The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is $10\,s^{-1}$. At, $t = 0$ the displacement is $5\, m$. What is the maximum acceleration ? The initial phase is $\frac{\pi }{4}$
JEE MAIN 2017, Diffcult
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Maximum velocity in $\mathrm{SHM}, \mathrm{v}_{\max }=\mathrm{a} \omega$
Maximum acceleration in $\mathrm{SHM}, \mathrm{A}_{\max }=\mathrm{a} \omega^{2}$
where $a$ and $\omega$ are maximum amplitude and angular frequency.
Given that, $\frac{A_{\max }}{v_{\max }}=10$
i.e., $\omega=10 \mathrm{s}^{-1}$
Displacement is given by
Maximum acceleration in $\mathrm{SHM}, \mathrm{A}_{\max }=\mathrm{a} \omega^{2}$
where $a$ and $\omega$ are maximum amplitude and angular frequency.
Given that, $\frac{A_{\max }}{v_{\max }}=10$
i.e., $\omega=10 \mathrm{s}^{-1}$
Displacement is given by
$x=a \sin (\omega t+\pi t 4)$
at $t=0, x=5$
$5=a \sin \pi / 4$
$5=a \sin 45^{\circ} \Rightarrow a=5 \sqrt{2}$
Maximum acceleration $\mathrm{A}_{\mathrm{max}}=\mathrm{a} \omega^{2}$
$=500\sqrt{2}\,m/s^2$
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