The coefficient swap between solid and hollow bodies is the trap that costs marks on moment of inertia (MOI) questions. A solid sphere uses 2/5, a hollow sphere uses 2/3, a disc uses 1/2, and a ring uses 1. The difference comes from how mass is distributed relative to the rotation axis, and mixing these coefficients is a common source of negative marking.
What moment of inertia is. MOI quantifies a rigid body's resistance to angular acceleration about a given axis — the rotational analogue of mass (NCERT Class 11 Physics, Chapter 7, page 8). For a system of point particles: I = Σ mᵢrᵢ², where rᵢ is the perpendicular distance of mass mᵢ from the axis.
Standard geometries. For uniform rigid bodies rotating about their symmetry axis:
The physical logic: a hollow body has all its mass at the maximum distance from the axis, so its MOI is always larger than the corresponding solid body of the same M and R.
Shifting axes — two theorems. When the rotation axis is not the symmetry axis:
Watch out. NEET questions frequently ask you to compare MOI ratios of two geometries (solid sphere vs hollow sphere, disc vs ring) or to apply the parallel axes theorem to shift a known symmetry-axis MOI to an edge or tangent. The coefficient swap trap — using 2/5 where 2/3 is needed, or vice versa — is a high-frequency error. Before substituting, confirm the geometry word-for-word from the stem.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
The SI unit of moment of inertia is:
Moment of inertia of a rigid body depends on:
The moment of inertia of a uniform solid sphere about an axis through its centre is (2/5)MR². What is the MOI of a uniform hollow sphere of the same mass and radius about a diameter?
A uniform disc has moment of inertia I_cm = (1/2)MR² about an axis through its centre and perpendicular to its plane. Using the parallel axes theorem, the MOI about a tangent axis parallel to this central axis is:
A uniform ring has mass M and radius R. Its moment of inertia about a diameter (in-plane axis through centre) is:
The ratio of moments of inertia of a solid sphere to a hollow sphere of the same mass and radius, each about a diameter, is:
A thin uniform rod of mass M and length L has MOI = (1/12)ML² about an axis through its centre perpendicular to its length. The MOI about a parallel axis through one end is:
Two solid spheres A and B have the same mass M. Sphere A has radius R and sphere B has radius 2R. The ratio I_A : I_B about their respective diameters is:
Pattern: MOI geometry ratio (based on pattern NEET pattern: moi geometry ratio, observed in NEET 2022 and 2023)
Given
- Solid sphere: mass M, radius R - Disc: mass M, radius R - Both rotate about their symmetry axes
Required
Ratio I_sphere : I_disc
Concept
Each standard uniform rigid body has a tabulated MOI about its symmetry axis. The coefficients differ because mass is distributed differently relative to the axis.
Formula
- Solid sphere: I_sphere = (2/5)MR² - Disc: I_disc = (1/2)MR²
Substitution
Ratio = I_sphere / I_disc = [(2/5)MR²] / [(1/2)MR²]
Calculation
MR² cancels: Ratio = (2/5) ÷ (1/2) = (2/5) × (2/1) = 4/5 Note: the coefficients 2, 5, and 1, 2 are exact mathematical fractions from integration over uniform mass distributions. They do not limit significant figures.
Final answer
I_sphere : I_disc = 4 : 5 The solid sphere has a slightly smaller MOI than the disc of the same mass and radius, because the sphere distributes mass in three dimensions (some mass lies closer to the axis), while the disc distributes mass in a plane.
Common trap
Swapping the coefficients (using 2/3 for the solid sphere instead of 2/5) would give 4/3 instead of 4/5 — a ratio greater than 1, which should immediately signal an error since the solid sphere must have less rotational inertia than a flat disc of the same M and R.
Similar NEET-style question
A ring and a solid cylinder have equal mass and equal radius. What is the ratio of their MOI about the symmetry axis? (Answer: I_ring : I_cylinder = MR² : (1/2)MR² = 2 : 1.) ---
Moment of inertia of a rigid body about an axis: I = Σ m_i r_i² (sum over all particles). For continuous distribution: I = ∫ r² dm. Plays the role of mass in rotational equations: τ = I α.
-- NCERT Class 11 Physics, Ch. 6, p. 8For a particle: L = r x p. For a rigid body about its rotation axis: L = I omega. Vector quantity.
| Symbol | Quantity | SI Unit |
|---|---|---|
| L | angular momentum | kg*m^2/s |
| I | moment of inertia | kg*m^2 |
| omega | angular velocity | rad/s |
The position of the centre of mass equals the mass-weighted average of particle positions. For continuous bodies use integral form.
| Symbol | Quantity | SI Unit |
|---|---|---|
| R_cm | CoM position | m |
| m_i | mass of i-th particle | kg |
| r_i | position of i-th particle | m |
Standard moments of inertia about the symmetry axis. For other axes use parallel/perpendicular axes theorems.
| Symbol | Quantity | SI Unit |
|---|---|---|
| M | mass | kg |
| R | radius | m |
| L | length | m |
| I | moment of inertia | kg*m^2 |
Moment of inertia about any axis = moment about parallel axis through CM + Md^2.
| Symbol | Quantity | SI Unit |
|---|---|---|
| I | MOI about given axis | kg*m^2 |
| I_cm | MOI about parallel CM axis | kg*m^2 |
| M | total mass | kg |
| d | perpendicular distance | m |
For planar lamina: MOI about axis perpendicular to plane = sum of MOI about two perpendicular in-plane axes through same point.
| Symbol | Quantity | SI Unit |
|---|---|---|
| I_z | MOI perp to plane | kg*m^2 |
| I_x, I_y | MOI in plane | kg*m^2 |
Rotational analogues of linear kinematic equations under constant angular acceleration.
| Symbol | Quantity | SI Unit |
|---|---|---|
| omega | angular velocity | rad/s |
| alpha | angular acceleration | rad/s^2 |
| theta | angular displacement | rad |
| t | time | s |
Energy of rotation about an axis. Adds to translational KE for rolling bodies.
| Symbol | Quantity | SI Unit |
|---|---|---|
| I | moment of inertia | kg*m^2 |
| omega | angular velocity | rad/s |
Cross product of position vector and force vector. Magnitude r F sin(theta).
| Symbol | Quantity | SI Unit |
|---|---|---|
| tau | torque | N*m |
| r | position from pivot | m |
| F | force | N |
| theta | angle between r and F | rad |
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Overthinking
Student answers L/2 for two-particle CoM regardless of mass ratio.
Question gives two masses on rigid rod and asks for CoM distance.
R_cm from m1 = m2*L/(m1+m2). Heavier mass pulls CoM closer to it.
Category: Similar Terms
Student conserves rotational KE when angular momentum is conserved (or vice versa). When I changes, L = Iω is conserved but KE = ½Iω² is NOT (it depends on I and ω together).
Question describes a body whose moment of inertia changes (skater pulling arms in, star collapsing).
L conservation requires zero external torque. KE conservation requires no work done — different criteria. When I changes via internal forces, L conserved, ω increases, KE increases.
Category: Similar Terms
Student confuses 2/5 (solid sphere) with 2/3 (hollow sphere) or 1/2 (disc) with 1 (ring).
Question gives a specific geometry and asks for I or radius of gyration.
Memorise: solid sphere 2/5, hollow sphere 2/3, disc/cylinder 1/2, ring/hoop 1, rod-centre 1/12, rod-end 1/3.
Category: Unit Conversion
Student plugs rpm directly into formulas requiring rad/s. 1 rpm = 2π/60 rad/s.
Question gives ω in rpm and asks for kinematic quantities in SI units.
Convert: ω(rad/s) = (2π/60) × rpm. Always check units before substituting.
Root cause: concept gap
L = Iω is conserved when external torque is zero. KE = ½Iω² is NOT conserved when I changes (since ω changes too). When skater pulls arms in, L conserved, ω increases, KE increases (work done by muscles).
Root cause: concept gap
Memorise standard moments of inertia. Solid sphere has more mass near axis (smaller MOI = 2MR²/5); hollow sphere has all mass at radius R (larger MOI = 2MR²/3).
Root cause: unit error
Convert: ω(rad/s) = (2π/60) × ω(rpm). For example, 1200 rpm = 1200 × 2π/60 = 125.66 rad/s.
The angular acceleration of a body, moving along the circumference of a circle, is
forgets conversion rpm to rad s
Treats rpm as rad/s without 2*pi/60 conversion
uses equal distribution
Default to L/2 regardless of mass ratio
uses energy conservation instead
Confusing L conservation with KE conservation
swap solid hollow coefficients
Confusing 2/5 (solid sphere) with 2/3 (hollow sphere)
Test yourself on this topic with real past-paper questions:
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