Step 1: Understanding the Concept:
The moment of inertia (MOI) of an object with a cavity can be calculated by subtracting the MOI of the removed part from the MOI of the original complete object.
However, we must be very careful with the mass distribution.
In this question, \(M\) represents the mass of the final object (the disc with the hole).
The axis of rotation passes through the geometric center and is perpendicular to the plane, so we use the formula for a disc: \(I = \frac{1}{2}mr^2\).
Step 2: Key Formula or Approach:
For an annular disc (a disc with a hole) of mass \(M\), inner radius \(r_i\) and outer radius \(r_o\), the moment of inertia about the central perpendicular axis is:
\[ I = \frac{1}{2} M (r_o^2 + r_i^2) \]
Alternatively, if \(M\) is the mass of the full disc *before* the hole was cut, we use the subtraction method: \(I = I_{full} - I_{removed}\).
Step 3: Detailed Explanation:
Let's analyze the phrasing. Usually, in such problems, \(M\) is the mass of the full disc before the hole is cut. Let's solve it both ways to see which matches the options.
**Method A: \(M\) is the mass of the original solid disc.**
Original Mass = \(M\), Original Radius = \(R\).
MOI of full disc \(I_1 = \frac{1}{2}MR^2\).
The hole has radius \(R/2\). Since mass is proportional to area (\(A = \pi r^2\)):
Mass of removed part \(m = M \times \frac{\pi(R/2)^2}{\pi R^2} = \frac{M}{4}\).
MOI of removed part \(I_2 = \frac{1}{2} m (R/2)^2 = \frac{1}{2} \left( \frac{M}{4} \right) \left( \frac{R^2}{4} \right) = \frac{MR^2}{32}\).
Net MOI \(I = I_1 - I_2 = \frac{1}{2}MR^2 - \frac{1}{32}MR^2\).
Using a common denominator of 32:
\[ I = \frac{16MR^2 - 1MR^2}{32} = \frac{15}{32}MR^2 \]
This matches Option (A).
**Method B: \(M\) is the mass of the final annular disc.**
If \(M\) were the mass of the actual object shown:
\(I = \frac{1}{2}M(R^2 + (R/2)^2) = \frac{1}{2}M(R^2 + R^2/4) = \frac{1}{2}M(\frac{5R^2}{4}) = \frac{5}{8}MR^2\).
Since \(\frac{5}{8} = \frac{20}{32}\), and this is not in the options, the question implies \(M\) is the mass of the original full disc.
Step 4: Final Answer:
By treating \(M\) as the initial mass and subtracting the moment of inertia of the smaller disc, we get \(\frac{15}{32}MR^2\).
Hence, the correct option is (A).