1 Simple harmonic motion with period \(\pi / \omega\)
2 Simple harmonic motion with period \(2 \pi / \omega\)
3 Non-periodic
4 Periodic but not simple harmonic
Explanation:
Given,
\(y = 3\cos \left( {\frac{\pi }{4} - 2\omega t} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
Velocity, \(v=\dfrac{d y}{d t}=3 \times 2 \omega \sin \left(\dfrac{\pi}{4}-2 \omega t\right)\)
Acceleration,
\(a=\dfrac{d v}{d t}=-4 \omega^{2} \times 3 \cos \left(\dfrac{\pi}{4}-2 \omega t\right)=-4 \omega^{2} y\)
As \(A \propto-y\) hence particle will execute SHM comparing Eq. (i) with equation
\(y=A \cos \left(\phi-\omega^{\prime} t\right)\)
we have, \(\omega^{\prime}=2 \omega\) or \(\dfrac{2 \pi}{T^{\prime}}=2 \omega\) or \(T^{\prime}=\dfrac{\pi}{\omega}\)