Question

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

• A. Only one of these is S.H.M
• B. Ratio of maximum speeds is $2:1$
• C. Ratio of maximum speeds is $1:1$
• D. Ratio of maximum accelerations is $1:4$

Answer 1

The correct option is C Ratio of maximum speeds is $1:1$

$y=2Aco{s}^2\omega t\Rightarrow y=A(\cos 2\omega t+1)\Rightarrow y-A=A\cos 2\omega t\Rightarrow A_1=A;{\omega }_1=2\omega y=A(sin\omega t+\sqrt{3}cos\omega t)\Rightarrow y=2A(\frac{1}{2}sin\omega t+\frac{\sqrt{3}}{2}cos\omega t)\Rightarrow y=2A\sin (\omega t+\frac{\pi }{3})\Rightarrow A_2=2A;{\omega }_2=\omega$
Ratio of maximum speeds$=\frac{A_1{\omega }_1}{A_2{\omega }_2}=\frac{A\left(2\omega \right)}{\left(2A\right)\omega }=\frac{1}{1}$