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Physics

Mechanics

Easy

Question

A thin rod of length 4/ and mass 4m is bent at the points as shown in Fig. What is the moment of inertia of the rod about the axis passing through point O and perpendicular to the plane of the paper

$\frac{M^{2}}{3}$
$\frac{10 M l^{2}}{3}$
$\frac{M l^{2}}{12}$
$\frac{M I^{2}}{24}$

The correct answer is: $fraction numerator 10 M l squared over denominator 3 end fraction$

$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell text Total end text MI end cell cell equals I subscript 1 plus I subscript 2 plus I subscript 3 plus I subscript 4 end cell row blank cell equals 2 I subscript 1 plus 2 I subscript 2 end cell row blank cell equals 2 left parenthesis I subscript 1 plus I subscript 2 right parenthesis space of 1em left square bracket I subscript 1 equals I subscript 3 semicolon space of 1em I subscript 1 equals I subscript 4 right square bracket end cell end table$
Now,
$I subscript 2 equals I subscript 3 equals fraction numerator M l squared over denominator 3 end fraction$
using parallel axis theorem, we have
$table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell I equals I subscript CM plus M x squared text and end text x equals square root of l squared plus left parenthesis l over 2 right parenthesis squared end root end cell row cell I subscript 1 equals I subscript 4 equals fraction numerator M l squared over denominator 12 end fraction plus M left square bracket square root of l squared plus left parenthesis l over 2 right parenthesis squared end root right square bracket squared end cell end table$
Putting all values, we get
$I equals fraction numerator 10 M l squared over denominator 3 end fraction$

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