The moment of inertia of a solid sphere about | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The moment of inertia of a solid sphere of mass $M$ and radius $R$ about its diameter is given by $I = \frac{2}{5}MR^2$. Let the radius of the lar

Vedclass · Accepted Answer

$\frac{I}{243}$

Vedclass · Answer

(D) The moment of inertia of a solid sphere of mass $M$ and radius $R$ about its diameter is given by $I = \frac{2}{5}MR^2$. Let the radius of the large sphere be $R$ and its mass be $M$. When it is recast into $27$ small spheres of radius $r$,the volume remains constant: $\frac{4}{3}\pi R^3 = 27 	imes \frac{4}{3}\pi r^3$. This implies $R^3 = 27r^3$,so $R = 3r$ or $r = \frac{R}{3}$. The mass of each small sphere $m$ is $\frac{M}{27}$. The moment of inertia of each small sphere about its diameter is $I' = \frac{2}{5}mr^2$. Substituting $m = \frac{M}{27}$ and $r = \frac{R}{3}$: $I' = \frac{2}{5} \left( \frac{M}{27} ight) \left( \frac{R}{3} ight)^2 = \frac{2}{5} \left( \frac{M}{27} ight) \left( \frac{R^2}{9} ight) = \frac{1}{243} \left( \frac{2}{5}MR^2 ight)$. Since $I = \frac{2}{5}MR^2$,we get $I' = \frac{I}{243}$.

The moment of inertia of a solid sphere about its diameter is $I$. It is then recast into $27$ small spheres of the same diameter. The moment of inertia of each small sphere about its diameter is:

Similar Questions

$A$ solid sphere of radius $r = 5 \text{ cm}$ is rotating about an axis at a distance $d = 10 \text{ cm}$ from its center, as shown in the figure. If the radius of gyration about that axis is $\sqrt{x} \text{ cm}$, then the value of $x$ is:

$A$ solid sphere of radius $10 \ cm$ is rotating about an axis which is at a distance $15 \ cm$ from its centre. The radius of gyration about this axis is $\sqrt{n} \ cm$. The value of $n$ is

Three thin uniform rods,each of mass $M$ and length $L$,are placed along the three Cartesian axes such that one end of each rod is at the origin. Find the moment of inertia of this system about the $z$-axis.

The radius of gyration of a body depends upon:

The moment of inertia of a body is $160 \ kg \ m^2$ and its mass is $10 \ kg$. What will be the radius of gyration in $m$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module