Question

A thin circular ring and a circular disc have the same mass and moment of inertia about their centers perpendicular to the plane. 
Then the ratio of their radii is:

• A. \(\frac{1}{\sqrt{2}}\)
• B. \(\frac{1}{\sqrt{3}}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{2}{3}\)

Hint:

When comparing moments of inertia, remember that the mass distribution relative to the rotation axis greatly affects the inertia.

Answer 1

The Correct Option is A

We are given that the mass and the moment of inertia of a thin circular ring and a circular disc are the same about their centers perpendicular to the plane. We need to find the ratio of their radii. The moment of inertia \( I \) of a thin circular ring and a circular disc are given by the following formulas: - For a thin circular ring with radius \( R \) and mass \( M \), the moment of inertia is: \[ I_{{ring}} = M R^2 \] - For a circular disc with radius \( r \) and mass \( M \), the moment of inertia is: \[ I_{{disc}} = \frac{1}{2} M r^2 \] We are told that the two moments of inertia are equal, so: \[ I_{{ring}} = I_{{disc}} \] Substitute the expressions for the moments of inertia: \[ M R^2 = \frac{1}{2} M r^2 \] Canceling \( M \) from both sides: \[ R^2 = \frac{1}{2} r^2 \] Taking the square root of both sides: \[ R = \frac{r}{\sqrt{2}} \] Thus, the ratio of the radii is: \[ \frac{R}{r} = \frac{1}{\sqrt{2}} \] Conclusion: The ratio of the radii is \( \frac{1}{\sqrt{2}} \), so the correct answer is (1).