Rotational Motion Notes for NEET and JEE Main 2027: Key Points and PDF

Systems of Particles and Rotational Motion is Chapter 6 of NCERT Class 11 Physics and Unit 5, Rotational Motion, in both the NEET UG and JEE Main syllabi. These notes cover centre of mass, vector product, torque, angular momentum, equilibrium of rigid bodies, moment of inertia, the axes theorems and rotation about a fixed axis, with a free PDF.

Systems of Particles and Rotational Motion Notes for NEET and JEE Main

Section-wise key points from NCERT Chapter 6, formula tables, high-yield facts and common mistakes for 2027 revision.

Systems of Particles and Rotational Motion at a Glance

Topics follow the NEET UG 2026 syllabus released by the National Medical Commission (NMC) on 22 December 2025 and the JEE Main 2026 syllabus published by NTA. Both stay the reference until the 2027 syllabi are out. Full lists: NEET syllabus and JEE Main syllabus.

What is a rigid body and what kinds of motion can it have?

A rigid body is one in which the distances between all pairs of particles do not change. No real body is truly rigid, but wheels, tops, steel beams, molecules and planets can be treated as rigid when deformation is negligible.

• Pure translation: at any instant every particle of the body has the same velocity (a block sliding down an incline).
• Rotation about a fixed axis: every particle moves in a circle in a plane perpendicular to the axis, with its centre on the axis. Particles on the axis stay at rest (ceiling fan, potter’s wheel).
• Rotation with one point fixed: the axis itself moves, as in a spinning top whose axis sweeps a cone (precession) or an oscillating table fan.
• A body that is not pivoted or fixed has either pure translation or translation plus rotation. Rolling of a cylinder down an incline is translation plus rotation; the point of contact has zero velocity if it rolls without slipping.

Centre of mass: where is it and how does it move?

The centre of mass (CM) is the mass-weighted mean position of a system: R = Σmᵢrᵢ / M, or R = (1/M)∫r dm for a continuous body.

• MA = F_ext: the CM moves as if all the mass were concentrated there and all external forces acted there. Internal forces cancel in pairs (Newton’s third law) and do not affect the CM.
• A projectile that explodes in mid-air: its fragments’ CM continues along the same parabola, because the explosion forces are internal.
• Total linear momentum P = MV (V = velocity of the CM), and dP/dt = F_ext.
• If F_ext = 0, P is constant and the CM moves with constant velocity, as in the decay of a radium nucleus into radon and an alpha particle, or a binary star system.
• Kinetic energy of a system = kinetic energy about the CM + MV²/2.

Vector Product, Angular Velocity and Angular Acceleration

Torque and angular momentum are vector (cross) products, so the rules of the vector product come first.

• c = a × b has magnitude ab sin θ and is perpendicular to the plane of a and b; its direction follows the right-handed screw or right-hand rule (rotate through the smaller angle from a to b).
• a × b = −b × a (not commutative); a × a = 0; a × b does not change sign under reflection; the product is distributive.
• î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ; reversing the cyclic order gives a negative sign.
• Every particle of a body rotating about a fixed axis has the same angular velocity ω = dθ/dt; ω is a vector along the axis (right-handed screw).
• Linear velocity of a particle: v = ω × r, magnitude v = ωr⊥, where r⊥ is its perpendicular distance from the axis.
• Angular acceleration α = dω/dt. For a fixed axis, the directions of ω and α stay fixed.

Torque and Angular Momentum

Torque (moment of force) is the rotational analogue of force, and angular momentum is the rotational analogue of linear momentum.

• Torque has the same dimensions as work but is a vector, while work is a scalar.
• τ = 0 if r = 0, F = 0, or the line of action of the force passes through the origin (θ = 0° or 180°).
• Internal torques cancel only if the internal forces obey Newton’s third law and act along the line joining the particles.
• Conservation of angular momentum: if τ_ext = 0, L is constant, and each of Lx, Ly and Lz is conserved.
• A particle moving with constant velocity has constant angular momentum about any point, since r⊥ of its line of motion does not change.

Equilibrium of a Rigid Body, Couple and Centre of Gravity

A rigid body is in mechanical equilibrium when both the total external force and the total external torque on it are zero.

• Translational equilibrium: ΣF = 0. Rotational equilibrium: Στ = 0. Together they give six scalar conditions; for coplanar forces only three are needed.
• A body can be in partial equilibrium: translational without rotational, or the reverse.
• If the body is in translational equilibrium, the rotational condition does not depend on the point about which torques are taken.
• A couple is a pair of equal and opposite forces with different lines of action. It produces rotation without translation, and its moment does not depend on the origin. Examples: turning a bottle lid, a compass needle in the earth’s magnetic field.
• Principle of moments for a lever: load arm × load = effort arm × effort (d₁F₁ = d₂F₂). Mechanical advantage = F₁/F₂ = d₂/d₁, which exceeds one when the effort arm is longer.
• Centre of gravity (CG) is the point where the total gravitational torque on the body is zero. It coincides with the CM only when g is the same over the whole body.
• To find the CG of an irregular lamina, suspend it from two or more points; the verticals through the points of suspension meet at the CG.

Moment of Inertia and Radius of Gyration

Moment of inertia I = Σmᵢrᵢ² is the rotational analogue of mass; the rotational kinetic energy is K = ½Iω².

• I depends on the mass, shape and size of the body, the distribution of mass about the axis, and the position and orientation of the axis. Unlike mass, it is not fixed for a body.
• SI unit kg m²; dimensions [ML²].
• Radius of gyration k: I = Mk², the distance from the axis of a point mass equal to the whole mass that has the same moment of inertia. For a rod about the perpendicular axis at its midpoint, k = L/√12; for a disc about a diameter, k = R/2.
• A flywheel has a large moment of inertia, so it resists sudden changes of speed and keeps an engine’s motion smooth.

Parallel and Perpendicular Axes Theorems (syllabus Topic, Not in the Rationalised NCERT)

Both the NEET UG and JEE Main syllabi list these theorems, but the rationalised NCERT chapter dropped them. The statements below follow the older NCERT edition.

• Disc about a diameter: I_z = MR²/2 and I_x = I_y by symmetry, so I_diameter = MR²/4.
• Rod about a perpendicular axis through one end: ML²/12 + M(L/2)² = ML²/3.
• Ring about a tangent perpendicular to its plane: MR² + MR² = 2MR².

Rotation About a Fixed Axis: Kinematics and Dynamics

Rotation about a fixed axis has one degree of freedom, so its equations mirror those of motion in a straight line.

• Uniform angular acceleration: ω = ω₀ + αt; θ = θ₀ + ω₀t + ½αt²; ω² = ω₀² + 2α(θ − θ₀).
• For a fixed axis, only forces in planes perpendicular to the axis and only the components of torque along the axis matter.
• Work done by a torque: dW = τ dθ. Power: P = τω.
• Newton’s second law for rotation: τ = Iα (from equating the rate of work done to the rate of increase of ½Iω²).

Angular Momentum About a Fixed Axis and Its Conservation

For a body rotating about a fixed axis that is also its axis of symmetry, L = Iω along the axis, and d(Iω)/dt = τ.

• For a single particle, or a body not symmetric about the axis, L and ω need not be parallel; only the component L_z = Iω lies along the axis.
• If the external torque about the axis is zero, Iω = constant.
• A person on a swivel chair slows down on stretching the arms (I increases) and speeds up on pulling them in (I decreases).
• Circus acrobats, divers, skaters and dancers doing a pirouette use the same principle.

Most-asked Facts for NEET and JEE Main: Systems of Particles and Rotational Motion

Revise these one-line facts from the NCERT chapter just before the exam.

• Internal forces do not change the motion of the centre of mass; only external forces do.
• CM of a uniform triangular lamina is at its centroid; three equal masses also have their CM at the centroid.
• a × b = −b × a; î × ĵ = k̂.
• v = ω × r for every particle of a rotating rigid body.
• Torque τ = r × F has unit N m and the same dimensions as work.
• dL/dt = τ_ext; L is conserved when τ_ext = 0.
• A couple gives zero net force but a non-zero torque that is the same about every point.
• Lever: mechanical advantage = effort arm/load arm.
• CG = CM only in a uniform gravitational field.
• Ring MR², disc MR²/2, rod (centre) ML²/12, solid sphere 2MR²/5, hollow cylinder MR², solid cylinder MR²/2.
• Radius of gyration: I = Mk²; rod about centre k = L/√12, disc about diameter k = R/2.
• Parallel axes: I = I_cm + Ma²; perpendicular axes (plane bodies): I_z = I_x + I_y.
• τ = Iα, P = τω, K = ½Iω², L = Iω.
• Stretching the arms on a rotating stool increases I and lowers ω (Iω constant).

Common Mistakes to Avoid

Watch for these mix-ups when you answer questions on this chapter.

• Using the perpendicular axes theorem for a three-dimensional body such as a sphere or cylinder. It holds only for plane bodies.
• Applying the parallel axes theorem between two axes when neither passes through the centre of mass. One axis must pass through the CM.
• Treating moment of inertia as a fixed property like mass. It changes with the axis chosen.
• Thinking torque and work are the same quantity because they share units. Torque is a vector; work is a scalar.
• Assuming the centre of mass must lie inside the body. For a ring it lies at the centre, where there is no material.
• Saying internal forces can change the momentum of a system. Only external forces can.
• Writing L = Iω as always parallel to ω. That holds for rotation about a symmetry axis, not in general.
• Assuming a body with zero net force is in full equilibrium. A couple gives zero net force but still rotates the body.
• Mixing up the disc values: MR²/2 about the perpendicular axis through the centre, MR²/4 about a diameter.

Systems of Particles and Rotational Motion Notes PDF and Related Pages

Download the AglaSem notes for Systems of Particles and Rotational Motion (key notes on centre of mass, moment of inertia, torque, angular momentum and rolling, with important questions and answers) free: View PDF

• All chapters: NEET and JEE Main Physics notes
• Full topic lists: NEET syllabus and JEE Main syllabus
• Practice: NEET question papers and JEE Main question papers
• Source text: NCERT Chapter 6 (NCERT, Class 11 Physics Part I)

Physics Notes for NEET and JEE Main: All Chapters

• Units and Measurement
• Motion in a Straight Line
• Motion in a Plane
• Laws of Motion
• Work, Energy and Power
• Systems of Particles and Rotational Motion
• Gravitation
• Mechanical Properties of Solids
• Mechanical Properties of Fluids
• Thermodynamics
• Kinetic Theory of Gases
• Oscillations
• Waves
• Thermal Properties of Matter
• Electric Charges and Fields
• Electrostatic Potential and Capacitance
• Current Electricity
• Moving Charges and Magnetism
• Magnetism and Matter
• Electromagnetic Induction
• Electromagnetic Waves
• Ray Optics and Optical Instruments
• Wave Optics
• Dual Nature of Radiation and Matter
• Atoms
• Nuclei
• Semiconductor Electronics

All chapters with key points and notes PDFs are on the Physics notes for NEET and JEE Main page.

Systems of Particles and Rotational Motion Notes for NEET and JEE Main FAQs