Class 11 Physics · NCERT notes for NEET

System of Particles and Rotational Motion: NCERT Notes, Formulas and Tricks for NEET

Newton's laws extended to systems and rigid bodies: centre of mass, torque, angular momentum, moment of inertia, equilibrium and rolling. NEET asks from it regularly, most often on moment of inertia, conservation of angular momentum and rolling down an incline.

• 1. Centre of mass
• 2. Torque and angular momentum 🔒
• 3. Moment of inertia, parallel and perpendicular axes 🔒
• 4. Rotational dynamics and equilibrium 🔒
• 5. Rolling motion 🔒

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Topic 1

Centre of mass

A firework shell bursts into a hundred glowing pieces, yet their centre of mass keeps flying along the same smooth parabola as if nothing happened.

• A rigid body keeps the distance between every pair of its particles fixed, whatever forces act. It is an idealisation, but it works well for most solids.
• A body that is not pivoted can show pure translation or translation + rotation. A body fixed along a line (axis) can only rotate about that axis; every particle moves in a circle centred on the axis.
• Centre of mass (CM): the point that moves as if the whole mass were placed there and all external forces acted there.
• Discrete particles: $x_{CM}=\frac{\sum m_i{x}_i}{\sum m_i}$ (same for $y$, $z$). Continuous body: $X=\frac{1}{M}\int xdm$.
• Two particles: the CM lies on the line joining them, nearer the heavier one, with $m_1{r}_1=m_2{r}_2$.
• Uniform symmetric bodies (rod, ring, disc, sphere, cube) have the CM at the geometric centre. It can lie where there is no mass (ring, hollow sphere). Three equal masses at a triangle's corners: CM at the centroid.
• Total momentum $\vec{P}=M{\vec{v}}_{CM}$ and $M{\vec{a}}_{CM}={\vec{F}}_{ext}$. Internal forces cancel in pairs and cannot change the motion of the CM.
• If ${\vec{F}}_{ext}=0$, $\vec{P}$ is constant and the CM moves with constant velocity. A nucleus at rest that decays: CM stays at rest, products fly apart in opposite directions.
• An exploding projectile: the CM continues on the original parabola until a piece hits the ground (gravity is the only external force).
• Precession: the axis of a spinning top sweeps out a cone about the vertical through its fixed point of contact.

Formulas

• $x_{CM}=\frac{m_1{x}_1+m_2{x}_2+\cdots +m_n{x}_n}{m_1+m_2+\cdots +m_n}$
  CM of particles; same form for $y_{CM}$, $z_{CM}$
• $X=\frac{1}{M}\int xdm$
  CM of a continuous body
• ${\vec{v}}_{CM}=\frac{\sum m_i{\vec{v}}_i}{M},{\vec{a}}_{CM}=\frac{\sum m_i{\vec{a}}_i}{M}$
  Velocity and acceleration of the CM
• $\vec{P}=M{\vec{v}}_{CM},M{\vec{a}}_{CM}={\vec{F}}_{ext}$
  Newton's second law for a system
• ${\vec{F}}_{ext}=0\Rightarrow \vec{P}=\text{constant},{\vec{v}}_{CM}=\text{constant}$
  Conservation of linear momentum
• $r_1=\frac{m_{2}d}{m_1+m_2},r_2=\frac{m_{1}d}{m_1+m_2}$
  Two masses a distance $d$ apart: distances from the CM

Tips, tricks and hacks

Common mistakes

NEET traps

"The CM of a system moves only under external forces" is TRUE; options saying friction between two parts of the system shifts the CM are wrong.

NEET asks the CM of a triangle of three equal masses or of a semicircle-type body: for equal masses use the centroid, and use symmetry to kill one coordinate first.

After one fragment of an exploding shell lands, the ground exerts a force on it, so the CM no longer follows the parabola. The parabola rule holds only while all pieces are in the air.

Practise System of Particles and Rotational Motion questions

Topic-wise practice from Level 1 to Level 3, NCERT-based questions and NEET PYQs, with a full solution after every answer.