(A) : Two circular discs of equal masses and thickness made of different material, will have same moment of inertia about their central axes of rotation.
(R ) : Moment of inertia depends upon the distribution of mass in the body.

Options

A. Both (A) and (R ) are true and (R ) is the correct explanation of (A)
B. Both (A) and (R ) are true and (R ) is not the correct explanation of (A)
C. (A) is true but (R ) is false
D. Both (A) and (R ) are false

Answer

To solve the question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step 1: Understanding the Assertion (A)
The assertion states that two circular discs of equal masses and thickness made of different materials will have the same moment of inertia about their central axes of rotation.

- **Analysis**:
- Moment of inertia (I) for a disc about its central axis is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
- Here, \(m\) is the mass and \(r\) is the radius of the disc.
- Since the discs have equal masses (m) but are made of different materials, their densities will differ. Given that density (\(\rho\)) is defined as mass per unit volume, if the thickness is constant, the radius must vary to maintain the same mass.

### Step 2: Relating Mass, Density, and Radius
- For a disc, the volume \(V\) can be expressed as:
\[ V = \pi r^2 t \]
- Therefore, the density can be expressed as:
\[ \rho = \frac{m}{\pi r^2 t} \]
- Since the thickness (t) is the same for both discs, and their masses are equal, the radii of the discs must differ to satisfy the density condition.

### Step 3: Moment of Inertia Calculation
- For disc 1 with radius \(r_1\):
\[ I_1 = \frac{1}{2} m r_1^2 \]
- For disc 2 with radius \(r_2\):
\[ I_2 = \frac{1}{2} m r_2^2 \]
- Since \(r_1\) and \(r_2\) are different (as derived from the density relationship), it follows that:
\[ I_1 \neq I_2 \]
- Thus, the assertion that both discs have the same moment of inertia is **false**.

### Step 4: Understanding the Reason (R)
The reason states that the moment of inertia depends upon the distribution of mass in the body.

- **Analysis**:
- Moment of inertia indeed depends on how mass is distributed relative to the axis of rotation. It is not just about the total mass but how far that mass is from the axis.
- The formula for moment of inertia shows that it is directly related to the square of the distance from the axis of rotation.

### Conclusion
- Both the assertion (A) and the reason (R) are false. Therefore, the correct answer is that both the assertion and reason are false.

### Final Answer
Both assertion and reason are false.

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