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Position of a $2 \mathrm{~kg}$ mass moving along the $X$-axis is given by $x=2 \cos [(2) t] \mathrm{m}$. Then maximum kinetic energy of the mass in joule is
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The correct answer is:
$16$
Position of particle, $x=2 \cos (2 t)$
$\Rightarrow$ Velocity of particle, $v=\frac{d x}{d t}=\frac{d}{d t} 2 \cos 2 t$
$\Rightarrow \quad v=-2 \sin 2 t \times \frac{d}{d t} 2 t \Rightarrow v=-4 \sin 2 t$
Velocity is maximum when $\sin 2 t=1$. Hence maximum kinetic energy of particle is
$K_{\max }=\frac{1}{2} m\left(v_{\max }\right)^2$
Substituting $m=2 \mathrm{~kg}$ and $v_{\max }=-4 \mathrm{~m} / \mathrm{s}$;
$\Rightarrow \quad K_{\max }=\frac{1}{2} \times 2 \times(-4)^2=16 \mathrm{~J}$
$\Rightarrow$ Velocity of particle, $v=\frac{d x}{d t}=\frac{d}{d t} 2 \cos 2 t$
$\Rightarrow \quad v=-2 \sin 2 t \times \frac{d}{d t} 2 t \Rightarrow v=-4 \sin 2 t$
Velocity is maximum when $\sin 2 t=1$. Hence maximum kinetic energy of particle is
$K_{\max }=\frac{1}{2} m\left(v_{\max }\right)^2$
Substituting $m=2 \mathrm{~kg}$ and $v_{\max }=-4 \mathrm{~m} / \mathrm{s}$;
$\Rightarrow \quad K_{\max }=\frac{1}{2} \times 2 \times(-4)^2=16 \mathrm{~J}$
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