Physics Q28 – JEE-MAIN 2025

Home Physics Rotational MotionQuestion #447

The moment of inertia of a circular ring of mass

M

and diameter

r

about a tangential axis lying in the plane of the ring is :

Clue 1 of 3: Moment of Inertia Basics

Recall the moment of inertia of a circular ring about an axis passing through its center and perpendicular to its plane, and then about a diameter.

🥷

Ninja Strategy•Careful Variable Interpretation

The key to quickly eliminating options is to correctly interpret 'r' as the diameter, not the radius, and to distinguish between moment of inertia about a diameter versus a tangential axis.

🧠 Full Solution Path

Step 1: Moment of Inertia about a Diameter✦ Active

For a circular ring of mass $M$ and radius $R$ , the moment of inertia about an axis passing through its center and perpendicular to its plane is $I_{C M, ⊥} = M R^{2}$ . By the perpendicular axis theorem, the moment of inertia about a diameter (an axis lying in the plane of the ring and passing through its center) is $I_{d i a m e t e r} = \frac{1}{2} M R^{2}$ .

Step 2: Apply Parallel Axis Theorem○ Expand

The tangential axis lying in the plane of the ring is parallel to a diameter and is at a distance $R$ from it. Using the parallel axis theorem, $I_{t a n g e n t i a l} = I_{d i a m e t e r} + M R^{2}$ .

I_{t a n g e n t i a l} = \frac{1}{2} M R^{2} + M R^{2} = \frac{3}{2} M R^{2}

💡 Teacher's Secret Hint

Remember that the distance 'd' in the parallel axis theorem is the radius 'R' in this case.

Step 3: Substitute Radius in terms of Diameter○ Expand

The problem states that the diameter is $r$ . Therefore, the radius of the ring is $R = \frac{r}{2}$ . Substitute this into the expression for $I_{t a n g e n t i a l}$ .

I_{t a n g e n t i a l} = \frac{3}{2} M {(\frac{r}{2})}^{2} = \frac{3}{2} M \frac{r^{2}}{4} = \frac{3}{8} M r^{2}

💡 Teacher's Secret Hint

A common mistake is to assume 'r' is the radius. Always double-check the definition of variables in the problem statement.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Physics Question 28 – JEE-MAIN 2025

Clue 1 of 3: Moment of Inertia Basics

🧠 Full Solution Path