3)

$\frac{dy}{dt}=\omega (Acos\omega t-Bsin\omega t)$

$\frac{d^{2}y}{d{t}^2}=-{\omega }^2(Asin\omega t+Bcos\omega t)$

$\frac{d^{2}y}{d{t}^2}=-{\omega }^{2}y$ (from equation 3)

This is the equation of S.H.M.

D) $y=Asi{n}^2\omega t$

$\frac{dy}{dt}=2\omega Asin\omega t.cos\omega t$

$\frac{d^{2}y}{d{t}^2}=2\omega A(-\omega sin\omega t.sin\omega t+\omega cos\omega t.cos\omega t)$

$\frac{d^{2}y}{d{t}^2}=-{\omega }^{2}y$

This is the equation of S.H.M.

The correct option is B.