Rotational Motion— Notes, Formulas & Revision

Angular kinematics describes rotation: θ (angle), ω (angular velocity), α (angular acceleration).

Same form as linear kinematics with substitutions: x → θ, v → ω, a → α.

Units: θ (rad), ω (rad/s), α (rad/s²). 1 rev = 2π rad = 360°.

For uniform angular acceleration: ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ.

Linear-angular relations (for rigid body, distance r from axis): v = rω, a_tangential = rα, a_centripetal = rω² = v²/r.

Total acceleration of a point in circular motion: a = √(a_t² + a_c²).

Sign convention: counter-clockwise positive (by right-hand rule, ω points along rotation axis).

Used in: rotating machinery, wheels, planets, gyroscopes.

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.

Torque τ is the rotational analog of force — measures the tendency of a force to rotate a body about an axis.

τ = r × F. Magnitude: τ = r·F·sin θ = F·r_perp = r·F_perp.

Units: N·m (same as energy units, but conceptually different — vector vs scalar).

Direction: perpendicular to plane of r and F (right-hand rule).

Rotational Newton's 2nd law: τ_net = I·α.

Equilibrium: a body is in mechanical equilibrium when Σ F = 0 AND Σ τ = 0 (both translational and rotational).

Couple: two equal-magnitude, opposite-direction forces with different lines of action — produces pure rotation (net F = 0 but τ ≠ 0).

Lever, wrench, door hinge — all use torque. Long lever arm = more rotational effect for same force.

Parallel-axis theorem: I about any parallel axis = I_CM + M·d², where I_CM is the moment of inertia about the parallel axis through the center of mass, and d is the distance between the two axes.

Lets you find I about any axis if you know I_CM.

I about COM is always MINIMUM (parallel-axis adds a positive term).

Perpendicular-axis theorem (for PLANAR bodies only): I_z = I_x + I_y, where x, y are in the plane and z is perpendicular.

Used together: parallel + perpendicular axis theorems let you compute I for many standard shapes.

Example: rod about end: I_end = I_CM + M·(L/2)² = ML²/12 + ML²/4 = ML²/3.

Example: disk about edge: I_edge = ½MR² + MR² = (3/2)MR².

Theorem is universal — applies to any rigid body.

Rotational kinetic energy: K_rot = ½Iω². Direct analog of K_trans = ½mv².

Total KE of a rolling body = K_trans + K_rot = ½Mv² + ½Iω². With v = rω (rolling condition).

For a rolling solid sphere: K_total = ½Mv² + ½·(2MR²/5)·(v/R)² = (7/10)Mv². So 71% translation, 29% rotation.

Disk rolling: K_total = (3/4)Mv². Hollow sphere: (5/6)Mv². Ring: Mv² (50/50 split).

Energy stored in a flywheel: K = ½Iω². Large I and large ω ⇒ huge storage. Used in regenerative braking.

Work done by torque: W = ∫τ·dθ = ΔK_rot.

Power delivered to rotating body: P = τ·ω.

Rotation kinetic energy is real KE — has the same units (J) and converts to/from translational KE in rolling.

Conservation of angular momentum: if net external TORQUE on a system is zero, total L is conserved.

L = Iω. If I changes, ω compensates so that Iω stays constant.

Examples: figure skater pulls in arms → I drops → ω rises. Diver tucks → spins faster.

Holds about any FIXED axis or about the COM.

In the absence of external torque, an isolated rotating body keeps spinning forever.

Earth's day length is slowly INCREASING (~2.3 ms per century) because tidal friction transfers angular momentum to the Moon's orbit.

Astronomical examples: pulsars (collapsed stars) spin extremely fast because of conservation as r shrinks during collapse.

Conservation of L is INDEPENDENT of conservation of energy or linear momentum.

Rolling without slipping: a rotating body translates AND rotates, with v_CM = R·ω (rolling condition).

The point of contact with the ground is INSTANTANEOUSLY AT REST.

Total KE of a rolling body: K = ½Mv² + ½Iω². For rolling condition: K = ½Mv²(1 + I/MR²).

Rolling friction acts at contact point — small (compared to sliding friction) because the contact area doesn't slip.

Static friction at contact provides the torque for rolling — it does NO WORK (point of contact at rest).

On an incline: a = g·sin θ / (1 + I/MR²). Different bodies have different accelerations.

Solid sphere rolls fastest on an incline, hollow ball slower, then ring slowest — depends on I/MR² ratio.

Pure rolling: no energy lost to friction. Slipping: kinetic friction acts and energy is lost.

Each particle on a rotating rigid body has linear velocity v = ωr.

Tangential acceleration a_t = αr; centripetal a_c = ω²r = v²/r.

Different points on the same body share ω and α, but have different v and a_c.

Lever balances when net torque about pivot is zero: m₁ g d₁ = m₂ g d₂.

Mechanical advantage = effort arm / load arm. First-class lever has pivot in the middle.

Most stable when COM is just above the pivot.

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.

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.

Useful in rolling motion: a = g sinθ / (1 + k²/R²).

Pure rolling: contact point is instantaneously at rest, so v = ωR.

Skidding: v > ωR (over-braking, locked wheel sliding).

Spinning: ωR > v (drive wheel spinning out, e.g. on ice).

Slipping speed at contact: v_slip = v − ωR.

Spinning top with weight off-vertical: gravity creates torque τ perpendicular to L.

Instead of falling, the angular momentum vector precesses around the vertical.

Precession rate Ω = τ / L = mgr / (Iω) — slows as spin ω increases.