Question

From the solid sphere of mass M and the radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is:
$A.{\text{ }}\dfrac{{M{R^2}}}{{32\sqrt {2\pi } }} \\\ B.{\text{ }}\dfrac{{M{R^2}}}{{16\sqrt {2\pi } }} \\\ C.\;\dfrac{{4M{R^2}}}{{9\sqrt {3\pi } }} \\\ D.{\text{ }}\dfrac{{4M{R^2}}}{{3\sqrt {3\pi } }} \\\$

Answer 1

Hint: The term used in this question diagonal of cube means that it is the segment that joins the two points that are not the end points that are not the end point of the edge. But the main diagonal cube is the one that cuts through the centre of the cube.Try to find the relation between dimensions of cube and sphere and then proceed.

Step By Step Answer:
As we can see our cube diagonal is

$\sqrt 3 a = 2R \\\ \Rightarrow a = \dfrac{{2R}}{3} \\\$

As we say ${M^ / }$is the mass of the cube so, material is same thus sphere density must equal to the cube density.

$\dfrac{M}{{\dfrac{4}{3}\pi {R^3}}} = \dfrac{{{M^ / }}}{{{a^3}}} \\\ \Rightarrow {M^ / } = \dfrac{{3M{a^3}}}{{4\pi {R^3}}}.......................................(a = 8) \\\ \Rightarrow \dfrac{{3M}}{{4\pi {R^3}}} \times \dfrac{{8{R^3}}}{{3\sqrt 3 }} = \dfrac{{2M}}{{\sqrt 3 \pi }} \\\$

AS WE KNOW THAT MOMENT OF INERTIA IS:

$I = \dfrac{1}{6}{M^ / }{a^2} \\\ \Rightarrow \dfrac{1}{6} \times \dfrac{{2M}}{{\sqrt 3 \pi }} \times \dfrac{{4{R^2}}}{3} \\\ \Rightarrow \dfrac{{4M{R^2}}}{{9\sqrt 3 \pi }} \\\$
So C is the correct option.

Additional Information: The term moment of inertia is also known as mass moment of inertia. It is defined as the ratio of net angular momentum of a system to its angular velocity around the principal axis. Moment of inertia plays a very important role in physics which means that in physics problems that involve the mass in rotation motion and that are calculated by angular momentum.

THERE ARE THREE TYPES OF INERTIA:

A.)INERTIA OF REST
B.)INERTIA OF MOTION
C.)INERTIA OF DIRECTION

If the moment of inertia is increased there will be a slowing down process of speed of rotation. We can also say that the moment of inertia of the body is directly proportional to the mass and it increases as the mass moves further from the axis of rotation.
Therefore option (C) is the right answer.

Note: Don’t get confused in saying that the moment of inertia and the inertia is the same in nature in physics. No it is not inertia means just the state of the body either it is in motion or rest whereas the moment of inertia is the measurement of resistances of the object against the rotation.