Class 11 Physics Chapter 7 | Rotational Motion - Full Chapter NCERT Solutions

Rotational Motion: Rigid Body Dynamics and Moment of Inertia

Rotational motion describes the movement of a rigid body around a fixed axis. While translational motion deals with objects moving from one point to another, rotational motion deals with objects spinning or turning. Every concept in translational mechanics — displacement, velocity, acceleration, force, mass, momentum, and energy — has a direct rotational analogue. In this chapter of the CBSE Class 11 Physics syllabus, students learn to describe rotational motion using angular quantities, understand the concept of moment of inertia and torque, and apply Newton's second law to rotating bodies. This knowledge is essential for understanding everything from the spinning of a ceiling fan and the rolling of wheels to the rotation of planets and the stability of gyroscopes.

In pure translational motion, position is described by linear displacement x; in rotation, the corresponding quantity is angular displacement θ, measured in radians. Angular velocity ω = dθ/dt describes how fast an object rotates (radians per second), and angular acceleration α = dω/dt describes how the angular velocity changes. The kinematic equations for rotational motion with constant angular acceleration mirror the linear equations: ω = ω₀ + αt, θ = ω₀t + ½αt², and ω² = ω₀² + 2αθ. The connection between linear and angular quantities is given by v = rω and a = rα, where r is the perpendicular distance from the axis of rotation. Torque (τ) is the rotational analogue of force — it is the turning effect produced by a force and is defined as τ = r × F, or τ = rF sin θ, where r is the position vector from the axis to the point of force application, F is the force, and θ is the angle between them. The SI unit of torque is the newton-metre (N·m).

Moment of inertia (I) is the rotational analogue of mass and represents a body's resistance to angular acceleration. It depends not only on the total mass but also on how that mass is distributed relative to the axis of rotation: I = Σmᵢrᵢ² for a system of particles, or I = ∫r²dm for a continuous body. The moment of inertia increases as mass is placed farther from the axis, which is why a flywheel with most of its mass at the rim is harder to spin up than a solid disc of the same mass. The parallel axis theorem states that I = Icm + Md², where Icm is the moment about the centre of mass and d is the distance to the parallel axis. The perpendicular axis theorem (for planar bodies) states that Iz = Ix + Iy. Newton's second law for rotation is τnet = Iα. Angular momentum L = Iω is the rotational analogue of linear momentum, and it is conserved when no external torque acts on the system — this principle explains why a spinning figure skater pulls in their arms to rotate faster (I decreases, so ω must increase to keep L = Iω constant). The kinetic energy of rotation is K = ½Iω², and for a body that is both rolling and translating, the total kinetic energy is K = ½Mv² + ½Iω². Pure rolling occurs when v = rω (no slipping).

• Angular quantities: θ (radians), ω = dθ/dt (rad/s), α = dω/dt (rad/s²); related to linear by v = rω, a = rα.
• Torque τ = rF sin θ causes rotation; moment of inertia I = Σmᵢrᵢ² resists angular acceleration.
• Newton's second law for rotation: τnet = Iα; parallel axis theorem: I = Icm + Md².
• Angular momentum L = Iω is conserved when no external torque acts; explains why skaters spin faster when they pull their arms in.
• Rolling without slipping: v = rω; total kinetic energy K = ½Mv² + ½Iω².