Question

A book of mass 0.5kg has its length 75cm and breadth 25cm. Find the moment of inertia about an axis perpendicular to the book and passing through the center of gravity of the book.
A.$\dfrac{{10}}{{289}}\,kg{m^2}$
B.$\dfrac{{289}}{{10}}\,kg{m^2}$
C.$\dfrac{{10}}{{384}}\,kg{m^2}$
D.$\dfrac{{10}}{{483}}\,kg{m^2}$

Answer 1

Moment of inertia of a body determines the torque required for producing the required angular acceleration about an axis of rotation. Moment of inertia of a rectangular plate is given by the formula $I = \dfrac{M}{{12}}\left( {{l^2} + {b^2}} \right)$. Where $M$ is the mass of the body, $l$ is length and $b$ is the breadth.

Complete step by step answer:
The moment of inertia of a rectangular plate is given by the formula $I = \dfrac{M}{{12}}\left( {{l^2} + {b^2}} \right)$
Given that,
Mass,$M = 0.5\,kg$
Length, $l = 75\,cm$
$\Rightarrow l = \dfrac{{75}}{{100}}\,m \\\ \Rightarrow l = 0.75\,m \\\$
Breadth, $b = 25\,cm$
$\Rightarrow b = \dfrac{{25}}{{100}}\,m \\\ \Rightarrow b = 0.25\,m \\\$
Substituting the given values in the equation. Then we get,
$I = \dfrac{M}{{12}}\left( {{l^2} + {b^2}} \right) \\\ \Rightarrow I = \dfrac{{0.5\,kg}}{{12}}\left( {{{\left( {0.75\,m} \right)}^2} + {{\left( {0.25\,m} \right)}^2}} \right)$
On simplification,
$\Rightarrow I = \dfrac{{\left( {\dfrac{1}{2}} \right)\,kg}}{{12}}\left( {{{\left( {\dfrac{3}{4}\,m} \right)}^2} + {{\left( {\dfrac{1}{4}\,m} \right)}^2}} \right)$
On further simplification,
$\Rightarrow I = \dfrac{{1\,}}{{24}}\left( {\dfrac{9}{{16}} + \dfrac{1}{{16}}} \right)\,kg{m^2} \\\ \Rightarrow I = \dfrac{{10}}{{384}}\,kg{m^2} \\\$

$\therefore$ The moment of inertia of the book about a perpendicular axis passing through its center of gravity is $\dfrac{{10}}{{384}}\,kg{m^2}$. Hence, the answer is an option (C).

Note:
The perpendicular axis theorem can be used to arrive at the equation for finding the moment of inertia of the book. According to this theorem, the moment of inertia of a planar lamina about a perpendicular axis is equal to the sum of the moment of inertia of two axes which are at right angles to each other, lying in the plane of the lamina and intersects at the point where the perpendicular axis passes through the plane. In equation form it can be written as ${I_Z} = {I_X} + {I_Y}$, where ${I_Z}$ is the required moment of inertia about the perpendicular axis and ${I_X}$ and ${I_Y}$ are the moment of inertia of two axes which are at right angles to each other, lying in the plane of the lamina and intersects at the point where the perpendicular axis passes through the plane of the lamina.