Question

A uniform circular disc of radius \( R \) and mass \( M \) is rotating about an axis perpendicular to its plane and passing through its center. A small circular part of radius \( R/2 \) is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

• A. \( \frac{7}{32} MR^2 \)
• B. \( \frac{9}{32} MR^2 \)
• C. \( \frac{17}{32} MR^2 \)
• D. \( \frac{13}{32} MR^2 \)

Hint:

To calculate the moment of inertia of the remaining part of a disc, subtract the moment of inertia of the removed portion from the original disc's moment of inertia.

Answer 1

The Correct Option is D

The moment of inertia of the original disc is: \[ I_{\text{original}} = \frac{MR^2}{2} \] \[ I = \frac{MR^2}{2} - \left[ \frac{\frac{M}{4} \left( \frac{R}{2} \right)^2}{2} + \frac{M}{4} \left( \frac{R}{2} \right)^2 \right] \]

Explanation: The first term, $\frac{MR^2}{2}$, denotes the moment of inertia of a solid disc about an axis perpendicular to it and passing through its center. The terms within the square brackets represent the moment of inertia of the removed portion of the disc. The expression $\frac{M}{4} \left( \frac{R}{2} \right)^2$ is associated with the mass and radius of the removed section. The division by 2 in the first term within the square brackets relates to the moment of inertia of the removed section about its own center. 

Step 1: Simplify terms within the square brackets \[ \frac{\frac{M}{4} \left( \frac{R}{2} \right)^2}{2} = \frac{M}{4} \cdot \frac{R^2}{4} \cdot \frac{1}{2} = \frac{MR^2}{32} \] \[ \frac{M}{4} \left( \frac{R}{2} \right)^2 = \frac{M}{4} \cdot \frac{R^2}{4} = \frac{MR^2}{16} \] 

Explanation: This step simplifies the fractions and squares inside the square brackets for easier manipulation. 

Step 2: Combine terms within the square brackets \[ \frac{MR^2}{32} + \frac{MR^2}{16} = \frac{MR^2}{32} + \frac{2MR^2}{32} = \frac{3MR^2}{32} \] Explanation: This step consolidates the simplified terms within the square brackets into a single fractional expression. 

Step 3: Substitute the combined term back into the original equation \[ I = \frac{MR^2}{2} - \frac{3MR^2}{32} \] 

Explanation: This step replaces the original terms within the square brackets with the result obtained in Step 2. 

Step 4: Find a common denominator and simplify \[ I = \frac{16MR^2}{32} - \frac{3MR^2}{32} = \frac{13MR^2}{32} \] 

Explanation: This step establishes a common denominator for the two fractions and performs the subtraction to derive the final expression for I. 

Final Result: \[ I = \frac{13}{32} MR^2 \]