Step 1: Understanding the Concept:
The moment of inertia (\(I\)) of a uniform solid disc rotating about its central perpendicular axis depends on its mass (\(M\)) and its radius (\(r\)).
If discs are made of the same material and have the same thickness, their masses are not constant; rather, mass depends on the volume, which in turn depends on the radius.
Step 2: Key Formula or Approach:
The standard formula for the moment of inertia of a disc is:
\[ I = \frac{1}{2} M r^2 \]
The mass \(M\) can be expressed in terms of density (\(\rho\)) and volume (\(V\)):
\[ M = \rho \times V \]
For a disc with radius \(r\) and uniform thickness \(t\), the volume is \(V = \pi r^2 t\).
Therefore, mass becomes:
\[ M = \rho (\pi r^2 t) \]
Step 3: Detailed Explanation:
Substitute the expression for mass into the moment of inertia formula:
\[ I = \frac{1}{2} (\rho \pi r^2 t) r^2 = \frac{1}{2} \rho \pi t r^4 \]
Since both discs A and B are made of the same material, they have the same density (\(\rho\)).
They are also given to have the same thickness (\(t\)).
Therefore, the entire term \((\frac{1}{2} \rho \pi t)\) is a constant for both discs.
This establishes a direct proportionality between the moment of inertia and the fourth power of the radius:
\[ I \propto r^4 \]
We are given the radii of the two discs: \(r_A = R\) and \(r_B = 3R\).
Now, we set up the ratio of their moments of inertia:
\[ \frac{I_A}{I_B} = \frac{r_A^4}{r_B^4} = \left( \frac{r_A}{r_B} \right)^4 \]
Substituting the given radii:
\[ \frac{I_A}{I_B} = \left( \frac{R}{3R} \right)^4 = \left( \frac{1}{3} \right)^4 = \frac{1}{81} \]
Step 4: Final Answer:
The ratio of their moments of inertia is \(1 : 81\).