MCQ
The equilateral triangle $ABC$ is cut from a thin solid sheet of wood. (See figure) $D, E$ and $F$ are the mid points of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_0.$ If the smaller triangle $DEF$ is removed from $ABC,$ the moment of inertia of the remaining figure about the same axis is $I.$ Then
  • $I = \frac{{15}}{{16}}{I_0}$
  • B
    $I = \frac{{3}}{{4}}{I_0}$
  • C
    $I = \frac{{9}}{{16}}{I_0}$
  • D
    $I = \frac{{{I_0}}}{4}$

Answer

Correct option: A.
$I = \frac{{15}}{{16}}{I_0}$
a
$I\, \propto \,m{\ell ^2}\,\,\,\,\,\,\left( {let\,\sigma  = mass\,present\,area} \right)$

$\therefore \,\,\,{I_1}\, \propto \,{\ell ^4}\,\,\,\,\,\,\,...\left( 1 \right)$

$and\,\,{I_2}\, \propto \,{\left( {\frac{\ell }{2}} \right)^4}\,\,\,\,\,\,\,\,...\left( 2 \right)$

$So,\,{I_2} = \frac{I}{{16}}$

Moment of inertia of remaining sheet = $I - \frac{1}{{16}}$

$ = \frac{{151}}{{16}}$

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