A uniform stick with mass m and length $l$ lies on a frictionless horizontal surface. It is pivoted at a point a distance x from its center. A spring (at its relaxed length) with spring constant k is attached to the far end of the stick, perpendicular to the stick as shown in figure. If the stick is given a tiny kick, what value of x yields the largest frequency of small oscillations? If $x = l / n$ , find n

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Detailed Solution

$τ = F d = (- k (\frac{l}{2} + x) θ) (\frac{l}{2} + x) = - k {(\frac{l}{2} + x)}^{2} θ$

$- k {(\frac{l}{2} + x)}^{2} θ = (\frac{m l^{2}}{12} + m x^{2}) \ddot{θ}$

$ω = \sqrt{\frac{k}{m}} \sqrt{\frac{{(l / 2 + x)}^{2}}{l^{2} 12 + x^{2}}}$

$0 = (\frac{l^{2}}{12} + x^{2}) \cdot 2 (\frac{l}{2} + x) - {(\frac{l}{2} + x)}^{2} \cdot 2 x$

$\Rightarrow (\frac{l}{2} + x) x = \frac{l^{2}}{12} + x^{2} \Rightarrow x = \frac{l}{6}$

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