Answer

Verified

51.9k+ views

**Hint:**To solve this question we have to apply a parallel axis theorem. According to this theorem, the moment of inertia along an axis parallel to the original axis will be the sum of the moment of inertia along the perpendicular axis and the product of mass and the distance between the perpendicular axis and parallel axis.

**Formulae used:**

${I_{parallel}} = {I_{perpendicular}} + M{R^2}$

Here ${I_{parallel}}$ is the moment of inertia along the parallel axis, ${I_{perpendicular}}$ is the moment of inertia along the axis through the centre of mass, $M$ is the mass of the object and $R$ is the distance between the centre of mass and the parallel axis.

**Complete step by step answer:**

In the question, a uniform square plate of side $a$ and mass $m$ is given. Let’s draw a figure.

From the above figure, we can easily find $R$ using the Pythagoras theorem,

$ \Rightarrow R = \sqrt {{a^2} - {{\left( {\dfrac{a}{2}} \right)}^2}} = \dfrac{a}{{\sqrt 2 }}$

We know that for a square plate, the moment of inertia along a perpendicular axis passing through the centre of mass is,

$ \Rightarrow {I_{perpendicular}} = \dfrac{{m{a^2}}}{6}$

So, using the parallel axis theorem, we get

$ \Rightarrow {I_{parallel}} = {I_{perpendicular}} + M{R^2}$

Here ${I_{parallel}}$ is the moment of inertia along the parallel axis, ${I_{perpendicular}}$ is the moment of inertia along the axis through the centre of mass, $M$ is the mass of the object and $R$ is the distance between the centre of mass and the parallel axis.

Substituting the value of $R$ and ${I_{perpendicular}}$ we get

$ \Rightarrow {I_{parallel}} = {I_{perpendicular}} + M{R^2}$

$ \therefore {I_{parallel}} = \dfrac{{m{a^2}}}{6} + \dfrac{{m{a^2}}}{{{{\left( {\sqrt 2 } \right)}^2}}} = \dfrac{2}{3}m{a^2}$

**So the required answer is $\dfrac{2}{3}m{a^2}$. Hence option (D) is correct.**

**Note:**While solving questions related to moment of inertia, make sure to apply the correct formulae. There are two different theorems i.e. parallel axis theorem and perpendicular axis theorem. Always use the correct theorem. The parallel axis theorem is used for axes parallel to the centroidal axis of the body. However, the perpendicular axis theorem is used for axes that are perpendicular to the centroidal axis of the body.

Recently Updated Pages

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

State two factors upon which the heat absorbed by a class 10 physics JEE_Main

What will be the halflife of a first order reaction class 12 chemistry JEE_Main

Which of the following amino acids is an essential class 12 chemistry JEE_Main

Which of the following is least basic A B C D class 12 chemistry JEE_Main

Out of the following hybrid orbitals the one which class 12 chemistry JEE_Main

Other Pages

The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main

According to classical free electron theory A There class 11 physics JEE_Main

In Bohrs model of the hydrogen atom the radius of the class 12 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main

In projectile motion the modulus of rate of change class 11 physics JEE_Main

Explain the construction and working of a GeigerMuller class 12 physics JEE_Main