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physics

mechanics

Easy

Question

A uniform thin rod of mass m and length L is standing on a smooth horizontal surface. A slight disturbance causes the lower end to slip on the smooth surface and the rod starts falling. Then velocity of centre of mass of the rod when it makes an angle $theta equals 45 to the power of o$ with horizontal is

$\sqrt{\frac{3 g L (\sqrt{2} + 1)}{5 (\sqrt{2} - 1)}}$
$\sqrt{\frac{3 g L (\sqrt{2} - 1)}{5 \sqrt{2}}}$
$\frac{\sqrt{3 g L (\sqrt{2} + 1)}}{5 \sqrt{2}}$
$\sqrt{\frac{3 g L (\sqrt{2} - 1)}{5}}$

The correct answer is: $\sqrt{\frac{3 g L (\sqrt{2} - 1)}{5 \sqrt{2}}}$

$table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell m g h equals 1 half I. omega squared end cell row cell I subscript ∗ equals open curly brackets fraction numerator m L squared over denominator 12 end fraction plus fraction numerator m left parenthesis L cos invisible function application theta right parenthesis squared over denominator 4 end fraction close curly brackets end cell row cell h equals L over 2 left parenthesis 1 minus sin invisible function application theta right parenthesis end cell row cell omega equals square root of fraction numerator 12 g left parenthesis 1 minus sin invisible function application theta right parenthesis over denominator L open parentheses 1 plus 3 cos squared invisible function application theta close parentheses end fraction end root end cell row cell text velocity of centre of mass end text equals open parentheses L over 2 cos invisible function application theta close parentheses omega end cell row cell equals square root of fraction numerator 3 g L left parenthesis 1 minus sin invisible function application theta right parenthesis cos squared invisible function application theta over denominator open parentheses 1 plus 3 cos squared invisible function application theta close parentheses end fraction end root end cell end table$

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