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Centre of Mass & Linear Momentum revision notes

Every idea in the Centre of Mass & Linear Momentum chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 29 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Centre of mass of particles

COM of discrete particles
5× asked, last 2026
r⃗cm=∑mir⃗i∑mi,xcm=∑mixiM\vec r_{cm}=\frac{\sum m_i\vec r_i}{\sum m_i},\qquad x_{cm}=\frac{\sum m_i x_i}{M}
Weighted average, one axis at a time. Lies inside the region spanned by the masses, not necessarily on any mass.
  • Equal masses: the COM is the plain average of the positions. A mass at the origin adds nothing to the numerator but still counts in MM.
  • Not the geometric centre when masses differ. Symmetric positions cancel only for equal masses; the COM shifts towards the heavier side.
  • Quick method. Fix the origin at the point the question names, write signed coordinates, and find xcmx_{cm} and ycmy_{cm} separately. Position vectors: add mir⃗im_i\vec r_i component by component.
Two particles
2× asked, last 2024
m1r1=m2r2,r1=m2 dm1+m2m_1r_1=m_2r_2,\qquad r_1=\frac{m_2\,d}{m_1+m_2}
r1,r2r_1,r_2 = distances from the COM, dd = separation. The COM is nearer the heavier mass.
mm3m3mCOMr1=3d/4r_1=3d/4r2=d/4r_2=d/4
m1r1=m2r2m_1r_1=m_2r_2: the COM sits nearer the heavier mass
  • To keep the COM fixed, m1Δx1=m2Δx2m_1\Delta x_1=m_2\Delta x_2 with opposite signs: if one mass moves towards the COM, the other must also move towards it.
Composite bodies
3× asked, last 2025
xcm=m1x1+m2x2+…m1+m2+…x_{cm}=\frac{m_1x_1+m_2x_2+\dots}{m_1+m_2+\dots}
Replace each regular part by its mass at its own COM. Uniform plate: mass ∝ area; uniform rod or wire: mass ∝ length.
A1=3A_1=3A2=2A_2=2COM (1.1, 1.1)(1.1,\,1.1)
Each rectangle at its own centre, weighted by area: xcm=1.1x_{cm}=1.1, ycm=1.1y_{cm}=1.1
  • Quick method. Bent rod or L-shaped plate: split into straight pieces or rectangles, put each mass at its centre, weight by length or area. Density and thickness cancel.
Balancing on a support
1× asked, last 2022
∑mig di (clockwise)=∑mjg dj (anticlockwise)\sum m_i g\,d_i\ (\text{clockwise})=\sum m_j g\,d_j\ (\text{anticlockwise})
Torques about the pivot. A body's whole weight acts at its COM; a uniform rod's COM is its mid-point.

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Practise Centre of Mass & Linear Momentum

25 more concepts in 6 topics

Centre of mass of continuous bodies, Motion of the centre of mass, Conservation of linear momentum, Impulse and Variable mass: rockets and conveyor belts, plus ideas from kinematics and angular momentum. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Centre of mass of continuous bodies

  • Continuous body3×
  • Non-uniform rod3×
  • Semicircular ring and arc1×
  • Semicircular and quarter disc2×
  • Hemispheres1×
  • Cones and triangle
  • Removed part (negative mass)4×

Motion of the centre of mass

  • Velocity and acceleration of COM2×
  • Newton's second law for a system1×
  • Projectile exploding in mid-air
  • Kinetic energy of a system1×

Conservation of linear momentum

  • Momentum and kinetic energy5×
  • Conservation of momentum7×
  • Splitting from rest and gun recoil5×
  • Explosion into three pieces2×
  • Man on a boat or plank
  • Sticking together2×

Impulse

  • Impulse–momentum theorem8×
  • Rebound4×
  • Oblique impact on a wall1×
  • Force–time and momentum–time graphs2×
  • Stream of bullets or balls3×

Variable mass: rockets and conveyor belts

  • Thrust2×
  • Rocket going up1×
  • Mass picked up at speed v1×

Also asked here: kinematics and angular momentum

  • Key points2

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