Two spheres of equal masses, one of which is a thin spherical shell and the other a solid, have the same moment of inertia about their respective diameters. The ratio of their radii will be

Question

Two spheres of equal masses, one of which is a thin spherical shell and the other a solid, have the same moment of inertia about their respective diameters. The ratio of their radii will be (A)  5: 7  (B)  3: 5  (C)  √3: √5  (D)  √3: √7

Tardigrade Admin · Accepted Answer

Let the radii of the thin spherical shell and the solid sphere are

R_{1}

and

R_{2}

respectively
Then the moment of inertia of the spherical shell about their diameter

I = \frac{2}{3} M R_{1}^{2}

......(i)
and the moment of inertia of the solid sphere is given by

I = \frac{2}{5} M R_{2}^{2}

......(ii)
It is given that the masses and moment of inertia for both the bodies are equal, then from Eqs. (i) and (ii)

\frac{2}{3} M R_{1}^{2} = \frac{2}{5} M R_{2}^{2}

\Rightarrow \frac{R _{1}^{2}}{R _{2}^{2}} = \frac{3}{5}

\Rightarrow \frac{R _{1}}{R _{2}} = \frac{3}{5}

\Rightarrow R_{1} : R_{2} = 3 : 5

Q. Two spheres of equal masses, one of which is a thin spherical shell and the other a solid, have the same moment of inertia about their respective diameters. The ratio of their radii will be

$5 : 7$

$3 : 5$

$3 : 5$

$3 : 7$

Q. Two spheres of equal masses, one of which is a thin spherical shell and the other a solid, have the same moment of inertia about their respective diameters. The ratio of their radii will be

5:7

3:5

3​:5​

3​:7​

$5 : 7$

$3 : 5$

$3 : 5$

$3 : 7$