System of Particles and Rotational MotionNEET Physics · Class 11 · NCERT Chapter 6

Introduction

Up to this point we have treated every object as a point particle. That works only for translation. The moment a body spins, every part of it moves with a different velocity, and we need new tools: the centre of mass, torque, angular momentum, and moment of inertia.

For NEET 2027, expect 2 to 4 questions from this chapter. Moment of inertia of standard shapes (memorise seven), the parallel axis theorem, conservation of angular momentum and rolling motion are the heavy hitters.

Centre of mass

For a system of particles with masses $m_i$ at positions ${\vec{r}}_i$, the centre of mass is the mass-weighted average position:

For a continuous body of density $\rho (\vec{r})$:

Quick results worth memorising:

• Two particles $m_1,m_2$ at separation $d$ → CoM divides the line in ratio $m_2:m_1$ from $m_1$.
• Uniform rod of length $L$ → CoM at the geometric centre.
• Triangle (uniform lamina) → CoM at the centroid (one-third of any median from the base).
• Solid hemisphere of radius $R$ → CoM at $3R/8$ from the flat face along the axis of symmetry.

Motion of the centre of mass

Differentiate the CoM definition twice and use Newton's second law on each particle:

Internal forces (Newton's third-law pairs) cancel out. The CoM moves as if all the mass and all the external forces acted at that single point.

Total linear momentum: $\vec{P}=M{\vec{v}}_{\text{cm}}$. If no net external force acts, $\vec{P}$ is conserved — the CoM moves in a straight line at constant velocity even if the individual parts go flying.

Torque and angular momentum

Torque (moment of force)

For a particle at position $\vec{r}$ from a chosen axis acted on by a force $\vec{F}$:

The torque has SI unit $\text{N}\cdot \text{m}$ and dimensions $[M{L}^2{T}^{-2}]$. Note: torque has the same dimensions as energy, but they are physically different (torque is a vector).

Angular momentum

Magnitude: $|\vec{L}|=rp\sin \theta =mv{r}_{\perp }$. SI unit $\text{kg}{\text{m}}^2/\text{s}$, dimensions $[M{L}^2{T}^{-1}]$— the same dimensions as Planck's constant.

The rotational analog of Newton's second law

This is the foundational equation for rotational motion. If torque is zero, angular momentum is constant — the conservation law you will use again and again.

Equilibrium of a rigid body

A rigid body is in mechanical equilibrium when both:

• Translational equilibrium: $\sum {\vec{F}}_i=\vec{0}$.
• Rotational equilibrium: $\sum {\vec{\tau }}_i=\vec{0}$ (about any axis).

A couple is a pair of equal and opposite forces with different lines of action. A couple produces zero net force but a non-zero net torque — it tries to rotate the body without translating it.

Centre of gravity is the point where the total gravitational torque vanishes. In a uniform gravitational field, it coincides with the centre of mass.

Moment of inertia

Moment of inertia plays the role of mass in rotational dynamics. For a system of particles about a chosen axis:

For a continuous body: $I=\int r^{2}dm$. SI unit $\text{kg}{\text{m}}^2$, dimensions $[M{L}^2]$.

The seven NEET classics

Bonus: solid cylinder about its axis is $M{R}^2/2$ (same as a disc); hollow cylinder about its axis is $M{R}^2$ (same as a ring).

Radius of gyration

Defined by $I=M{k}^2$ so that $k=\sqrt{I/M}$. It is the distance from the axis at which the body's entire mass would have to sit to give the same moment of inertia.

Uniform disc, axis through centre, perpendicular to plane.

Theorems of parallel and perpendicular axes

Parallel axis theorem

For any axis parallel to an axis through the centre of mass:

where $d$ is the perpendicular distance between the two axes. Use it to compute moments of inertia about edges or pivots far from the CoM. Example: rod about its end $=\frac{M{L}^2}{12}+M{\left(\frac{L}{2}\right)}^2=\frac{M{L}^2}{3}$.

Perpendicular axis theorem (planar bodies only)

For a flat lamina lying in the xy plane:

Example: a ring has $I_z=M{R}^2$ about the central axis. By symmetry $I_x=I_y$, so each diameter has $M{R}^2/2$.

Rotational kinematics

For a rigid body rotating about a fixed axis with constant angular acceleration $\alpha$, the equations are direct analogs of linear kinematics:

Linear-angular link: $v=r\omega$, $a_{\text{tangential}}=r\alpha$, $a_{\text{centripetal}}=r{\omega }^2$.

Rotational dynamics

For a rigid body rotating about a fixed axis with moment of inertia $I$:

These three are the rotational versions of $F=ma$, $p=mv$, and $K=\frac{1}{2}m{v}^2$.

Conservation of angular momentum

If the net external torque is zero, the total angular momentum is conserved:

Classic NEET examples:

• Figure skater pulls arms in. $I$ drops, $\omega$ rises to keep $L$ the same.
• Planet in elliptical orbit. Gravity is a central force (torque about Sun = 0), so $L$ is conserved — that is Kepler's second law.
• Collapsing star core (neutron star). Radius shrinks dramatically, $\omega$ rises by a huge factor — neutron stars spin hundreds of times per second.

Rolling motion

Rolling combines translation of the centre of mass and rotation about an axis through the CoM. For pure rolling without slipping, the contact point with the ground is momentarily at rest. This gives the constraint:

Kinetic energy of a rolling body

Using $I_{\text{cm}}=M{k}^2$ and $\omega =v_{\text{cm}}/R$:

Speed at the bottom of an incline

A body of moment of inertia $I=M{k}^2$ released from rest on an incline of height $h$. Energy conservation gives:

Smaller $k^2/R^2$ means more KE in translation and a higher $v$. Order of finish on an incline (slowest to fastest):

• Hollow ring ($k^2/R^2=1$): slowest.
• Solid disc / cylinder ($k^2/R^2=1/2$).
• Hollow sphere ($k^2/R^2=2/3$).
• Solid sphere ($k^2/R^2=2/5$): fastest.

Four bodies start from rest at the top of an incline of height h and roll down without slipping. The smaller the rotational inertia, the more energy goes into translation and the higher the linear speed at the bottom.

How energy splits between translation and rotation

Worked NEET problems

Summary cheat sheet

• Centre of mass: ${\vec{R}}_{\text{cm}}=\frac{1}{M}\sum m_i{\vec{r}}_i$.
• Motion of CoM: $M{\vec{a}}_{\text{cm}}=\sum {\vec{F}}^{\text{ext}}$. Internal forces cancel.
• Torque: $\vec{\tau }=\vec{r}\times \vec{F}$. Angular momentum: $\vec{L}=\vec{r}\times \vec{p}$.
• Rotational Newton II: ${\vec{\tau }}_{\text{net}}=d\vec{L}/dt=I\vec{\alpha }$ (fixed axis).
• Moment of inertia (memorise 7): rod centre $M{L}^2/12$, rod end $M{L}^2/3$, ring axis $M{R}^2$, ring diameter $M{R}^2/2$, disc axis $M{R}^2/2$, solid sphere $2M{R}^2/5$, hollow sphere $2M{R}^2/3$.
• Parallel axis: $I=I_{\text{cm}}+M{d}^2$.
• Perpendicular axis (lamina): $I_z=I_x+I_y$.
• Rolling without slipping: $v_{\text{cm}}=R\omega$, $K=\frac{1}{2}M{v}^2(1+k^2/R^2)$.
• Down an incline: $v^2=\frac{2gh}{1+k^2/R^2}$. Solid sphere fastest, hollow ring slowest.
• Conservation of L: $I_1{\omega }_1=I_2{\omega }_2$ when external torque is zero.

Next: try the interactive widgets for centre of mass, moment of inertia, parallel axis theorem and rolling motion, or work through the 30+ NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.