Question

The displacement of a particle is represented by the equation $y=3cos[\frac{\pi }{4}-2\omega t]$.
The motion is:

• A. non - periodic
• B. periodic but not simple harmonic
• C. simple harmonic with period $2pi\omega$
• D. simple harmonic with period $\left(\frac{\pi }{\omega }\right)$

Answer 1

Answer: (D)

simple harmonic with period $\left(\frac{\pi }{\omega }\right)$

$y=3\cos (\frac{\pi }{4}-2wt)$

$\frac{dy}{dt}=3(-)(-2w)\sin (\frac{\pi }{4}-2wt)$

$\frac{d^{2}y}{d{t}^2}=6w(-2w)\cos (\frac{\pi }{4}-2wt)=-12w^{2}y$

$\therefore$ $Force\propto (-)displacement$

$\therefore$ The motion is simple harmonic motion.

$\therefore \text{Time period}T=\frac{2\pi }{w_{effective}}=\frac{2\pi }{2w}=\frac{\pi }{w}$

Option $D$ is the correct answer.