Systems of Particles and Rotational Motion

NCERT Class 11 Physics Chapter 6

Systems of Particles and Rotational Motion for Class 11 Physics, JEE and NEET: centre of mass, torque, angular momentum, moment of inertia, axis theorems.

14 topics Updated 8 Nov 2026

For: Class 11, JEE and NEET, unless a topic says otherwise

In the syllabus: JEE Class 11 syllabus, NEET Class 11 syllabus

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NCERT Physics Part I (Class 11), Chapter 6 “Systems of Particles and Rotational Motion”, reprint 2026-27.

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Checked on 19 Oct 2026.

Learn in this order

Rigid bodies: translation and rotation
What a rigid body is, translation, rotation about a fixed axis or point, and rolling without slipping.

NCERT Section 6.1 16 min read 3 solved examples 7 practice questions
Centre of mass
Two particles, many particles and rigid bodies, symmetry, composite bodies and cut-outs, with solved examples.

NCERT Section 6.2 14 min read 3 solved examples 6 practice questions
Motion of the centre of mass
Velocity and acceleration of the centre of mass, MA = F_ext, explosions and boats, with examples.

NCERT Section 6.3 13 min read 3 solved examples 6 practice questions
Linear momentum of a system of particles
P = MV, the second law for a system, conservation and the centre-of-mass frame.

NCERT Section 6.4 13 min read 3 solved examples 6 practice questions
Vector (cross) product of two vectors
Ab sin θ, the right-hand rule, properties, unit vectors and the determinant form.

NCERT Section 6.5 13 min read 3 solved examples 6 practice questions
Angular velocity and linear velocity
Angular displacement, ω = dθ/dt, v = ωr, the vector v = ω × r and angular acceleration, with examples.

NCERT Section 6.6 12 min read 3 solved examples 6 practice questions
Torque and angular momentum
τ = r × F, the moment arm, l = r × p, dL/dt = τ and conservation of angular momentum, solved.

NCERT Section 6.7 13 min read 3 solved examples 6 practice questions
Equilibrium of a rigid body
Zero force and zero torque, couples, the principle of moments and the centre of gravity, solved.

NCERT Section 6.8 15 min read 3 solved examples 6 practice questions
Experiment: mass of an object with a metre scale
Finding a mass with a metre scale and a knife-edge, with the theory, procedure, precautions and examples.

JEE and NEET only 11 min read 3 solved examples 6 practice questions
Moment of inertia
I = Σmr^², rotational kinetic energy, radius of gyration and I of a ring, disc, rod and sphere, with examples.

NCERT Section 6.9 13 min read 3 solved examples 6 practice questions
Parallel and perpendicular axes theorems
Statements, proofs and uses for rods, rings, discs, plates and spheres, with solved examples.

JEE and NEET only 12 min read 3 solved examples 6 practice questions
Kinematics of rotation about a fixed axis
Angular displacement, velocity and acceleration, the equations for constant α and the linear analogy.

NCERT Section 6.10 11 min read 3 solved examples 6 practice questions
Dynamics of rotation about a fixed axis
Torque and angular acceleration, τ = Iα, work and power in rotation and the linear comparison, solved.

NCERT Section 6.11 13 min read 3 solved examples 6 practice questions
Angular momentum about a fixed axis
L = Iω about a fixed axis, I_₁ω_₁ = I_₂ω_₂, skaters, divers and locking discs, with examples.

NCERT Section 6.12 14 min read 3 solved examples 6 practice questions

Key formulas in this chapter

Rigid body motion

v = ωR (rolling without slipping on fixed ground)
v_top = 2v, v_contact = 0 (pure rolling on fixed ground)
v_side = √2 v (point level with the centre)

Centre of mass

X = (m_₁x_₁ + m_₂x_₂)/(m_₁ + m_₂) (two particles)
m_₁r_₁ = m_₂r_₂ (distances from the centre of mass)
X = Σm_ᵢx_ᵢ/M, Y = Σm_ᵢy_ᵢ/M, Z = Σm_ᵢz_ᵢ/M
R = Σm_ᵢr_ᵢ/M (vector form)
X = (1/M)∫x dm (continuous body)

Motion of the centre of mass

MV = Σm_ᵢv_ᵢ (velocity of the centre of mass)
MA = Σm_ᵢa_ᵢ (acceleration of the centre of mass)
MA = F_ext (motion of the centre of mass)
F_ext = 0 ⇒ V = constant

Momentum of a system

P = Σm_ᵢv_ᵢ = MV (total momentum)
dP/dt = F_ext (second law for a system)
F_ext = 0 ⇒ P = constant (P_x, P_y, P_z each)
K = K′ + ½MV^² (kinetic energy of a system)
K_₁/K_₂ = m_₂/m_₁ (two pieces from rest)

Vector product

|a × b| = ab sin θ
b × a = −a × b
a × a = 0
i × j = k, j × k = i, k × i = j
a × b = (a_y b_z − a_z b_y)i + (a_z b_x − a_x b_z)j + (a_x b_y − a_y b_x)k
area of parallelogram = |a × b|

Angular velocity

ω = dθ/dt (angular velocity)
ω = 2πN/60 (N in rpm)
v = ωr (r from the axis)
v = ω × r (origin on the axis)
α = dω/dt (angular acceleration)
a_t = αr, a_c = ω²r

Torque and angular momentum

τ = r × F (torque)
τ = rF sin θ = r⊥F
l = r × p (angular momentum)
l = rp sin θ = r⊥p
dl/dt = τ (particle)
dL/dt = τ_ext (system)
τ_ext = 0 ⇒ L = constant

Equilibrium of a rigid body

ΣF = 0 and Στ = 0 (equilibrium of a rigid body)
moment of a couple = Fd
d_₁F_₁ = d_₂F_₂ (principle of moments)
M.A. = F_₁/F_₂ = d_₂/d_₁

Experiment: mass with a metre scale

m d_₁ = M d_₂ (balance about G)
m = M d_₂/d_₁
Δm/m ≈ ΔM/M + Δd_₁/d_₁ + Δd_₂/d_₂ (first-order worst-case bound)

Moment of inertia

I = Σm_ᵢr_ᵢ^² (system of particles)
K = ½Iω^² (kinetic energy of rotation)
I = Mk^² (radius of gyration)
ring MR^², disc MR^²/2 (central axis); rod ML^²/12 (middle); solid sphere 2MR^²/5 (diameter)

Parallel and perpendicular axes

I_z = I_x + I_y (flat body)
I = I_cm + Md² (any body)
rod about an end: ML^²/3
ring about a diameter: MR^²/2; disc about a diameter: MR^²/4
disc about a tangent in its plane: 5MR^²/4

Rotational kinematics

ω = ω_₀ + αt
θ − θ_₀ = ω_₀t + ½αt^²
ω^² = ω_₀^² + 2α(θ − θ_₀)
θ − θ_₀ = ½(ω_₀ + ω)t
s = rθ, v_t = rω, a_t = rα (signed); speed = r|ω|

Rotational dynamics

τ = Iα (fixed axis)
dW = τ dθ; W = τθ (constant τ)
P = τω
W = ½Iω^² − ½Iω_₀^² (work-energy theorem)
a = Rα (string on a wheel, no slipping)

Angular momentum about an axis

L = Iω (about a fixed axis)
d(Iω)/dt = τ
τ = 0 ⇒ Iω = constant; I_₁ω_₁ = I_₂ω_₂
K = ½Iω^² = L^²/2I
ω = (I_₁ω_₁ + I_₂ω_₂)/(I_₁ + I_₂) (two discs locking together; ω signed)

Learn in this order

Key formulas in this chapter

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