Question

A uniform solid sphere of mass m, radius r, and having moment of inertia about an axis passing through center of mass is I, is recast into a uniform circular disc of thickness t, whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains I. Then the radius of the disc will be: _______

Answer 1

The radius of the disc recast from the solid sphere, maintaining the same moment of inertia I, will be R = r sqrt(4/15).

Step-by-step explanation:

To find the radius of the disc, we need to equate the moment of inertia of the recast disc to the original sphere.

For a solid sphere, the moment of inertia about an axis through its center of mass is I = 2/5 m r^2.

When recast into a disc with thickness t, assuming that the mass remains the same, the moment of inertia of the disc about an axis through its edge and perpendicular to its plane is I (given to be equal to the moment of inertia of the sphere).

The moment of inertia of a disc about an axis through its center is I = 1/2 m R^2, where R is the radius of the disc.

Using the parallel-axis theorem, the moment of inertia about an axis through its edge will be I = 1/2 m R^2 + m R^2 (since the distance from the center to the edge axis is R). Equating this to the original moment of inertia of the sphere, we get 3/2 m R^2 = 2/5 m r^2.

Simplifying, we find that R^2 = (4/15) r^2, hence the radius of the disc will be R = r sqrt(4/15).