Angular Kinematics
ω = ω₀ + αt, θ = ω₀t + ½αt² — see disk spin with live ω(t) and θ(t) plots.
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Class 11 · Notes
Rotational Motion— Notes, Formulas & Revision
Complete revision notes and formulas for Rotational Motion (Class 11). Curated for JEE, NEET, AP Physics, SAT, and CUET. Tap any topic to open the live simulation and full PYQ set.
Moment of Inertia
I formulas for 6 common shapes — solid/hollow sphere, cylinder, rod.
Notes coming soon — open the simulation for the live walkthrough.
Notes coming soon — open the simulation for the live walkthrough.
Parallel Axis Theorem
I = I_cm + Md² — shift the axis and see I grow quadratically with d.
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Rotational Kinetic Energy
KE_rot = ½Iω² — compare with translational KE for rolling objects.
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Conservation of Angular Momentum
L = Iω conserved — pull string shorter, ω grows as (r₀/r)².
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Rolling Motion
a = g sinθ/(1 + k²) — 4 shapes race down an incline; sphere wins.
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Linear ↔ Angular Variables
v = ωr, a_t = αr, a_c = ω²r — drag a particle on a rotating disc to see all three.
01
Each particle on a rotating rigid body has linear velocity v = ωr.
02
Tangential acceleration a_t = αr; centripetal a_c = ω²r = v²/r.
03
Different points on the same body share ω and α, but have different v and a_c.
Linear-angular
Connects circular and rotational kinematics.
Total acceleration = √(a_t² + a_c²).
On a wheel rolling without slipping, the topmost point moves at 2v relative to ground.
Lever Balance (Torque)
Two masses on a beam — adjust weights and arms; see torque balance live.
01
Lever balances when net torque about pivot is zero: m₁ g d₁ = m₂ g d₂.
02
Mechanical advantage = effort arm / load arm. First-class lever has pivot in the middle.
03
Most stable when COM is just above the pivot.
Balance
Torque balance about pivot.
Mechanical advantage
Force amplification.
Crowbar, scissors, seesaw — all first-class levers.
Wheelbarrow is a second-class lever (load between effort and pivot).
Radius of Gyration
k = √(I/M) — see equivalent ring radius for ring, disc, rod, sphere, shell.
01
k = √(I/M) is the distance from the axis at which all the mass, if concentrated as a thin ring, would give the same I.
02
Ring: k = R. Disc: k = R/√2. Solid sphere: k = R√(2/5). Hollow shell: k = R√(2/3). Rod about center: k = L/√12.
03
Useful in rolling motion: a = g sinθ / (1 + k²/R²).
Definition
Equivalent ring radius.
Rolling acceleration
Smaller k/R → faster rolling.
Hollow objects have larger k than solid ones of the same mass and radius.
k depends on the chosen axis — use parallel-axis theorem if shifted.
Rolling With Slipping
Independent v and ω — detect skidding (v > ωR) vs spinning (ωR > v) vs pure rolling.
01
Pure rolling: contact point is instantaneously at rest, so v = ωR.
02
Skidding: v > ωR (over-braking, locked wheel sliding).
03
Spinning: ωR > v (drive wheel spinning out, e.g. on ice).
04
Slipping speed at contact: v_slip = v − ωR.
Pure rolling condition
Contact point at rest.
Slip velocity
Relative speed at contact.
Friction acts to oppose slip: forward friction during spinning, backward during skidding.
Once pure rolling is reached on a flat surface, friction does no work.
Gyroscopic Precession
Spinning top — see L precess about vertical at Ω = mgr/(Iω). Real animatronic feel.
01
Spinning top with weight off-vertical: gravity creates torque τ perpendicular to L.
02
Instead of falling, the angular momentum vector precesses around the vertical.
03
Precession rate Ω = τ / L = mgr / (Iω) — slows as spin ω increases.
Precession rate
Independent of tilt for small angles.
Angular momentum
Magnitude (along spin axis).
Used in inertial navigation (gyroscopes), bicycles (stability via wheels' L), and toy tops.
Faster spin → slower precession; this is why a tossed coin doesn't tumble much.
Rotational Motion on sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs). Free physics revision for Class 11, JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT Subject Physics, and CUET-UG.