Moment of Inertia
I formulas for 6 common shapes — solid/hollow sphere, cylinder, rod.
Key Notes
Moment of inertia I is the rotational analog of mass — measures resistance to angular acceleration.
For a point mass at distance r from axis: I = mr².
For a collection of particles: I = Σ m_i r_i². For continuous bodies: I = ∫r²·dm.
I depends on (i) mass distribution, (ii) chosen rotation AXIS — different axes give different I.
Common bodies (about symmetry axes): rod (about center, length L): ML²/12. Disk (about center): MR²/2. Solid sphere: 2MR²/5. Hollow sphere: 2MR²/3. Cylinder (about axis): MR²/2. Ring: MR².
Newton's 2nd law for rotation: τ = Iα.
Rotational KE: K = ½Iω². Higher I ⇒ more KE for same ω.
Engineering: flywheels store rotational energy via large I.
Formulas
Point mass
r = perpendicular distance from rotation axis.
Discrete system
Sum over all particles.
Continuous body
Integral form for extended objects.
Common bodies (about symmetry axis)
Standard values worth memorising.
Rotational Newton's 2nd law
Direct analog of F = ma.
Important Points
I depends on AXIS — not just mass. Same body, different axis ⇒ different I.
I is larger when mass is far from axis (r² weighting).
I for a ring (all mass at R) is double that of a disk (mass spread): both = MR² but disk = ½MR² because mass is closer in.
Flywheels: heavy and large radius ⇒ high I ⇒ stores lots of rotational KE.
I for a non-symmetric body about an arbitrary axis: use inertia tensor (advanced — not in NCERT).
Common mistake: using I = MR² for a disk (it's for a ring). Disks: I = ½MR².
Moment of Inertia notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.