Class 11 · Chapter 7
System of Particles & Rotational Motion
Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.
System of Particles & Rotational Motion Short Notes
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Condensed revision points for System of Particles & Rotational Motion — for quick recall before exams, not a substitute for the full notes.
Angular Kinematics
- θ in radians (dimensionless). ω = dθ/dt (rad/s). α = dω/dt (rad/s²).
- v = ωr, a_t = αr, a_r = ω²r — same particle, three different speeds/accelerations depending on what you're asking.
- ω = ω₀ + αt | θ = ω₀t + ½αt² | ω² = ω₀² + 2αθ — direct rotational twins of v = u+at etc.
- All particles of a rigid body share the same ω and α at a given instant; only v, a_t differ (since r differs).
Moment of Inertia — Core Idea
- I = Σmr² (discrete) or ∫r² dm (continuous). Rotational analogue of mass.
- Depends on: mass, mass distribution, axis chosen. Does NOT depend on ω, α, τ, or L.
- Same mass spread farther from axis → larger I.
- K (radius of gyration): I = MK². Depends on axis + mass distribution, not on total mass value.
Two Theorems
- Parallel axis (any body): I = I_CM + Md².
- Perpendicular axis (laminae/2-D bodies ONLY): I_z = I_x + I_y.
Standard Moments of Inertia
- Ring, ⊥ axis through centre: MR² | Ring, diameter: MR²/2
- Disc, ⊥ axis through centre: MR²/2 | Disc, diameter: MR²/4
- Solid sphere, diameter: (2/5)MR² | Hollow shell, diameter: (2/3)MR²
- Rod, ⊥ axis through centre: ML²/12 | Rod, ⊥ axis through end: ML²/3
- Solid cylinder, own axis: MR²/2 | Hollow cylinder, own axis: MR²
- Rectangular plate, ⊥ through centre: M(a²+b²)/12
Torque
- τ = r × F, magnitude rF sinθ. Unit: N·m (never write as joule).
- Only the force component ⊥ to r contributes — force along r or through the pivot gives zero torque.
- τ = Iα — rotational version of F = ma.
- Couple: two equal, opposite, non-collinear forces → moment = F × b (b = perpendicular distance between lines), zero net force.
Equilibrium
- Full equilibrium needs BOTH ΣF = 0 AND Στ = 0.
- A single force can't give rotational equilibrium unless its line of action passes through the axis.
- Cyclist bending on a turn: tanθ = v²/(rg).
Angular Momentum
- L = r × mv (particle) | L = Iω (rigid body).
- L is max when r ⊥ v (e.g. circular motion); L = 0 when v is along r (motion passes through the reference point).
- τ = dL/dt — rotational version of F = dp/dt.
- Angular impulse = τ·Δt = ΔL.
Conservation of Angular Momentum (COAM)
- τ_net = 0 ⇒ I₁ω₁ = I₂ω₂ (L stays constant).
- Arms in → I decreases → ω increases (skater, diver). Arms out → reverse.
- COAM conserves L, NOT rotational KE — KE_rot can change even while L stays fixed.
- Ice melting at poles → mass shifts toward equator → Earth's I increases → ω decreases → day lengthens.
Rotational Kinetic Energy
- KE_rot = ½Iω² = L²/(2I).
- W = τΔθ (constant torque) = ΔKE_rot — rotational work-energy theorem.
- Rotational power: P = τω.
Rolling Motion
- Pure rolling condition: v_cm = ωR (no slipping at contact point).
- Contact point velocity = 0. Top point velocity = 2v_cm. Centre velocity = v_cm.
- Total KE while rolling = ½mv² (1 + K²/R²).
- Pure rolling ⇒ equivalent to pure rotation about the instantaneous axis through the contact point.
Rolling on an Inclined Plane
- v at bottom = √[2gh / (1 + K²/R²)]
- a down the slope = g sinθ / (1 + K²/R²)
- t to reach bottom = (1/sinθ)·√[(2h/g)(1 + K²/R²)]
- Smaller K²/R² → reaches first (solid sphere fastest; ring/hollow cylinder slowest of standard shapes).
- Sliding (frictionless) body always beats a rolling body of the same shape down the same incline.
- μ_min for pure rolling on incline of angle θ: tanθ / (1 + R²/K²).
Common Exam Traps
- Perpendicular axis theorem fails for 3-D bodies (sphere, cylinder) — parallel axis theorem works for all.
- I is NOT a fixed number for a body — always state the axis.
- Torque's SI unit is N·m, never joule, even though dimensions match work.
- Don't confuse 'sliding' (K²/R² = 0, frictionless) with 'rolling' (K²/R² > 0, friction provides spin-up torque).
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