Question 45: Oscillations

Question

Function $x=\mathrm{A} \sin ^2 \omega t+\mathrm{B} \cos ^2 \omega t+\mathrm{C} \sin \omega t \cos \omega t$ represents SHM

Accepted Answer

$$ \begin{aligned} & x=A \sin ^2 \omega t+B \cos ^2 \omega t+C \sin \omega t \cos \omega t \ & x=\frac{A}{2}(1-\cos 2 \omega t)+\frac{B}{2}(1+\cos 2 \omega t)+\frac{C}{2} \sin 2 \omega t \ & x=\frac{A+B}{2}+\left(\frac{B}{2}-\frac{A}{2}ight)+\cos 2 \omega t+\frac{C}{2} \sin 2 \omega t \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)\end{aligned} $$
$$ \begin{aligned} & x=\frac{A+B}{2} \sqrt{\left(\frac{B-A}{2}ight)^2+\left(\frac{C}{2}ight)^2} \sin (2 \omega t+\alpha) \ & \qquad 	an \alpha=\frac{B-A}{C} \ & x-\frac{A+B}{2}=\sqrt{\left(\frac{B-A}{2}ight)^2+\left(\frac{C}{2}ight)^2} \sin (2 \omega t+\alpha) \ & x-x_0=a_0 \sin (\omega t+\alpha) \ & 	ext { for } A=-B, C=2 B . \ & 	ext { Amplitude } a_0=\sqrt{\left(\frac{B-A}{2}ight)^2+\left(\frac{C}{2}ight)^2}=\sqrt{2} B \end{aligned} $$
From (i)
(A) For $\mathrm{A}=\mathrm{B}=0, x=\frac{\mathrm{C}}{2} \quad \sin 2 \omega t(\mathrm{SHM})$
(B) For $\mathrm{A}=-\mathrm{B}, \mathrm{C}=2 \mathrm{~B}$
$$ \begin{aligned} & x=\frac{-\mathrm{B}+\mathrm{B}}{2}+\left(+\frac{\mathrm{B}}{2}+\frac{\mathrm{B}}{2}ight) \cos 2 \omega t+\frac{2 \mathrm{~B}}{2} \sin 2 \omega t \ & x=\cos 2 \omega t+\mathrm{B} \sin 2 \omega t \ & x=\sqrt{2} \mathrm{~B} \sin \left(2 \omega t+\frac{\pi}{4}ight)(\mathrm{SHM}) \end{aligned} $$
Resultant amplitude $=\sqrt{2}$
(C) For $\mathrm{A}=\mathrm{B}, \mathrm{C}=0$,
$$ \begin{aligned} x & =\frac{A+A}{2}+\left(\frac{A}{2}-\frac{A}{2}ight) \cos 2 \omega t+0 \ & =\frac{A+A}{2}=A(	ext { no SHM }) \end{aligned} $$
$$ \begin{aligned} & 	ext { (D) } \mathrm{A}=\mathrm{B}, \mathrm{C}=2 \mathrm{~B} \ & \begin{aligned} x=\frac{\mathrm{A}+\mathrm{A}}{2}+\left(\frac{\mathrm{A}}{2}-\frac{\mathrm{A}}{2}ight) \cos 2 \omega t+\frac{2 \mathrm{~B}}{2} \sin 2 \omega t & \  =\mathrm{A}+\mathrm{B} \sin 2 \omega t(\mathrm{OR}) \ x=\mathrm{B}+\mathrm{B} \sin 2 \omega t(\mathrm{SHM}),  	ext { Amplitude }=|\mathrm{B}| \end{aligned} \end{aligned} $$

Answer

for any value ol $\mathrm{A}, \mathrm{B}$ and C (except $\mathrm{C}=0$ ).

Answer

if $\mathrm{A}=-\mathrm{B} ; \mathrm{C}=2 \mathrm{~B}$, amplitude $=|\mathrm{B} \sqrt{2}|$.

Answer

if $\mathrm{A}=\mathrm{B} ; \mathrm{C}=0$.

Answer

if $\mathrm{A}=\mathrm{B} ; \mathrm{C}=2 \mathrm{~B}$, amplitude $=|\mathrm{B}|$

Oscillations

Function x=\mathrm{A} \sin ^2 \omega t+\mathrm{B} \cos ^2 \omega t+\mathrm{C} \sin \omega t \cos \omega t represents SHM

Choose the correct answer: