Question

A solid sphere and a hollow sphere of the same mass have the same moment of inertia about their respective diameters. The ratio of their radii will be
A. $1:2$
B. $\sqrt 3 :\sqrt 5$
C. $\sqrt 5 :\sqrt 3$
D. $5:4$

Answer 1

By writing the equation for moment of inertia for a solid sphere and the equation for moment of inertia for a hollow sphere about the axis of rotation and by equating both these equations we can find the ratio of their radii.

Step by step solution:
It is given that a solid sphere and a hollow sphere of same mass have the same moment of inertia about their diameter. We need to find the ratio of their radii.
We know that moment of inertia of a solid sphere is given as
${I_s} = \dfrac{2}{5}M{R^2}$..............(1)
Where M is the mass and R is the radius of a solid sphere.
In the case of a hollow sphere the moment of inertia is given as
${I_h} = \dfrac{2}{3}M'{R'^2}$................(2)
Where,$M'$ is the mass of the sphere and $R'$ is the radius of the hollow sphere.
It is given that the moment of inertia is the same for both solid sphere and hollow sphere.
So, let us equate equation 1 and equation 2.
$\Rightarrow \dfrac{2}{5}M{R^2} = \dfrac{2}{3}M'{R'^2}$
It is given that the mass of the solid sphere and hollow sphere are the same so they will get cancelled.
$\Rightarrow \dfrac{2}{5}{R^2} = \dfrac{2}{3}{R'^2}$
From this we get
$\Rightarrow \dfrac{{{R^2}}}{{{{R'}^2}}} = \dfrac{5}{3}$
$\therefore \dfrac{R}{{R'}} = \sqrt {\dfrac{5}{3}}$
This is the ratio of their radii.

So, the correct answer is option C.

Note: Moment of inertia depends not only on mass but also on the distance at which particles are located with respect to the axis of rotation. The moment of inertia of a solid sphere will be different from the moment of a hollow sphere even if the mass of both spheres is the same because the distribution of mass is different.