Question

M.I of a solid sphere about its diameter is $64\,kg\,{m^2}$. If that sphere is recast into $8$identical small spheres, then M.I. of such small sphere about its diameter is
$\left( A \right)\,\,8\,Kg\,{m^2}$
$\left( B \right)\,\,4\,kg\,{m^2}$
$\left( C \right)\,\,3\,kg\,{m^2}$
$\left( D \right)\,\,2\,kg\,{m^2}$

Answer 1

In the question, the diameter of the solid sphere is given. But the sphere is divided into eight equal parts. By substituting the known values in the equation of moment, we get the value of the moment of inertia of such a small sphere of the diameter.
Moment of Inertia of the sphere is $I = \dfrac{2}{5}m{r^2}$
Where, $m$be the mass of the given sphere and $r$be the radius of the sphere.

Complete step by step solution:
Given that Diameter of the solid sphere$d = 64\,kg\,{m^2}$
Let the mass of given sphere is $m$and the radius of the sphere is $r.$
Moment of Inertia of the sphere is $I = \dfrac{2}{5}m{r^2}$.
Here, now we divide the sphere into eight equal small spheres
So, the equation of the sphere is,
$\Rightarrow 8\left( {\dfrac{4}{3}\pi {R^3}} \right) = \dfrac{4}{3}\pi {r^3}$
Simplify the above equation, we get the value of $r$,
$\Rightarrow r = \dfrac{R}{2}$
Now we have to find the moment of inertia of the each small spheres, we get
$\Rightarrow I = \dfrac{2}{5}m{r^2}$
Substitute the value of $r$in the equation, we get
$\Rightarrow I = \dfrac{2}{5}m{\left( {\dfrac{R}{2}} \right)^2}$
There are eight small spheres, so we written the equation of the moment of the inertia as,
$\Rightarrow I = \dfrac{2}{5} \times \dfrac{m}{8}{\left( {\dfrac{R}{2}} \right)^2}$
Simplify the above equation, we get
$\Rightarrow I = \dfrac{{64}}{{32}}$
Perform the divide operations in the above equation, we get
$\Rightarrow I = 2\,Kg\,{m^2}$
Therefore, the moment of the inertia of the small diameter is $2\,Kg\,{m^2}$.
Hence from the above options, option (D) is correct.

Note:
In the question, the sphere is used to find the moment of inertia. Spheres are divided into identical spheres which means it splits into equal spheres. But, So, diameter of the sphere is about the recast of the spheres. Consider those values, we get the moment of inertia of the sphere.