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A square plate, of side a and mass M is placed in $(x - z)$ plane as shown in figure and hinged at A to rotate about x-axis. The plate is allowed to fall upto $(x - y)$ plane. Find the final angular velocity.

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$\sqrt{\frac{24 g}{7 \sqrt{2 a}}}$

${[\frac{24 g}{7 a}]}^{1 / 2}$

${[\frac{12 g}{7 a}]}^{1 / 2}$

${[\frac{12 g}{7 \sqrt{2} a}]}^{1 / 2}$

answer is A.

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Detailed Solution

$M g \frac{1}{\sqrt{2}} = \frac{1}{2} I w^{2} = \frac{1}{2} (\frac{2}{3}) M a^{2} ω^{2}$

Which gives $ω = {[\frac{3 g}{\sqrt{2} a}]}^{1 / 2} = {[\frac{3 g}{a \sqrt{2}}]}^{1 / 2}$

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