Question

Find the Moment of inertia about AB axis.

& A.\dfrac{2}{3}M{{L}^{2}} \\\ & B.\dfrac{4}{3}M{{L}^{2}} \\\ & C.\dfrac{3}{2}M{{L}^{2}} \\\ & D.\dfrac{3}{4}M{{L}^{2}} \\\ \end{aligned}$$

Answer 1

Moment of inertia is the property of a body to resist angular acceleration. We know that the moment of inertia depends on the density of the material, the axis of rotation and the dimensions of the body, i.e. the shape and the size of the body. We also know that, the moment of inertia depends mainly on the mass of the object and its distance from the axis of rotation.

Complete step by step answer:
Moment of inertia is the resistance offered by an object against rotational acceleration. Let us consider the given wheel of mass of the to be $m$, and $r$ its radius. Then the moment of inertia $I$ experienced by the wheel during a rotation along its own axis is given by $I=mk^{2}$, where $k$ is the radius of gyration .
Let us assume that ABC is an equilateral triangle. Here, we can take which is the distance of the mass from the point of consideration.
To find the MOI about AB axis consider O as the midpoint of the length AB, let thus also be the origin for our reference. Then the perpendicular bisector of AB to C is as shown in the figure below.

Clearly, $OC=AC sin(60^{\circ})=\dfrac{\sqrt 3}{2}L$
Then, the $I_{ab}=I_{a}+I_{b}+I_{c}$
$\implies I_{ab}=I_{c}$
$\implies I_{ab}=\left(\dfrac{\sqrt 3}{2}L\right)^{2}\times M$
$\implies I_{ab}=\dfrac{3}{4}ML^{2}$
Hence the correct answer is option $D.\dfrac{3}{4}M{{L}^{2}}$

Note:
Rotational inertia of the wheel is $I=mk^{2}$, where $k$ is the radius of gyration, this value is unique for each object. Thus it is useful to remember the value of $k$ or $I$ for all the objects. Also, the question might look complex, but it can be solved easily, if the formulas are known. The moment of inertia depends on the density of the material, the axis of rotation and the dimensions of the body, i.e. the shape and the size of the body. Its dimensional formula is given as $[M^{1}L^{2}T^{0}]$ with SI unit is $kgm^{2}$.