A $3\ kg$ sphere dropped through air has a terminal speed of $25\ m/s$. (Assume that the drag force is $-bv$.) Now suppose the sphere is attached to a spring of force constant $k = 300\ N/m$, and that it oscillates with an initial amplitude of $20\ cm$. What is the angular frequencu of its damped $SHM$? ..... $rad/s$
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$m g=b \times 25$
$\frac{b}{m}=\frac{g}{25}$
$\omega=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}-\frac{\mathrm{b}^{2}}{4 \mathrm{m}^{2}}}=\sqrt{\frac{300}{3}-\frac{1}{4} \times\left(\frac{10}{25}\right)^{2}}$
$=\sqrt{100-\frac{1}{25}}=10\left(1-\frac{1}{2} \times \frac{1}{2500}\right)$
$=9.998 \mathrm{rad} / \mathrm{s}$
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