Moment of Inertia of a Circle Explained for JEE & Boards

Moment of inertia of a circle is a fundamental concept in rotational motion, reflecting how mass is distributed about an axis passing through the circle. For JEE Main, understanding this topic is crucial to solving problems involving rotating bodies, solid discs, and circular rings.

In Physics, the moment of inertia (often denoted as I) is a measure of an object's resistance to changes in its rotational motion about an axis. For a circle, this depends not just on mass and radius but also on the axis chosen.

Consider a uniform, solid disc or circular lamina with mass M and radius R. To calculate its moment of inertia, we must specify the axis. The two most important axes are the centre (perpendicular to the plane) and the diameter (in-plane axis passing through the centre).

Moment of Inertia of a Circle: Key Formulas

For a solid disc or circle, the moment of inertia about its central axis (perpendicular to plane) is given by:
I = (1/2)MR², where M is mass and R is radius.

When the axis is along the diameter (lying in the plane), the formula becomes I = (1/4)MR². Both are essential variants for JEE questions involving rotation about diverse axes.

In summary, the formula changes depending on whether the axis is at the centroid, across the diameter, or another orientation. Always check the axis in exam questions.

Derivation and Axis Variations for Moment of Inertia of a Circle

Let’s derive the principal formula for a solid circle about its central axis. A small mass element at a radial distance r from the center experiences I = ∫r²dm. For a uniform disc:

Divide the disc into concentric rings of radius r and width dr. The mass of each ring is dm = (M/πR²)(2πr dr). Applying the definition:

I = ∫₀^R r² dm = ∫₀^R r² (M/πR²)(2πr dr) = (2M/R²) ∫₀^R r³ dr = (2M/R²) × (R⁴/4) = (1/2)MR²

For the diameter axis, apply the Perpendicular Axis Theorem which states: I_perp = I_x + I_y. Since the disc is symmetric, each diameter axis shares half of the perpendicular axis value.

Beware: choosing the wrong axis or missing mass distribution is a common trap in JEE questions. Always verify both the axis and whether it is a solid or hollow circle.

Hollow Circles, Rings, and Special Cases

For a thin ring or hollow circle, all mass is at radius R. The moment of inertia about the central axis (perpendicular to plane) is:
I = MR².

If you have an annular disc (circle with a hole, inner radius r, outer radius R), the moment of inertia is:
I = (1/2)M(R² + r²).

• For a circle with a hole, subtract the inertia of the missing part from the full disc.
• For a solid disc, never use ring’s formula; check “thick” or “thin” in the question.

To compare, the moment of inertia of a rectangle or square about centroidal axes is lower for similar dimensions. This distinction may appear in JEE Main conceptual MCQs or assertion-reason types.

Solved Example: Applying the Moment of Inertia of a Circle

Question: A solid disc of mass 2 kg and radius 0.4 m rotates about its central axis. What is its moment of inertia?

Solution: Using I = (1/2)MR², substitute M = 2 kg, R = 0.4 m:

I = (1/2) × 2 × (0.4)² = 1 × 0.16 = 0.16 kg·m²

• This formula is ONLY for a solid disc, axis through center perpendicular to plane.
• If the axis was along the diameter, answer would be (1/4)MR² = 0.08 kg·m².

Many rotational motion problems, like discs rolling or oscillating, require identifying the right formula for the situation. Mastering these calculations will help with topics like rotational motion of a rigid body.

Tricks, Applications, and Exam Tips on Moment of Inertia of a Circle

JEE Main often tests whether you know which axis to use. Never assume “central axis” unless stated. Pay special attention to units and the difference between area moment and mass moment of inertia.

• Always check “solid,” “hollow,” or “annular” in the problem statement.
• Apply the Perpendicular Axis Theorem for planar bodies as needed.
• Compare circular disc, ring, and rectangle quickly with their standard formulas.
• Relate examples to other rotational systems using the moment of inertia basics page.
• Review moment of inertia of a disc for similarities and differences with the circle.
• Try recent numericals from JEE question papers for practice.

Applications include flywheels, pulleys, gyroscopes, and in structural engineering. Problems may require combining moments of inertia using the parallel axis theorem or for composite bodies.

For further clarity, compare the values for a circle to those of a square and a cube. Knowing these basics aids with ready-reckoner style MCQs often seen in JEE Main.

Vedantu’s expert team ensures that all moment of inertia formulas provided here are verified for competitive exam accuracy and are fully aligned with the latest JEE Main syllabus. Keep revising with rotational motion mock tests to cement your understanding.