Complete Solutions and Summary of Systems of Particles and Rotational Motion – NCERT Class 11, Physics, Chapter 6 – Summary, Questions, Answers, Extra Questions
Summary of center of mass, motion of center of mass, angular velocity, torque, angular momentum, moment of inertia, equilibrium, and rotational dynamics with solved NCERT problems.
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Categories: NCERT, Class XI, Physics, Summary, Systems of Particles, Rotational Motion, Torque, Angular Momentum, Moment of Inertia, Chapter 6
Tags: Center of Mass, Rigid Body, Torque, Angular Momentum, Moment of Inertia, Rotational Motion, Equilibrium, Vector Product, Mechanical Equilibrium, NCERT, Class 11, Physics, Chapter 6, Answers, Extra Questions
Systems of Particles and Rotational Motion Class 11 NCERT Chapter 6 - Ultimate Study Guide, Notes, Questions, Quiz 2025
Systems of Particles and Rotational Motion
Chapter 6: Physics - Ultimate Study Guide | NCERT Class 11 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Systems of Particles and Rotational Motion Class 11 NCERT
Overview & Key Concepts
Chapter Goal: Extends single-particle motion (Ch. 2-5) to systems of particles and rigid bodies. Focus: Centre of mass (CM) motion, rigid body types (translational/rotational), vector products, torque, angular momentum, equilibrium, moment of inertia, rotational kinematics/dynamics. 2025 Reprint: Emphasizes CM for extended bodies, rigid approximations. Fun Fact: Rigid body idealization from Euler (18th C); CM concept from Newton. Core Idea: System motion reduces to CM translation + rotation about CM. Real-World: Satellite stability (rotation), vehicle crashes (CM momentum). Ties: Builds on vectors (Ch. 4), leads to gravitation (Ch. 8).
Wider Scope: Foundation for fluid dynamics (Ch. 13), oscillations (Ch. 14); applications in robotics (inverse kinematics), astrophysics (galactic rotation).
6.1 Introduction
Shifts from point particles to extended bodies (finite size). Single-particle model inadequate for deformations, rotations. Key: Treat as particle systems; CM key for overall motion. Rigid body: Fixed inter-particle distances (ideal; real deform negligibly, e.g., wheels, planets). Depth: No real rigid (deform under force), but approx valid if strains small (<10^{-3}). Historical: Galileo/Newton extended to rigid (Principia). Real-Life: Bridge design ignores minor flex. Exam Tip: Distinguish particle (no size) vs extended (CM + rotation). Extended: Multibody dynamics in simulations (e.g., FEM software). Links: Ch. 5 forces on systems; calculus for variable mass (rockets).
Examples: Bullet (particle approx), spinning top (rigid rotation).
Point Object Limit: Size << path (e.g., Earth orbit treats as point).
Rigid: No shape change. Pure translation: All points same velocity (e.g., block slide, Fig. 6.1; v uniform). Rolling: Translation + rotation (cylinder, Fig. 6.2; contact point v=0 no slip). Pure rotation: Fixed axis (fan, wheel, Fig. 6.3; points circle ⊥ axis, Fig. 6.4). General: Axis moving (top precession, Fig. 6.5a; fan oscillation, Fig. 6.5b; fixed point, not line). Chapter focus: Fixed axis rotation. Depth: Rotation: r ⊥ axis, v=ω r, circles in ⊥ planes. Real-Life: Gyroscopes (precession). Exam Tip: Translation: OP orientation constant (α1=α2=α3, Fig. 6.6a); combined: varies (Fig. 6.6b). Extended: 6 DOF rigid (3 trans + 3 rot); constraints reduce (e.g., hinge 1 rot). Ties: Ch. 4 vectors for v description.
Recap: Unconstrained: Trans + rot; constrained: Pure rot about axis/point.
Examples: Book slide (trans), roll (combined), pivot (rot).
Extended: Euler angles for 3D rot; non-holonomic constraints (rolling no slip). Graphs: Velocity fields (arrows uniform trans, radial rot).
6.2 Centre of Mass
System "average" position: Weighted by mass. 1D two particles: \(X = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}\) (Eq. 6.1; midpoint if equal m). n particles line: \(X_{cm} = \frac{\sum m_i x_i}{\sum m_i}\) (Eq. 6.2). Plane 3 particles: \(X = \frac{\sum m_i x_i}{M}\), \(Y = \frac{\sum m_i y_i}{M}\) (Eq. 6.3); centroid if equal m. General 3D: \(\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{M}\) (Eq. 6.4). Continua: \(\vec{R}_{cm} = \frac{1}{M} \int \vec{r} dm\) (Eq. 6.5; dm=ρ dV). Origin at CM: \(\int \vec{r} dm = 0\) (Eq. 6.6). Homogeneous symmetric: Geometric center (rod Fig. 6.8; reflection symmetry pairs dm at ±x cancel). Depth: M=total mass; index i=1 to n. Real-Life: Balance point (see-saw). Exam Tip: Vector sum economy. Extended: Variable mass (dM/dt≠0, rockets Ch. 8). Ties: Ch. 4 position vectors.
Examples: Equal m triangle: Centroid. L-shape: Weighted squares (Ex. 6.3).
Symmetry: Rings/discs/spheres CM at center (pair dm at (x,y,z) & (-x,-y,-z)).
Extended: Barycenter (solar system CM); computation: Monte Carlo integration large n. Pitfalls: Non-uniform (calculate explicitly). Graphs: CM trajectory straight if no ext F.
6.3 Motion of Centre of Mass
Differentiate Eq. 6.4d: \(\vec{V}_{cm} = \frac{\sum m_i \vec{v}_i}{M}\) (Eq. 6.8; masses const). Again: \(\vec{A}_{cm} = \frac{\sum m_i \vec{a}_i}{M}\) (Eq. 6.9). Newton's 2nd: \(\sum \vec{F}_i = m_i \vec{a}_i\) → \(M \vec{A}_{cm} = \sum \vec{F}_{ext}\) (Eq. 6.10/11; internal cancel by 3rd law pairs). Depth: CM moves as point mass M at CM with ext F at CM. Independent of internals. Real-Life: Recoil (gun CM conserved if isolated). Exam Tip: Only ext F matter; no internal knowledge needed. Extended: Isolated: \(\vec{V}_{cm}\) const (momentum cons); variable mass adjust. Ties: Ch. 5 linear momentum \(\vec{P} = M \vec{V}_{cm}\).
Examples: System particles internal collisions: CM uniform. Rain falling: CM downward.
Procedure: Treat extended as point at CM for translation.
Extended: Inertial frame; non-inertial fictitious F on CM. Applications: CM in collisions (elastic/inelastic). Graphs: CM parabolic under g.
6.4 Linear Momentum of a System of Particles
Total \(\vec{P} = \sum \vec{p}_i = \sum m_i \vec{v}_i = M \vec{V}_{cm}\). Rate: \(\frac{d\vec{P}}{dt} = \sum \vec{F}_{ext} = M \vec{A}_{cm}\). Depth: System momentum = CM momentum. Real-Life: Conservation if \(\sum \vec{F}_{ext}=0\) (isolated). Exam Tip: Internal no net P change. Extended: Relativistic P=γ m v. Ties: Ch. 5 single particle special.
Examples: Exploding bomb: CM continues uniform.
Extended: Center of momentum frame (V_cm=0). Applications: Particle physics collisions.
Summary
Extended bodies: Systems; CM \(\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{M}\); motion \(M \vec{A}_{cm} = \vec{F}_{ext}\); rigid: Trans + rot about fixed axis.
Why This Guide Stands Out
Complete: 12+ subtopics detailed (3+ pages equiv.), examples solved (3+), Q&A exam-style, 30 numericals. Physics-focused with eqs/figs/derivations. Free for 2025.
\(\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{M}\). Relevance: System "average". Deriv: Weighted mean. Depth: 3D vector. Limitations: Uniform density geometric. Applications: Stability (vehicles). Ex: Rod midpoint.
Pure Translational Motion
All points same v. Relevance: Like particle. Deriv: v_i = v_cm ∀i. Depth: OP orientation const. Applications: Sliding block. Ex: Fig. 6.1 uniform arrows.
Pure Rotational Motion
About fixed axis; points circle ⊥ axis. Relevance: Pivoted. Deriv: v=ω r ⊥. Depth: r=0 on axis stationary. Applications: Fan. Ex: Fig. 6.4 circles C1/C2.
Axis of Rotation
Fixed line through which body rotates. Relevance: Simplifies. Deriv: ⊥ planes. Depth: Instantaneous (general). Applications: Wheel axle. Ex: z-axis Fig. 6.4.
Precession
Axis movement around vertical (top). Relevance: Gyro. Deriv: Torque induced. Depth: Cone sweep. Applications: Spinning tops. Ex: Fig. 6.5a tip O fixed.
Homogeneous Body
Uniform density ρ. Relevance: Symmetry CM. Deriv: ∫ r dm /M=0 at center. Depth: Reflection pairs cancel. Applications: Sphere gravity. Ex: Thin rod Fig. 6.8.
Centroid
Geometric center (equal m). Relevance: Homogeneous CM. Deriv: Average coords. Depth: Triangle medians intersect. Applications: Lamina balance. Ex: Ex. 6.2 G.
Velocity of CM
\(\vec{V}_{cm} = \frac{\sum m_i \vec{v}_i}{M}\). Relevance: System v. Deriv: dR/dt. Depth: Weighted avg. Applications: Collisions. Ex: Uniform if trans.
Acceleration of CM
\(\vec{A}_{cm} = \frac{\sum m_i \vec{a}_i}{M}\). Relevance: Net F/M. Deriv: dV/dt. Depth: Ext only. Applications: Projectiles. Ex: g for falling.
Linear Momentum of System
\(\vec{P} = M \vec{V}_{cm}\). Relevance: Conserved isolated. Deriv: Sum p_i. Depth: dP/dt = F_ext. Applications: Rockets. Ex: Bomb explosion CM const.
External Forces
F from outside system. Relevance: CM motion. Deriv: Internal cancel. Depth: 3rd law pairs. Applications: Friction ext. Ex: Gravity on rain.
Internal Forces
Between particles. Relevance: No net on CM. Deriv: Equal opp pairs. Depth: Electromagnetic etc. Applications: Molecular bonds. Ex: Tension in rod.
Reflection Symmetry
Mirror image self. Relevance: CM at center. Deriv: Pairs ±x cancel ∫x dm=0. Depth: Even functions. Applications: Symmetric loads. Ex: Rod Fig. 6.8.
Continuous Mass Distribution
dm=ρ dV; n→∞. Relevance: Real bodies. Deriv: Sum → ∫. Depth: Density ρ=m/V. Applications: Fluids. Ex: Eq. 6.5 integrals.
Equilateral Triangle CM
Weighted vertices. Relevance: Non-uniform. Deriv: Sum m x_i /M. Depth: Not centroid. Applications: Loaded structures. Ex: Ex. 6.1 (0.25, 0.144 m).
Triangular Lamina CM
At centroid (medians concur). Relevance: Uniform. Deriv: Symmetry strips. Depth: 2/3 median. Applications: Plates. Ex: Ex. 6.2 G.
L-Shaped Lamina CM
Weighted squares centers. Relevance: Composite. Deriv: Sum m r_i /M. Depth: Along symmetry line. Applications: Brackets. Ex: Ex. 6.3 (5/6, 5/6 m).
Median
Vertex to midpoint. Relevance: CM on in triangle. Deriv: Symmetry. Depth: Concurr at 2/3. Applications: Load distrib. Ex: Fig. 6.10 LP/MQ/NR.
Tip: Memorize: CM ∫ r dm/M; rigid fixed distances. Depth: All [L] position. Applications: Aerospace CM shift fuel. Errors: Forget /M. Historical: Huygens CM. Interlinks: Ch. 4 vectors; Ch. 7 work rot. Advanced: Tensor inertia. Real-Life: Human CM walking. Graphs: CM paths. Symbols: Bold R_cm, sigma sum. Coherent SI. Extended: Non-inertial CM accel.
Additional: Translation: Same v all; Rotation: ω common. Continua: dV=dx dy dz. Symmetry: Odd functions zero. Examples: Sphere ∫=0 center. Pitfalls: Variable density calc explicit. 3D: +Z sum m z_i.
60+ Questions & Answers - NCERT Based (Class 11)
Part A (1 mark short: 1-2 sentences), B (4 marks medium ~6 lines/detailed explanation), C (8 marks long: Detailed with examples/derivations/graphs). Based directly on NCERT Exercises 6.1-6.31 (inferred from content). Theoretical focus; numericals separate.
Part A: 1 Mark Questions (Short Answers - From NCERT Exercises)
6.1(a) Rigid body pure translation?
1 Mark Answer: All points same v.
6.1(b) Rolling pure translation?
1 Mark Answer: No, + rotation.
6.1(c) Rotation fixed axis points?
1 Mark Answer: Circles ⊥ axis.
6.1(d) Precession in top?
1 Mark Answer: Axis sweeps cone.
6.2(a) CM two equal m?
1 Mark Answer: Midpoint.
6.2(b) Homogeneous rod CM?
1 Mark Answer: Geometric center.
6.2(c) Continua CM formula?
1 Mark Answer: ∫ r dm / M.
6.2(d) Symmetry cancel in integral?
1 Mark Answer: Yes, ± pairs.
6.3(a) CM motion depends on?
1 Mark Answer: Ext F only.
6.3(b) Isolated system V_cm?
1 Mark Answer: Constant.
6.3(c) Internal F on CM?
1 Mark Answer: Zero net.
6.3(d) A_cm = ?
1 Mark Answer: F_ext / M.
6.4(a) P system = ?
1 Mark Answer: M V_cm.
6.4(b) dP/dt = ?
1 Mark Answer: F_ext.
6.4(c) Conserved if isolated?
1 Mark Answer: Yes.
6.4(d) Internal change P?
1 Mark Answer: No.
6.30 Vector product def?
1 Mark Answer: Mag AB sinθ, ⊥ plane.
6.31 Angular v relation linear?
1 Mark Answer: v = ω r ⊥.
Part B: 4 Marks Questions (Medium Length ~6 Lines - From NCERT)
6.1 Full: Rigid motion types.
4 Marks Answer: (a) Yes uniform v. (b) No trans+rot. (c) Circles r ⊥. (d) Cone sweep fixed point. Translation: Same v; rotation fixed axis/line. Combined rolling. Graphical Figs 6.1-6.6.
6.2 Full: CM def 1D/2D/3D.
4 Marks Answer: (a) Mid equal. (b) Center sym. (c) Sum/∫. (d) Pairs zero. Eq 6.1-6.5 vector. Continua limit. Ex equal m centroid.
6.3 Full: CM motion eqs.
4 Marks Answer: (a) Ext only. (b) Const isolated. (c) Cancel pairs. (d) F/M. Diff V_cm= sum m v_i /M. Internal zero net.
Part C: 8 Marks Questions (Detailed Long Answers - From NCERT)
6.1 Detailed: Rigid body motions with ex/figs.
8 Marks Answer: Rigid: Fixed distances. Translation: Uniform v all points, OP const orient (Fig 6.6a block slide). Rotation: Fixed axis, circles r ⊥, v=ω r (Fig 6.4 P1 r1 C1). Combined: Rolling v_contact=0 (Fig 6.2 cylinder). General: Fixed point precession (Fig 6.5 top cone, fan oscillate). Deriv: Diff pos for v. Ex: Fan pure rot, book roll combined. Physical: 6 DOF constrained. Graphs: Velocity fields. Errors: Confuse trans rot. Ties: Ch4 vectors describe.
6.2 Detailed: CM calc discrete/continua sym.
8 Marks Answer: Weighted avg \(\vec{R} = \sum m \vec{r}/M\) (Eq6.4). 1D: X=(m1x1+m2x2)/(m1+m2) midpoint equal. 3D vector sum. Continua: ∫ r dm/M (Eq6.5 dm=ρdV). Sym: Reflection pairs ± cancel ∫=0 center (Fig6.8 rod). Deriv: Limit n→∞ Δm→dm. Ex: Triangle Ex6.1 weighted (0.25,0.144); lamina Ex6.2 centroid medians. Physical: Balance. Graphs: Integral approx sums. Advanced: Variable ρ numerical. Interlinks: Ch5 momentum M V_cm.
Tip: Diagrams/eqs in long; practice ex.
Key Concepts - In-Depth Exploration
Core ideas with derivations, examples, pitfalls, interlinks (4+ pages sim). Emphasize CM reduction, rigid constraints.
Extended Bodies
Finite size systems. Deriv: Particle sum. Pitfall: Treat as point always no. Ex: Deform under F. Interlink: Ch2 1D special.
Advanced: CM in relativity. Pitfalls: Non-sym calc full. Interlinks: Ch7 torque. Real: CM in sports (bat swing). Depth: Continua density. Examples: Deriv Eq6.5 limit. Graphs: CM vs geometric. Calculus: dR/dt=V. Errors: Internal F on CM. Tips: Symmetry first; verify M total.
Mnemonics: CM "Mass Weighted Mean" (MWM). Rigid "Fixed Distances No Deform" (FDND). Motion "Ext Force Mass Accel" (EFMA). Symmetry "Pairs Cancel Integral Zero" (PCIZ).
