Moment of inertia of a uniform quarter disc of radius `R` and mass `M` about an axis through its centre of mass and perpendicular to its plane is `:`

Options

A. `(MR^2)/2 - M((4R)/(3pi))^2`
B. `(MR^2)/2 - M(sqrt(2)(4R)/(3pi))^2`
C. `(MR^2)/2 - M((4R)/(2pi))^2`
D. `(MR^2)/2 - M(sqrt(2)(4R)/(3pi))^2 9/(32 pi)`

Answer

To find the moment of inertia of a uniform quarter disc of radius \( R \) and mass \( M \) about an axis through its center of mass and perpendicular to its plane, we can follow these steps:

### Step 1: Understand the Moment of Inertia of a Full Disc
The moment of inertia \( I \) of a full uniform disc about an axis through its center and perpendicular to its plane is given by the formula:
\[ I = \frac{1}{2} M R^2 \]

### Step 2: Use the Parallel Axis Theorem
Since we are dealing with a quarter disc, we will use the parallel axis theorem to find the moment of inertia about the center of mass of the quarter disc. The parallel axis theorem states:
\[ I = I_{cm} + M d^2 \]
where \( I_{cm} \) is the moment of inertia about the center of mass, \( M \) is the mass, and \( d \) is the distance between the two axes.

### Step 3: Calculate the Moment of Inertia of the Quarter Disc
The moment of inertia of the quarter disc about its center of mass \( I_{cm} \) can be derived from the moment of inertia of the full disc. Since the quarter disc is one-fourth of the full disc, we have:
\[ I_{cm} = \frac{1}{4} \left( \frac{1}{2} M R^2 \right) = \frac{1}{8} M R^2 \]

### Step 4: Determine the Distance \( d \)
To find the distance \( d \) between the center of the full disc and the center of mass of the quarter disc, we can use the coordinates of the center of mass of the quarter disc. The center of mass of a quarter disc is located at:
\[ \left( \frac{4R}{3\pi}, \frac{4R}{3\pi} \right) \]
The distance \( d \) from the center of the full disc (which is at the origin) to the center of mass of the quarter disc can be calculated using the Pythagorean theorem:
\[ d = \sqrt{\left(\frac{4R}{3\pi}\right)^2 + \left(\frac{4R}{3\pi}\right)^2} = \frac{4R}{3\pi} \sqrt{2} \]

### Step 5: Substitute Values into the Parallel Axis Theorem
Now we can substitute \( I_{cm} \) and \( d \) into the parallel axis theorem:
\[ I = I_{cm} + M d^2 \]
Substituting the values:
\[ I = \frac{1}{8} M R^2 + M \left(\frac{4R}{3\pi} \sqrt{2}\right)^2 \]
Calculating \( d^2 \):
\[ d^2 = \left(\frac{4R}{3\pi} \sqrt{2}\right)^2 = \frac{32R^2}{9\pi^2} \]
Thus,
\[ I = \frac{1}{8} M R^2 + M \cdot \frac{32R^2}{9\pi^2} \]

### Step 6: Final Calculation
Now we can combine these terms:
\[ I = \frac{1}{8} M R^2 + \frac{32M R^2}{9\pi^2} \]
To find a common denominator and simplify, we can calculate the final moment of inertia.

### Conclusion
The moment of inertia of a uniform quarter disc of radius \( R \) and mass \( M \) about an axis through its center of mass and perpendicular to its plane is given by the final expression derived above.