Directions : In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:
Which of the following functions represent simple harmonic motion? (1) \[sin\text{ }2\text{ }\omega t\] (2) \[sin\text{ }2\omega t+2\text{ }cos\text{ }\omega t\] (3) \[si{{n}^{2}}\text{ }\omega t\] (4) \[sin\text{ }\omega t+cos\text{ }2\text{ }\omega t\]A) 1 and 2 are correct
B) 2 and 3 are correct
C) 1 and 4 are correct
D) 1, 2 and 3 are correct
Correct Answer: A
Solution :
A motion will be simple harmonic if\[a\propto -y\]. (1) Let \[y=\sin 2\omega t\] velocity, \[v=\frac{dy}{dt}=2\omega \,\cos \,2\omega t\] Acceleration, \[v=\frac{dv}{dt}=-4{{\omega }^{2}}\sin 2\omega t\] or \[a=-4{{\omega }^{2}}y\] or\[a\propto -y\] So, it is a simple harmonic motion. (2) Let \[y=\sin \omega t+2\cos \omega t\] \[v=\frac{dy}{dt}=\omega \cos \omega t-2\omega \sin \omega t\] \[a=\frac{av}{dt}=-{{\omega }^{2}}\sin \omega t-2{{\omega }^{2}}\cos \omega t\] Or \[a=-{{\omega }^{2}}(\sin \omega t+2\cos \omega t)\] Or \[a\propto -y\] So, it is also a simple harmonic motion, The remaining two parts cannot represent simple harmonic motion.
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