Question

The moment of inertia of a big drop is $I$. If $8$ droplets are formed from the big drop, then the moment of inertia of the small droplet is.
(A) $\dfrac{I}{{32}}$
(B) $\dfrac{I}{{16}}$
(C) $\dfrac{I}{8}$
(D) $\dfrac{I}{4}$

Answer 1

The moment of inertia of a particle about an axis of rotation is the product of the mass of the particle and the square of the distance of the particle from the axis. Here the moment of inertia of a big drop is given. If we combine $8$ such drops to form a big drop then the moment of inertia of one drop should be found.
Formula used:
The moment of inertia of a sphere,
$I = \dfrac{2}{5}M{R^2}$
where $M$ stands for the mass of the sphere and $R$ stands for the radius of curvature of the sphere.

Complete step by step solution:
Let us assume the drop to be spherical.
The moment of inertia of a sphere is given by, $I = \dfrac{2}{5}M{R^2}$.
A big drop is formed by combining small drops. The volume will remain the same after the formation of the big drop from the small drops.
Let $r$ be the radius of the small drops,
Then the volume of $n$ small drops will be, ${V_n} = n\dfrac{4}{3}\pi {r^3}$
Let $R$ be the radius of the big drop,
Then the volume of the big drop will be $V = \dfrac{4}{3}\pi {R^3}$
The volume of the big drop will be equal to the volume of $n$ small drops.
$n\dfrac{4}{3}\pi {r^3} = \dfrac{4}{3}\pi {R^3}$
Canceling the common terms we get
$n{r^3} = {R^3}$
Taking the cube root of the above equation,
${n^{\dfrac{1}{3}}}r = R$
We know that $n = 8$. Putting this value in the above equation$M$
${8^{\dfrac{1}{3}}}r = R$
From this, the radius of the small drop can be written as,
$r = \dfrac{R}{2}$
The mass of the big drop is $M$.
Then the mass of one small drop will be $\dfrac{M}{8}$.
Therefore, the moment of inertia of the small droplet will be
${I_s} = \dfrac{2}{5}\left[ {\dfrac{M}{8}} \right]{\left[ {\dfrac{R}{2}} \right]^2}$
Solving the above equation,
${I_s} = \dfrac{1}{{32}}\left[ {\dfrac{2}{5}M{R^2}} \right]$
That is
${I_s} = \dfrac{I}{{32}}$

The answer is: Option (A): $\dfrac{I}{{32}}$

Note:
Rotational inertia is the inability of a body at rest to rotate by itself, and a body in uniform rotational motion to stop by itself is called the rotational inertia of a body. The moment of inertia is a measure of rotational inertia. Mass is a measure of inertia.