NCERT Solutions for Class 11 Physics Chapter 7 – System of Particles and Rotational Motion
Centre of mass, torque, moment of inertia, angular momentum, rolling motion, and a free PDF for Chapter 7.
About Chapter 7 – System of Particles and Rotational Motion
Chapter 7, System of Particles and Rotational Motion, is the most mathematically rich chapter in Class 11 Physics and forms the rotational counterpart of linear dynamics studied in Chapters 3–6. The chapter begins by introducing the concept of a system of particles and defines the centre of mass (CM) – the point that moves as though all external forces on the system act through it. For systems with uniform symmetry, the CM coincides with the geometric centre, making many problems simpler.
The chapter then develops the full rotational mechanics framework. Students learn about torque (τ = r × F), the rotational analogue of force, which produces angular acceleration in a rotating body. The key quantity in rotation is the moment of inertia (I = Σmr²), which is the rotational analogue of mass – it measures the resistance of an object to angular acceleration. Important theorems for calculating moment of inertia (the parallel axis theorem and perpendicular axis theorem) are derived and applied.
Angular momentum (L = Iω) is the rotational equivalent of linear momentum, and its conservation (when no external torque acts) explains phenomena like a figure skater spinning faster when arms are pulled in. The chapter concludes with rolling motion – a combination of translational and rotational motion – and the conditions for it. For CBSE boards, moment of inertia, torque, angular momentum, and rolling motion are the most tested topics. JEE aspirants need to master the parallel axis theorem, rolling without slipping conditions, and energy of rolling bodies, all of which appear regularly in competitive entrance exams.
NCERT Solutions for Class 11 Physics Chapter 7 – Free PDF Download
NCERT Solutions – Chapter 7: System of Particles and Rotational Motion (All Exercises)
Download solved PDF with full exercise solutions including centre of mass, rotational dynamics, angular momentum, and rolling motion problems.
Important Formulas – Chapter 7: System of Particles and Rotational Motion
| Concept | Formula | Description |
|---|---|---|
| Centre of Mass (2 particles) | x_cm = (m₁x₁ + m₂x₂)/(m₁+m₂) | Weighted average of positions |
| Centre of Mass (system) | R⃗_cm = Σmᵢrᵢ / M | M = total mass of system |
| Velocity of CM | v⃗_cm = Σmᵢv⃗ᵢ / M = p⃗_total / M | CM moves as if all mass is concentrated there |
| Torque | τ⃗ = r⃗ × F⃗ ; |τ| = rF sinθ | Rotational equivalent of force; unit = N·m |
| Angular Momentum | L⃗ = r⃗ × p⃗ = Iω | Rotational equivalent of linear momentum |
| Rotational Newton's 2nd Law | τ = dL/dt = Iα | Net torque = moment of inertia × angular acceleration |
| Moment of Inertia | I = Σmᵢrᵢ² | Sum of (mass × square of distance from axis) |
| Parallel Axis Theorem | I = I_cm + Md² | I about any parallel axis = I_cm + Md² |
| Perpendicular Axis Theorem | I_z = I_x + I_y | For laminar bodies only; z ⊥ plane of body |
| Conservation of Angular Momentum | Iω = constant (if τ_ext = 0) | Explains spinning skater, gyroscope |
| Kinetic Energy of Rotation | KE_rot = ½Iω² | Analogue of ½mv² for rotation |
| Rolling (KE total) | KE = ½mv² + ½Iω² = ½mv²(1 + k²/R²) | k = radius of gyration; v = Rω for rolling without slip |
| Angular Velocity | ω = dθ/dt | Rate of change of angular displacement; unit = rad/s |
| Angular Acceleration | α = dω/dt | Rate of change of angular velocity; unit = rad/s² |
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|---|---|---|
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Subtopics of Chapter 7 – System of Particles and Rotational Motion
7.1 Centre of Mass
Point representing average position of all masses. Moves as per Newton's second law for external forces. CM of uniform objects = geometric centre.
7.2 Motion of Centre of Mass
The CM of an isolated system moves with constant velocity. Internal forces do not affect CM motion. Key for explosion and collision problems.
7.3 Linear Momentum of a System
Total momentum = M·v_cm. If external force is zero, total momentum is conserved. Foundation for collision analysis in multiple dimensions.
7.4 Vector Product (Cross Product)
A⃗ × B⃗ = AB sinθ n̂. Direction given by right-hand rule. Used to define torque (τ = r × F) and angular momentum (L = r × p).
7.5 Angular Velocity and Angular Acceleration
ω = dθ/dt; α = dω/dt. For uniform angular acceleration: ω = ω₀+αt, θ = ω₀t+½αt², ω² = ω₀²+2αθ (rotational kinematics).
7.6 Torque and Angular Momentum
τ = Iα = dL/dt. For a system: net external torque = rate of change of total angular momentum. Zero torque → L conserved.
7.7 Moment of Inertia and Theorems
I depends on mass distribution and axis of rotation. Parallel axis: I = I_cm + Md². Perpendicular axis (lamina): I_z = I_x + I_y.
7.8 Kinematics of Rotation and Rolling Motion
Rolling without slipping: v = Rω. Total KE = translational KE + rotational KE. Condition: friction acts but does no work.
Quick Reference – Moments of Inertia for Common Bodies
| Body | Axis | Moment of Inertia (I) |
|---|---|---|
| Thin Ring / Hoop (mass M, radius R) | Central axis (perpendicular to plane) | MR² |
| Thin Ring / Hoop | Diameter | ½MR² |
| Solid Disc / Cylinder (mass M, radius R) | Central axis (through centre, perpendicular to face) | ½MR² |
| Solid Disc | Diameter | ¼MR² |
| Solid Sphere (mass M, radius R) | Diameter (any) | 2MR²/5 |
| Hollow Sphere (thin shell) | Diameter | 2MR²/3 |
| Thin Rod (mass M, length L) | Perpendicular through centre | ML²/12 |
| Thin Rod | Perpendicular through one end | ML²/3 |
| Rectangular Plate (a × b) | Through centre, perpendicular to plane | M(a²+b²)/12 |
| Hollow Cylinder (inner r, outer R) | Central longitudinal axis | ½M(R²+r²) |
NCERT Solutions for Class 11 Physics Chapter 7: System of Particles and Rotational Motion
Translational motion occurs when every point of a body moves the same distance in the same direction during a given time interval. Rotational motion, however, involves the movement of a body around a fixed axis. In this chapter, students learn how objects can simultaneously undergo both motions, such as a rolling wheel. Understanding this distinction is important because different physical quantities and equations are used to describe each type of motion. Concepts like angular displacement, angular velocity, and angular acceleration are introduced to analyze rotational systems and solve numerical problems effectively.
The center of mass is the point where the entire mass of a system can be assumed to be concentrated for analyzing motion. This concept simplifies the study of complex systems containing multiple particles. In rotational motion, the behavior of a body often depends on the location of its center of mass. Students use this concept to understand equilibrium, momentum, and motion of rigid bodies. Questions based on center of mass frequently appear in examinations because they test conceptual understanding and problem-solving skills. A clear understanding helps students tackle advanced mechanics topics with confidence.
Torque is the rotational equivalent of force and determines the tendency of an object to rotate about an axis. It depends on both the magnitude of the applied force and its perpendicular distance from the axis of rotation. This chapter explains how torque influences rotational equilibrium and angular acceleration. Everyday examples include opening a door, tightening a bolt, or using a wrench. Numerical problems involving torque require careful application of formulas and sign conventions. Understanding torque helps students connect theoretical concepts with practical situations involving rotational motion.
Moment of inertia measures the resistance of a body to changes in its rotational motion. It depends on both the mass of the object and how the mass is distributed relative to the axis of rotation. Objects with larger moments of inertia require greater torque to achieve the same angular acceleration. This concept is fundamental for understanding rotating bodies such as wheels, discs, and rods. Students often encounter derivations and numerical problems related to moment of inertia. Mastering this topic strengthens the foundation needed for higher-level mechanics and engineering applications.
Angular momentum describes rotational motion, while linear momentum describes translational motion. Angular momentum depends on the moment of inertia and angular velocity of a rotating object. One of the most important principles discussed in this chapter is the conservation of angular momentum. This principle explains phenomena such as a spinning skater increasing speed by pulling in their arms. Examination questions frequently test conceptual understanding and numerical applications of angular momentum. Learning this concept helps students understand many natural and technological systems involving rotational dynamics.