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Physics

Mechanics

Easy

Question

A uniform rod of mass m and length L is smoothly pivoted at one of its ends. When it is released from rest from horizontal position, it swings in a vertical circle. When the rod swings through an angle $theta$ with horizontal, reaction force offered by the rod along normal and tangential directions are respectively

$R_{n} = \frac{m g \cos θ}{4}, R_{1} = \frac{5 m g \sin θ}{4}$
$R_{n} = \frac{5 m g \cos θ}{4}, R_{1} = \frac{m g \sin θ}{4}$
$R_{n} = \frac{m g \cos θ}{4}, R_{1} = \frac{5 m g \sin θ}{2}$
$R_{n} = \frac{5 m g \sin θ}{2}, R_{1} = \frac{m g \cos θ}{4}$

The correct answer is: $R subscript n equals fraction numerator 5 m g sin invisible function application theta over denominator 2 end fraction comma R subscript 1 equals fraction numerator m g cos invisible function application theta over denominator 4 end fraction$

$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell text From the figure, end text m g cos invisible function application theta minus R subscript t equals m a subscript t end cell row cell R subscript n minus m g sin invisible function application theta equals m a subscript n end cell row cell text About hinge, end text m g L over 2 cos invisible function application theta equals fraction numerator m L squared over denominator 3 end fraction alpha end cell end table$ $table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell alpha equals fraction numerator 3 g cos invisible function application theta over denominator 2 L end fraction end cell row cell a subscript n equals L over 2 omega squared text and end text a subscript t equals L over 2 alpha end cell row cell text As end text alpha text is changing with end text theta comma alpha equals fraction numerator d omega over denominator d t end fraction equals open parentheses fraction numerator d omega over denominator d theta end fraction close parentheses open parentheses fraction numerator d theta over denominator d t end fraction close parentheses equals fraction numerator omega d omega over denominator d theta end fraction end cell row blank row cell table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell not stretchy rightwards double arrow omega d omega equals fraction numerator 3 g over denominator 2 L end fraction cos invisible function application theta d theta end cell row cell integral subscript 0 superscript straight infinity omega d omega equals fraction numerator 3 g over denominator 2 L end fraction integral subscript 0 superscript theta cos invisible function application theta d theta not stretchy rightwards double arrow omega equals square root of fraction numerator 3 g sin invisible function application theta over denominator L end fraction end root end cell row cell text so end text a subscript n equals fraction numerator 3 g sin invisible function application theta over denominator 2 end fraction text and end text a subscript t equals fraction numerator 3 g cos invisible function application theta over denominator 4 end fraction end cell row cell R subscript t equals fraction numerator m g cos invisible function application theta over denominator 4 end fraction end cell end table end cell row cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell R subscript n equals fraction numerator 5 m g sin invisible function application theta over denominator 2 end fraction end cell row cell text Net reaction at hinge end text R equals square root of R subscript t superscript 2 plus R subscript n superscript 2 end root end cell end table end cell end table$

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