Question

Two discs having the same moment of inertia about their axis. Thickness is \( t_1 \) and \( t_2 \), and they have the same density. If \( \frac{R_1}{R_2} = \frac{1}{2} \), then find \( \frac{t_1}{t_2} \).

• A. \( \frac{1}{16} \)
• B. 16
• C. \( \frac{1}{4} \)
• D. 4

Hint:

For two discs with the same moment of inertia and density, the ratio of thicknesses is proportional to the square of the ratio of radii.

Answer 1

The Correct Option is B

Step 1: Moment of inertia for a disc.
The moment of inertia of a disc is given by the formula \( I = \frac{1}{2} M R^2 \), where \( M \) is the mass and \( R \) is the radius. Since both discs have the same moment of inertia, we set their moments equal to each other. Step 2: Relating the thickness and radius.
Since the discs have the same density, the mass of each disc is proportional to its volume. Therefore, \( M \propto t R^2 \). Given \( \frac{R_1}{R_2} = \frac{1}{2} \), we can solve for the ratio \( \frac{t_1}{t_2} \) based on the equality of moments of inertia. Step 3: Conclusion.
The ratio \( \frac{t_1}{t_2} \) simplifies to 16. Final Answer: \[ \boxed{16} \]