Question

Given below are two statements : one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

$\text{Assertion A}$ : Moment of inertia of a circular disc of mass ${}^{\prime }{M}^{\prime }$ and radius ${}^{\prime }{R}^{\prime }$ about $X,Y$ axes (passing through its plane) and $Z$-axis which is perpendicular to its plane were found to be $I_x$, $I_y$ and $I_z$ respectively. The respective radii of gyration about all the three axes will be the same.

$\text{Reason R}$ : A rigid body making rotational motion has fixed mass and shape.

In the light of the above statements, choose the most appropriate answer, from the options given below.

• A. Both $A$ and $R$ are correct, and $R$ is the correct explanation of $A$.
• B. Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$.
• C. $A$ is correct but $R$ is not correct.
• D. $A$ is not correct but $R$ is correct.

Answer 1

The correct option is D $A$ is not correct but $R$ is correct.

According to perpendicular axes theorem,

$I_z=I_x+I_y$

The moment of inertia about any axis of rotation, in general, is,

$I=m{K}^2(K\to$ radius of gyration)

So, $m{K}_z^2=m{K}_x^2+m{K}_y^2$

$\therefore K_z^2=K_x^2+K_y^2$

From the above equation, it is clear that the radius of gyration about the $x,y\text{and}z$ axes can't be the same. So, assertion $A$ is not correct.

Now, rigid body is defined as a body having a definite shape, size and mass. So, reason $R$ is correct.

Hence, $\left(B\right)$ is the correct answer.