Question

13) The Moment of inertia of a rod about axis perpendicular to length about one
end is how many times than the moment of inertia about axis passing through
centre and perpendicular to length
a) 4
b) 3
c) 2
d) 6​

Answer 1

The moment of inertia of a rod of length $\sf{L}$ and mass $\sf{M}$ about an axis perpendicular to the length and passing through the center is,

$\longrightarrow\sf{I_c=\dfrac{ML^2}{12}\quad\quad\dots(1)}$

Each end of the rod is at a distance $\sf{\dfrac{L}{2}}$ from the center.

So the moment of inertia of the rod about an axis perpendicular to the length and passing through one of its ends is, by theorem of parallel axes,

$\longrightarrow\sf{I_e=I_c+M\left(\dfrac{L}{2}\right)^2}$

$\longrightarrow\sf{I_e=\dfrac{ML^2}{12}+\dfrac{ML^2}{4}}$

$\longrightarrow\sf{I_e=\dfrac{ML^2}{4}\left(\dfrac{1}{3}+1\right)}$

$\longrightarrow\sf{I_e=\dfrac{ML^2}{4}\cdot\dfrac{4}{3}}$

$\longrightarrow\sf{I_e=\dfrac{ML^2}{3}\quad\quad\dots(2)}$

Dividing (2) by (1),

$\longrightarrow\sf{\dfrac{I_e}{I_c}=\dfrac{ML^2}{3}\cdot\dfrac{12}{ML^2}}$

$\longrightarrow\sf{\dfrac{I_e}{I_c}=4}$

$\longrightarrow\underline{\underline{\sf{I_e=4I_c}}}$

Hence (a) is the answer.