The displacement of a particle executing SHM is given by $x=10 \sin \left(\omega t+\frac{\pi}{3}\right) \mathrm{m}$. The time period of motion is $3.14 \mathrm{~s}$. The velocity of the particle at $\mathrm{t}=0$is_________. $\mathrm{m} / \mathrm{s}$.
JEE MAIN 2024, Diffcult
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Given,
$\mathrm{T}=3.14=\frac{2 \pi}{\omega}$
$\omega=2 \mathrm{rad} / \mathrm{s}$
$\mathrm{x}=10 \sin \left(\omega \mathrm{t}+\frac{\pi}{3}\right)$
$\mathrm{v}=\frac{\mathrm{dx}}{\mathrm{dt}}=10 \omega \cos \left(\omega \mathrm{t}+\frac{\pi}{3}\right)$
$\text { at } \mathrm{t}=0$
$\mathrm{v}=10 \omega \cos \left(\frac{\pi}{3}\right)=10 \times 2 \times \frac{1}{2}[\text { using } \omega=2 \mathrm{rad} / \mathrm{s}]$
$\mathrm{v}=10 \mathrm{~m} / \mathrm{s}$
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