Question:
The ratio of moment of inertia with respect to their diameters of a circular disc and a solid sphere having same radii and same masses is:
The ratio of moment of inertia with respect to their diameters of a circular disc and a solid sphere having same radii and same masses is:
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Double check the axis specified in the question.
For a disc, the moment of inertia about its perpendicular axis is $\frac{1}{2}MR^2$, but about its diameter it is $\frac{1}{4}MR^2$.
A common mistake is using the perpendicular axis formula by mistake.
For a disc, the moment of inertia about its perpendicular axis is $\frac{1}{2}MR^2$, but about its diameter it is $\frac{1}{4}MR^2$.
A common mistake is using the perpendicular axis formula by mistake.
Updated On: Jul 22, 2026
- $4:5$
- $5:4$
- $8:5$
- $5:8$
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the ratio of the moments of inertia about their diameters of two different rigid bodies (a circular disc and a solid sphere) of equal mass $M$ and radius $R$.
Step 2: Key Formula and Approach:
We will use the standard formulas for moments of inertia:
- Moment of inertia of a circular disc about its diametrical axis:
\[ I_{\text{disc}} = \frac{1}{4}MR^2 \] - Moment of inertia of a solid sphere about its diametrical axis:
\[ I_{\text{sphere}} = \frac{2}{5}MR^2 \]
Step 3: Detailed Explanation:
• Moment of inertia of the disc ($I_{\text{disc}}$):
The moment of inertia of a disc about its central perpendicular axis is $\frac{1}{2}MR^2$.
By the perpendicular axis theorem, the moment of inertia about any diameter is half of that:
\[ I_{\text{disc}} = \frac{1}{4}MR^2 \]
• Moment of inertia of the sphere ($I_{\text{sphere}}$):
A uniform solid sphere has a symmetric mass distribution, with its moment of inertia about any diameter being:
\[ I_{\text{sphere}} = \frac{2}{5}MR^2 \]
• Calculate the ratio:
\[ \frac{I_{\text{disc}}}{I_{\text{sphere}}} = \frac{\frac{1}{4}MR^2}{\frac{2}{5}MR^2} = \frac{1}{4} \times \frac{5}{2} = \frac{5}{8} \] \[ I_{\text{disc}} : I_{\text{sphere}} = 5 : 8 \]
Step 4: Final Answer:
The ratio of their moments of inertia is $5:8$, which corresponds to Option (D).
The question asks for the ratio of the moments of inertia about their diameters of two different rigid bodies (a circular disc and a solid sphere) of equal mass $M$ and radius $R$.
Step 2: Key Formula and Approach:
We will use the standard formulas for moments of inertia:
- Moment of inertia of a circular disc about its diametrical axis:
\[ I_{\text{disc}} = \frac{1}{4}MR^2 \] - Moment of inertia of a solid sphere about its diametrical axis:
\[ I_{\text{sphere}} = \frac{2}{5}MR^2 \]
Step 3: Detailed Explanation:
• Moment of inertia of the disc ($I_{\text{disc}}$):
The moment of inertia of a disc about its central perpendicular axis is $\frac{1}{2}MR^2$.
By the perpendicular axis theorem, the moment of inertia about any diameter is half of that:
\[ I_{\text{disc}} = \frac{1}{4}MR^2 \]
• Moment of inertia of the sphere ($I_{\text{sphere}}$):
A uniform solid sphere has a symmetric mass distribution, with its moment of inertia about any diameter being:
\[ I_{\text{sphere}} = \frac{2}{5}MR^2 \]
• Calculate the ratio:
\[ \frac{I_{\text{disc}}}{I_{\text{sphere}}} = \frac{\frac{1}{4}MR^2}{\frac{2}{5}MR^2} = \frac{1}{4} \times \frac{5}{2} = \frac{5}{8} \] \[ I_{\text{disc}} : I_{\text{sphere}} = 5 : 8 \]
Step 4: Final Answer:
The ratio of their moments of inertia is $5:8$, which corresponds to Option (D).
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