Moment of Inertia for JEE Main: Complete Guide

Moment of inertia is the rotational analogue of mass — it measures how resistant a body is to angular acceleration. JEE Main tests it through standard results, two key theorems, and composite-body questions that combine both. Mastering this topic pays dividends across the entire rotational dynamics chapter, which contributes four to six questions every year.

Standard Results You Must Memorise

For a uniform rod of mass M and length L: I about its centre = ML²/12; about one end = ML²/3. For a uniform disc of mass M and radius R: I about its central axis = MR²/2; about a diameter = MR²/4. For a ring of the same mass and radius: I about central axis = MR², about a diameter = MR²/2. Solid sphere: 2MR²/5. Hollow sphere: 2MR²/3. Solid cylinder about its axis: MR²/2. These eight values are the building blocks of every exam question. For the broader rotational dynamics context see our rotational motion guide.

The Parallel Axis Theorem

For any axis parallel to a centre-of-mass axis, I = I_cm + Md² where d is the perpendicular distance between the axes. This theorem is the key that unlocks every non-standard axis. To verify: the rod's end-axis value should be ML²/12 + M(L/2)² = ML²/12 + ML²/4 = ML²/3 — exactly matching the memorised result. Always find I_cm first; then shift using Md². The most common error is forgetting the Md² term when the problem specifies an edge or surface axis.

The Perpendicular Axis Theorem

For any flat lamina lying in the xy-plane: I_z = I_x + I_y. This theorem applies only to 2D objects but is extremely powerful for discs and rectangular plates. For a uniform disc: I about the central perpendicular axis = MR²/2, and by symmetry I about each diameter = MR²/4 (since both are equal and must sum to MR²/2). Recognising the laminar shape is the trigger — the theorem then gives any missing axis in one step.

Composite Bodies and the Subtraction Method

For a disc with a circular hole, treat the hole as a negative-mass disc: I_total = I_full_disc − I_removed_disc. Use the parallel axis theorem to shift each piece to the required axis before summing. The same logic applies to rods with attached discs, rings on rods, and any assembly of standard parts. Always identify every piece, find each piece's own I_cm, shift via Md², and sum. Take a free mock test to practise composite problems under timed conditions.

Exam Approach

On exam day: identify the axis, list all pieces, apply the parallel axis theorem to each if needed, and sum. Verify by sanity-checking limiting cases — as the hole radius approaches zero, the disc formula must recover. For the complete mechanics picture including torque, angular momentum, and rolling, see our mechanics master guide. Moment of inertia is one of the few topics where careful memorisation of eight results plus two theorems handles every question the exam can create.