centre of mass

System of Particles

A system of particles is a collection of masses studied together. Internal forces act between particles of the system, while external forces are applied from outside.

Why COM Is Useful

It replaces a complex system by one representative point.
Internal forces cancel in total motion.
Extended body motion can be separated into translation plus rotation.
A particle has negligible size; an extended body has mass spread over space.

Centre of mass is the average position of mass. It may lie outside a body, and it differs from geometrical centre when mass distribution is non-uniform.

Coordinates

x_cm = Σm_ix_i / Σm_iy_cm = Σm_iy_i / Σm_iz_cm = Σm_iz_i / Σm_i

Physical Meaning

COM is the point at which total mass can be imagined concentrated for translational motion.

COM of Two Particles

For two masses on a line, COM divides the distance inversely in the ratio of masses.

x_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂)

Equal masses: midpoint.
Unequal masses: COM is closer to heavier mass.
If one particle is at origin, substitute x=0.
Distance ratio method: d₁/d₂ = m₂/m₁.

COM of Multiple Particles

Use tabular method for 2D/3D particles and symmetry method for regular arrangements.

Particle	Mass	x	y	mx	my
1	1 kg	0	0	0	0
2	2 kg	3	0	6	0
3	3 kg	0	4	0	12
Total	6 kg			6	12

Therefore x_cm=1 m and y_cm=2 m.

COM of Rod

A uniform rod has COM at L/2 by symmetry. For integration, take dm = λ dx.

x_cm = ∫x dm / ∫dm = ∫₀^Lxλdx / ∫₀^Lλdx = L/2

For non-uniform rod, COM shifts toward the denser side.

COM of Ring

For a uniform ring, every mass element has an opposite element, so COM lies at the centre. A semicircular ring has COM on its symmetry axis.

COM of Disc

For a uniform circular lamina or disc, symmetry in all directions places COM at the centre.

Removed Part / Cut-Out COM Formula

Very important: treat removed part as negative mass. Full body is positive mass, removed body is negative mass, and remaining mass is M_F-M_R.

x_cm = (M_Fx_F - M_Rx_R) / (M_F - M_R)y_cm = (M_Fy_F - M_Ry_R) / (M_F - M_R)z_cm = (M_Fz_F - M_Rz_R) / (M_F - M_R)

Always use the same coordinate origin for full body and removed body.

Disc With Circular Hole

Square Plate Corner Removed

Rod With Part Removed

Solved Example: disc of mass M has hole M/4 at x=R/2. Find remaining COM.

Given: full COM at 0, removed COM at R/2, removed mass M/4.
Formula: x=(MxF-mxR)/(M-m).
Calculation: x=(0-(M/4)(R/2))/(3M/4)=-R/6.
Final Answer: COM shifts R/6 away from the hole.

Motion of Centre of Mass

Internal forces do not affect COM motion. The COM behaves as if total external force acts on total mass.

v_cm = Σm_iv_i / Σm_ia_cm = Σm_ia_i / Σm_iF_ext = Ma_cm

Explosion: COM keeps moving as before if external impulse is zero.
Projectile fragments: COM follows projectile path.
Internal forces can separate parts but cannot move system COM by themselves.

Important Formula Table

Use this table as the quick formula sheet for exams.

System	COM Position	Formula	Exam Tip
Two particles	Between masses	(m1x1+m2x2)/(m1+m2)	Closer to heavier mass
Multiple particles	Weighted average	Σmiri/Σmi	Use table
Uniform rod	Midpoint	L/2	Symmetry
Uniform ring	Centre	By symmetry	COM may be empty space
Uniform disc	Centre	By symmetry	Circular lamina
Disc with hole	Away from hole	(MFxF-MRxR)/(MF-MR)	Negative mass
Composite bodies	Component average	ΣMiri/ΣMi	Area as mass for lamina
Motion of COM	System motion	Fext=Macm	Internal forces cancel

Searching for a Physics Tutor? If Centre of Mass, cut-out COM, rigid body COM or NEET/JEE numericals are not clear, contact Kumar Sir.

Phone: +91-9958461445 | Email: kumarsirphysics@gmail.com | Website: kumarphysicsclasses.com

WhatsApp Kumar Sir Visit Website Email Kumar Sir

High-Quality Numericals

Solved bank covering CBSE, NEET, JEE Main, JEE Advanced, IB, IGCSE and A-Level patterns.

1. CBSE two-particle COM: Masses 2 kg and 6 kg are kept at x=0 and x=8 m. Find COM.

Question: Masses 2 kg and 6 kg are kept at x=0 and x=8 m. Find COM.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: x_cm=(2x0+6x8)/8=6 m.
Final Answer: x_cm=(2x0+6x8)/8=6 m.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

2. NEET rod COM: Find COM of uniform rod length 2 m from one end.

Question: Find COM of uniform rod length 2 m from one end.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: x=L/2=1 m.
Final Answer: x=L/2=1 m.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

3. JEE Main cut-out disc: A disc mass M has a hole mass M/4 removed at x=R/2. Find x shift if full COM is origin.

Question: A disc mass M has a hole mass M/4 removed at x=R/2. Find x shift if full COM is origin.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: x=(0-(M/4)(R/2))/(3M/4)=-R/6.
Final Answer: x=(0-(M/4)(R/2))/(3M/4)=-R/6.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

4. JEE Advanced variable rod: Rod density lambda=kx from 0 to L. Find COM.

Question: Rod density lambda=kx from 0 to L. Find COM.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: x_cm=∫x(kx)dx/∫kx dx=2L/3.
Final Answer: x_cm=∫x(kx)dx/∫kx dx=2L/3.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

5. IB motion of COM: Masses 1 kg and 3 kg move with velocities 8 and 0 m/s. Find v_cm.

Question: Masses 1 kg and 3 kg move with velocities 8 and 0 m/s. Find v_cm.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: v_cm=8/4=2 m/s.
Final Answer: v_cm=8/4=2 m/s.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

6. IGCSE balance point: Two equal masses are at 20 cm and 80 cm. Find balance point.

Question: Two equal masses are at 20 cm and 80 cm. Find balance point.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: Midpoint = 50 cm.
Final Answer: Midpoint = 50 cm.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

7. A-Level explosion: A stationary 3 kg body splits into 1 kg moving 6 m/s and 2 kg. Find velocity of 2 kg.

Question: A stationary 3 kg body splits into 1 kg moving 6 m/s and 2 kg. Find velocity of 2 kg.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: 0=1x6+2v, so v=-3 m/s.
Final Answer: 0=1x6+2v, so v=-3 m/s.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

8. Composite lamina: Two square plates masses M and 2M have centres at x=0 and x=9 cm. Find COM.

Question: Two square plates masses M and 2M have centres at x=0 and x=9 cm. Find COM.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: x=(0+18M)/(3M)=6 cm.
Final Answer: x=(0+18M)/(3M)=6 cm.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

9. Square corner removed: Square side a, small square side a/2 removed at lower-left corner. Find coordinate of remaining COM.

Question: Square side a, small square side a/2 removed at lower-left corner. Find coordinate of remaining COM.
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: Full at (a/2,a/2), removed mass M/4 at (a/4,a/4). Result: (7a/12,7a/12).
Final Answer: Full at (a/2,a/2), removed mass M/4 at (a/4,a/4). Result: (7a/12,7a/12).
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

10. Projectile system: A projectile explodes into fragments at highest point. What path does COM follow?

Question: A projectile explodes into fragments at highest point. What path does COM follow?
Diagram: shown above.
Given: Read masses and coordinates from the question with one common origin.
Formula: r_cm=Σm_ir_i/Σm_i; for cut-out, r=(M_Fr_F-M_Rr_R)/(M_F-M_R).
Calculation: The original projectile path because external force is only gravity.
Final Answer: The original projectile path because external force is only gravity.
Exam Tip: COM is closer to heavier mass and shifts away from removed mass.
Common Mistake: Mixing origins or forgetting coordinate signs.

NEET Question Bank

50 NEET-style MCQs. Authentic years are not invented; these are labelled exam-style.

1. NEET Exam-style Question: Masses 2 kg and 3 kg are at x=0 and x=10 m. Find x_cm. A 4 m B 5 m C 6 m D 8 m

Correct Answer: C. x_cm=(2x0+3x10)/5=6 m.
Detailed Explanation: This tests two-particle COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

2. NEET Exam-style Question: Two equal masses are at x=2 m and x=8 m. COM is at: A 2 m B 5 m C 8 m D 10 m

Correct Answer: B. Equal masses give midpoint.
Detailed Explanation: This tests equal masses. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

3. NEET Exam-style Question: COM can lie outside the material body for: A solid sphere B ring C cube D uniform rod

Correct Answer: B. Ring COM is at centre, where there may be no material.
Detailed Explanation: This tests COM outside body. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

4. NEET Exam-style Question: Uniform rod of length L has COM at: A L/4 B L/2 C L D 2L

Correct Answer: B. By symmetry.
Detailed Explanation: This tests rod COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

5. NEET Exam-style Question: Uniform disc COM lies at: A rim B centre C outside D any point

Correct Answer: B. Symmetry gives centre.
Detailed Explanation: This tests disc COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

6. NEET Exam-style Question: In cut-out problems, removed part is treated as: A zero mass B negative mass C double mass D infinite mass

Correct Answer: B. Removed part contributes with minus sign.
Detailed Explanation: This tests negative mass method. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

7. NEET Exam-style Question: For an isolated exploding system, COM motion is affected by: A internal forces B external force C shape only D temperature

Correct Answer: B. Internal forces cancel.
Detailed Explanation: This tests motion of COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

8. NEET Exam-style Question: Equation for COM acceleration is: A F_ext=Ma_cm B F=mv C K=mv D p=Fx

Correct Answer: A.
Detailed Explanation: This tests external force. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

9. NEET Exam-style Question: A mass at x=-2 m must be substituted with coordinate: A +2 B -2 C 0 D ignored

Correct Answer: B. Coordinates are signed.
Detailed Explanation: This tests coordinate sign. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

10. NEET Exam-style Question: For a uniform square plate, COM is at: A corner B centre C edge midpoint D outside

Correct Answer: B. Symmetry.
Detailed Explanation: This tests symmetry. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

11. NEET Exam-style Question: Masses 2 kg and 3 kg are at x=0 and x=10 m. Find x_cm. A 4 m B 5 m C 6 m D 8 m

12. NEET Exam-style Question: Two equal masses are at x=2 m and x=8 m. COM is at: A 2 m B 5 m C 8 m D 10 m

13. NEET Exam-style Question: COM can lie outside the material body for: A solid sphere B ring C cube D uniform rod

14. NEET Exam-style Question: Uniform rod of length L has COM at: A L/4 B L/2 C L D 2L

15. NEET Exam-style Question: Uniform disc COM lies at: A rim B centre C outside D any point

16. NEET Exam-style Question: In cut-out problems, removed part is treated as: A zero mass B negative mass C double mass D infinite mass

17. NEET Exam-style Question: For an isolated exploding system, COM motion is affected by: A internal forces B external force C shape only D temperature

18. NEET Exam-style Question: Equation for COM acceleration is: A F_ext=Ma_cm B F=mv C K=mv D p=Fx

19. NEET Exam-style Question: A mass at x=-2 m must be substituted with coordinate: A +2 B -2 C 0 D ignored

20. NEET Exam-style Question: For a uniform square plate, COM is at: A corner B centre C edge midpoint D outside

21. NEET Exam-style Question: Masses 2 kg and 3 kg are at x=0 and x=10 m. Find x_cm. A 4 m B 5 m C 6 m D 8 m

22. NEET Exam-style Question: Two equal masses are at x=2 m and x=8 m. COM is at: A 2 m B 5 m C 8 m D 10 m

23. NEET Exam-style Question: COM can lie outside the material body for: A solid sphere B ring C cube D uniform rod

24. NEET Exam-style Question: Uniform rod of length L has COM at: A L/4 B L/2 C L D 2L

25. NEET Exam-style Question: Uniform disc COM lies at: A rim B centre C outside D any point

26. NEET Exam-style Question: In cut-out problems, removed part is treated as: A zero mass B negative mass C double mass D infinite mass

27. NEET Exam-style Question: For an isolated exploding system, COM motion is affected by: A internal forces B external force C shape only D temperature

28. NEET Exam-style Question: Equation for COM acceleration is: A F_ext=Ma_cm B F=mv C K=mv D p=Fx

29. NEET Exam-style Question: A mass at x=-2 m must be substituted with coordinate: A +2 B -2 C 0 D ignored

30. NEET Exam-style Question: For a uniform square plate, COM is at: A corner B centre C edge midpoint D outside

31. NEET Exam-style Question: Masses 2 kg and 3 kg are at x=0 and x=10 m. Find x_cm. A 4 m B 5 m C 6 m D 8 m

32. NEET Exam-style Question: Two equal masses are at x=2 m and x=8 m. COM is at: A 2 m B 5 m C 8 m D 10 m

33. NEET Exam-style Question: COM can lie outside the material body for: A solid sphere B ring C cube D uniform rod

34. NEET Exam-style Question: Uniform rod of length L has COM at: A L/4 B L/2 C L D 2L

35. NEET Exam-style Question: Uniform disc COM lies at: A rim B centre C outside D any point

36. NEET Exam-style Question: In cut-out problems, removed part is treated as: A zero mass B negative mass C double mass D infinite mass

37. NEET Exam-style Question: For an isolated exploding system, COM motion is affected by: A internal forces B external force C shape only D temperature

38. NEET Exam-style Question: Equation for COM acceleration is: A F_ext=Ma_cm B F=mv C K=mv D p=Fx

39. NEET Exam-style Question: A mass at x=-2 m must be substituted with coordinate: A +2 B -2 C 0 D ignored

40. NEET Exam-style Question: For a uniform square plate, COM is at: A corner B centre C edge midpoint D outside

41. NEET Exam-style Question: Masses 2 kg and 3 kg are at x=0 and x=10 m. Find x_cm. A 4 m B 5 m C 6 m D 8 m

42. NEET Exam-style Question: Two equal masses are at x=2 m and x=8 m. COM is at: A 2 m B 5 m C 8 m D 10 m

43. NEET Exam-style Question: COM can lie outside the material body for: A solid sphere B ring C cube D uniform rod

44. NEET Exam-style Question: Uniform rod of length L has COM at: A L/4 B L/2 C L D 2L

45. NEET Exam-style Question: Uniform disc COM lies at: A rim B centre C outside D any point

46. NEET Exam-style Question: In cut-out problems, removed part is treated as: A zero mass B negative mass C double mass D infinite mass

47. NEET Exam-style Question: For an isolated exploding system, COM motion is affected by: A internal forces B external force C shape only D temperature

48. NEET Exam-style Question: Equation for COM acceleration is: A F_ext=Ma_cm B F=mv C K=mv D p=Fx

49. NEET Exam-style Question: A mass at x=-2 m must be substituted with coordinate: A +2 B -2 C 0 D ignored

50. NEET Exam-style Question: For a uniform square plate, COM is at: A corner B centre C edge midpoint D outside

JEE Main Question Bank

50 difficult JEE Main-style questions on cut-out COM, composite bodies, motion of COM and symmetry.

1. JEE Main Exam-style Question: A disc of radius R has a circular hole of radius R/2 cut at x=R/2. Where does remaining COM shift?

Correct Answer: Away from the removed hole. Use x=(M xF - m xR)/(M-m), with m=M/4.
Detailed Explanation: This tests cut-out COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

2. JEE Main Exam-style Question: Two rods of equal length L and masses m,2m are joined at right angle. Find COM idea.

Correct Answer: Find each rod COM at L/2 from joint, then use weighted average of two point masses m and 2m.
Detailed Explanation: This tests composite body. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

3. JEE Main Exam-style Question: Two particles have masses 1 kg, 3 kg and velocities 4i, -2i m/s. Find v_cm.

Correct Answer: v_cm=(1x4+3x(-2))/4=-0.5 m/s.
Detailed Explanation: This tests motion of COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

4. JEE Main Exam-style Question: Three equal masses are at vertices of an equilateral triangle. COM location?

Correct Answer: At centroid of triangle.
Detailed Explanation: This tests symmetry method. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

5. JEE Main Exam-style Question: Masses 1,2,3 kg are at x=0,2,4 m. Find x_cm.

Correct Answer: x_cm=(0+4+12)/6=8/3 m.
Detailed Explanation: This tests multiple particles. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

6. JEE Main Exam-style Question: Uniform rod from x=a to x=b has COM at:

Correct Answer: (a+b)/2.
Detailed Explanation: This tests rod. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

7. JEE Main Exam-style Question: If a central hole is removed from a uniform disc, COM is:

Correct Answer: Still at original centre by symmetry.
Detailed Explanation: This tests disc hole. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

8. JEE Main Exam-style Question: A projectile explodes midair without external impulse beyond gravity. COM follows:

Correct Answer: Original projectile trajectory under gravity.
Detailed Explanation: This tests projectile COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

9. JEE Main Exam-style Question: For rod with density proportional to x from 0 to L, COM is:

Correct Answer: x_cm=2L/3.
Detailed Explanation: This tests variable density rod. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

10. JEE Main Exam-style Question: Changing origin changes coordinates but physical COM:

Correct Answer: Remains same physical point.
Detailed Explanation: This tests coordinate origin. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

11. JEE Main Exam-style Question: A disc of radius R has a circular hole of radius R/2 cut at x=R/2. Where does remaining COM shift?

12. JEE Main Exam-style Question: Two rods of equal length L and masses m,2m are joined at right angle. Find COM idea.

13. JEE Main Exam-style Question: Two particles have masses 1 kg, 3 kg and velocities 4i, -2i m/s. Find v_cm.

14. JEE Main Exam-style Question: Three equal masses are at vertices of an equilateral triangle. COM location?

15. JEE Main Exam-style Question: Masses 1,2,3 kg are at x=0,2,4 m. Find x_cm.

16. JEE Main Exam-style Question: Uniform rod from x=a to x=b has COM at:

17. JEE Main Exam-style Question: If a central hole is removed from a uniform disc, COM is:

18. JEE Main Exam-style Question: A projectile explodes midair without external impulse beyond gravity. COM follows:

19. JEE Main Exam-style Question: For rod with density proportional to x from 0 to L, COM is:

20. JEE Main Exam-style Question: Changing origin changes coordinates but physical COM:

21. JEE Main Exam-style Question: A disc of radius R has a circular hole of radius R/2 cut at x=R/2. Where does remaining COM shift?

22. JEE Main Exam-style Question: Two rods of equal length L and masses m,2m are joined at right angle. Find COM idea.

23. JEE Main Exam-style Question: Two particles have masses 1 kg, 3 kg and velocities 4i, -2i m/s. Find v_cm.

24. JEE Main Exam-style Question: Three equal masses are at vertices of an equilateral triangle. COM location?

25. JEE Main Exam-style Question: Masses 1,2,3 kg are at x=0,2,4 m. Find x_cm.

26. JEE Main Exam-style Question: Uniform rod from x=a to x=b has COM at:

27. JEE Main Exam-style Question: If a central hole is removed from a uniform disc, COM is:

28. JEE Main Exam-style Question: A projectile explodes midair without external impulse beyond gravity. COM follows:

29. JEE Main Exam-style Question: For rod with density proportional to x from 0 to L, COM is:

30. JEE Main Exam-style Question: Changing origin changes coordinates but physical COM:

31. JEE Main Exam-style Question: A disc of radius R has a circular hole of radius R/2 cut at x=R/2. Where does remaining COM shift?

32. JEE Main Exam-style Question: Two rods of equal length L and masses m,2m are joined at right angle. Find COM idea.

33. JEE Main Exam-style Question: Two particles have masses 1 kg, 3 kg and velocities 4i, -2i m/s. Find v_cm.

34. JEE Main Exam-style Question: Three equal masses are at vertices of an equilateral triangle. COM location?

35. JEE Main Exam-style Question: Masses 1,2,3 kg are at x=0,2,4 m. Find x_cm.

36. JEE Main Exam-style Question: Uniform rod from x=a to x=b has COM at:

37. JEE Main Exam-style Question: If a central hole is removed from a uniform disc, COM is:

38. JEE Main Exam-style Question: A projectile explodes midair without external impulse beyond gravity. COM follows:

39. JEE Main Exam-style Question: For rod with density proportional to x from 0 to L, COM is:

40. JEE Main Exam-style Question: Changing origin changes coordinates but physical COM:

41. JEE Main Exam-style Question: A disc of radius R has a circular hole of radius R/2 cut at x=R/2. Where does remaining COM shift?

42. JEE Main Exam-style Question: Two rods of equal length L and masses m,2m are joined at right angle. Find COM idea.

43. JEE Main Exam-style Question: Two particles have masses 1 kg, 3 kg and velocities 4i, -2i m/s. Find v_cm.

44. JEE Main Exam-style Question: Three equal masses are at vertices of an equilateral triangle. COM location?

45. JEE Main Exam-style Question: Masses 1,2,3 kg are at x=0,2,4 m. Find x_cm.

46. JEE Main Exam-style Question: Uniform rod from x=a to x=b has COM at:

47. JEE Main Exam-style Question: If a central hole is removed from a uniform disc, COM is:

48. JEE Main Exam-style Question: A projectile explodes midair without external impulse beyond gravity. COM follows:

49. JEE Main Exam-style Question: For rod with density proportional to x from 0 to L, COM is:

50. JEE Main Exam-style Question: Changing origin changes coordinates but physical COM:

JEE Advanced Question Bank

50 advanced questions on removed mass method, composite lamina, explosions, projectile COM and variable density rods.

1. JEE Advanced Exam-style Question: A uniform square plate side a has a square of side a/2 removed from one corner. Set up remaining COM.

Correct Answer: Treat full square mass M at (a/2,a/2), removed mass M/4 at (a/4,a/4). Remaining COM = [(M full COM)-(M/4 removed COM)]/(3M/4).
Detailed Explanation: This tests removed mass method. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

2. JEE Advanced Exam-style Question: A semicircular lamina is attached to a rectangle. What is the method?

Correct Answer: Use known COM of each part and area as mass for uniform lamina.
Detailed Explanation: This tests composite lamina. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

3. JEE Advanced Exam-style Question: Three masses m,2m,3m are at (0,0),(a,0),(0,a). Find COM.

Correct Answer: x=a/3, y=a/2.
Detailed Explanation: This tests system of particles. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

4. JEE Advanced Exam-style Question: A shell at rest explodes into masses m and 2m. If m moves with v, velocity of 2m?

Correct Answer: 2m body has speed v/2 opposite direction.
Detailed Explanation: This tests explosion problems. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

5. JEE Advanced Exam-style Question: A projectile splits into fragments in air. What path does COM follow?

Correct Answer: The same parabolic path dictated by external gravity.
Detailed Explanation: This tests projectile COM. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

6. JEE Advanced Exam-style Question: For lambda=kx² on 0 to L, x_cm?

Correct Answer: x_cm=3L/4.
Detailed Explanation: This tests variable density rod. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

7. JEE Advanced Exam-style Question: Disc radius R has small hole mass m at coordinate x=a removed from full mass M. Formula?

Correct Answer: x=(0 - ma)/(M-m), if full disc COM is origin.
Detailed Explanation: This tests hole in disc. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

8. JEE Advanced Exam-style Question: Two equal holes at x=+a and x=-a are removed. COM shift?

Correct Answer: No x-shift by symmetry.
Detailed Explanation: This tests negative coordinates. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

9. JEE Advanced Exam-style Question: Internal spring pushes two blocks apart on smooth table. COM acceleration?

Correct Answer: Zero if no external horizontal force.
Detailed Explanation: This tests relative motion. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

10. JEE Advanced Exam-style Question: Why same origin is mandatory in cut-out method?

Correct Answer: Weighted averages are algebraic; mixing origins makes coordinates incompatible.
Detailed Explanation: This tests axis choice. Choose one origin, write the weighted average formula, substitute signed coordinates, and simplify. For removed parts, use negative mass with the same origin.

11. JEE Advanced Exam-style Question: A uniform square plate side a has a square of side a/2 removed from one corner. Set up remaining COM.

12. JEE Advanced Exam-style Question: A semicircular lamina is attached to a rectangle. What is the method?

13. JEE Advanced Exam-style Question: Three masses m,2m,3m are at (0,0),(a,0),(0,a). Find COM.

14. JEE Advanced Exam-style Question: A shell at rest explodes into masses m and 2m. If m moves with v, velocity of 2m?

15. JEE Advanced Exam-style Question: A projectile splits into fragments in air. What path does COM follow?

16. JEE Advanced Exam-style Question: For lambda=kx² on 0 to L, x_cm?

17. JEE Advanced Exam-style Question: Disc radius R has small hole mass m at coordinate x=a removed from full mass M. Formula?

18. JEE Advanced Exam-style Question: Two equal holes at x=+a and x=-a are removed. COM shift?

19. JEE Advanced Exam-style Question: Internal spring pushes two blocks apart on smooth table. COM acceleration?

20. JEE Advanced Exam-style Question: Why same origin is mandatory in cut-out method?

21. JEE Advanced Exam-style Question: A uniform square plate side a has a square of side a/2 removed from one corner. Set up remaining COM.

22. JEE Advanced Exam-style Question: A semicircular lamina is attached to a rectangle. What is the method?

23. JEE Advanced Exam-style Question: Three masses m,2m,3m are at (0,0),(a,0),(0,a). Find COM.

24. JEE Advanced Exam-style Question: A shell at rest explodes into masses m and 2m. If m moves with v, velocity of 2m?

25. JEE Advanced Exam-style Question: A projectile splits into fragments in air. What path does COM follow?

26. JEE Advanced Exam-style Question: For lambda=kx² on 0 to L, x_cm?

27. JEE Advanced Exam-style Question: Disc radius R has small hole mass m at coordinate x=a removed from full mass M. Formula?

28. JEE Advanced Exam-style Question: Two equal holes at x=+a and x=-a are removed. COM shift?

29. JEE Advanced Exam-style Question: Internal spring pushes two blocks apart on smooth table. COM acceleration?

30. JEE Advanced Exam-style Question: Why same origin is mandatory in cut-out method?

31. JEE Advanced Exam-style Question: A uniform square plate side a has a square of side a/2 removed from one corner. Set up remaining COM.

32. JEE Advanced Exam-style Question: A semicircular lamina is attached to a rectangle. What is the method?

33. JEE Advanced Exam-style Question: Three masses m,2m,3m are at (0,0),(a,0),(0,a). Find COM.

34. JEE Advanced Exam-style Question: A shell at rest explodes into masses m and 2m. If m moves with v, velocity of 2m?

35. JEE Advanced Exam-style Question: A projectile splits into fragments in air. What path does COM follow?

36. JEE Advanced Exam-style Question: For lambda=kx² on 0 to L, x_cm?

37. JEE Advanced Exam-style Question: Disc radius R has small hole mass m at coordinate x=a removed from full mass M. Formula?

38. JEE Advanced Exam-style Question: Two equal holes at x=+a and x=-a are removed. COM shift?

39. JEE Advanced Exam-style Question: Internal spring pushes two blocks apart on smooth table. COM acceleration?

40. JEE Advanced Exam-style Question: Why same origin is mandatory in cut-out method?

41. JEE Advanced Exam-style Question: A uniform square plate side a has a square of side a/2 removed from one corner. Set up remaining COM.

42. JEE Advanced Exam-style Question: A semicircular lamina is attached to a rectangle. What is the method?

43. JEE Advanced Exam-style Question: Three masses m,2m,3m are at (0,0),(a,0),(0,a). Find COM.

44. JEE Advanced Exam-style Question: A shell at rest explodes into masses m and 2m. If m moves with v, velocity of 2m?

45. JEE Advanced Exam-style Question: A projectile splits into fragments in air. What path does COM follow?

46. JEE Advanced Exam-style Question: For lambda=kx² on 0 to L, x_cm?

47. JEE Advanced Exam-style Question: Disc radius R has small hole mass m at coordinate x=a removed from full mass M. Formula?

48. JEE Advanced Exam-style Question: Two equal holes at x=+a and x=-a are removed. COM shift?

49. JEE Advanced Exam-style Question: Internal spring pushes two blocks apart on smooth table. COM acceleration?

50. JEE Advanced Exam-style Question: Why same origin is mandatory in cut-out method?

IB / IGCSE / A-Level Questions

Separate international banks with answers and explanations.

IB Questions

IB 1. Define centre of mass.

Point whose motion represents average mass position of a system.

IB 2. Why do internal forces not affect COM motion?

They occur in equal and opposite pairs and cancel in total force.

IB 3. State x-coordinate formula for COM.

x_cm=Σm_ix_i/Σm_i.

IB 4. COM of uniform rod.

At midpoint.

IB 5. COM of uniform ring.

At centre.

IB 6. COM of uniform disc.

At centre.

IB 7. Can COM lie outside object?

Yes, for ring or crescent-like bodies.

IB 8. What is removed mass method?

Treat removed part as negative mass.

IB 9. COM velocity formula.

v_cm=Σm_iv_i/Σm_i.

IB 10. Relation between external force and COM.

F_ext=Ma_cm.

IB 11. Why use symmetry?

Symmetry locates COM without integration.

IB 12. Equal masses at ends of rod-like line.

COM is midpoint.

IB 13. Explosion in space with no external force.

COM moves with constant velocity.

IB 14. Projectile breaks into fragments.

COM continues projectile trajectory.

IB 15. Difference between COM and geometric centre.

COM depends on mass distribution; geometric centre depends on shape.

IB 16. Uniform density means?

Mass per unit length/area/volume is constant.

IB 17. Non-uniform rod COM shifts toward?

Heavier side.

IB 18. A symmetric circular lamina COM.

At centre.

IB 19. Cut-out hole shifts COM away from?

The removed part.

IB 20. What is total mass in denominator?

Sum of all actual masses; for cut-outs full mass minus removed mass.

IB 21. Use of COM in rigid body motion.

Separates translation of COM from rotation about COM.

IB 22. A two-particle COM is closer to which mass?

The heavier mass.

IB 23. Unit of COM coordinate.

Metre.

IB 24. Does COM need material there?

No.

IB 25. Main condition for constant COM velocity.

Zero net external force.

IGCSE Questions

IGCSE 1. What is centre of mass?

The point where the mass of an object may be considered concentrated.

IGCSE 2. COM of a uniform ruler.

At the middle.

IGCSE 3. COM of a uniform circular disc.

At the centre.

IGCSE 4. Why does a body balance at COM?

Clockwise and anticlockwise moments balance.

IGCSE 5. A ring COM is where?

At its centre.

IGCSE 6. Can COM be in empty space?

Yes, for a ring.

IGCSE 7. Heavier side shifts COM where?

Toward heavier side.

IGCSE 8. What happens to COM if a corner is removed?

It shifts away from removed corner.

IGCSE 9. Two equal masses at 0 and 10 cm. COM?

5 cm.

IGCSE 10. Two masses 1 kg and 3 kg at 0 and 8 cm. COM?

6 cm.

IGCSE 11. Why use same origin?

So positions can be averaged correctly.

IGCSE 12. Stable object has lower or higher COM?

Lower COM is usually more stable.

IGCSE 13. Wide base improves?

Stability.

IGCSE 14. COM of symmetric square.

Centre.

IGCSE 15. COM of triangle basic idea.

At centroid for uniform triangular lamina.

IGCSE 16. Does internal force move COM?

No, not by itself.

IGCSE 17. External force changes COM?

Yes.

IGCSE 18. COM of moving system depends on?

Masses and velocities for v_cm.

IGCSE 19. Unit of mass in formula.

kg.

IGCSE 20. Unit of coordinate.

IGCSE 21. What is a system of particles?

A collection of particles considered together.

IGCSE 22. Uniform rod of 2 m COM from one end.

1 m.

IGCSE 23. Removed part is counted as?

Negative mass in formula method.

IGCSE 24. COM of two equal masses at opposite sides of origin.

At origin.

IGCSE 25. Balance point of irregular lamina can be found by?

Suspension/plumb-line method.

A-Level Questions

A-Level 1. Derive x<sub>cm</sub> for discrete masses.

Use weighted average x_cm=Σmx/Σm.

A-Level 2. COM of continuous rod formula.

x_cm=∫x dm/∫dm.

A-Level 3. Uniform rod integration result.

x_cm=L/2.

A-Level 4. Rod with lambda=kx result.

x_cm=2L/3.

A-Level 5. Rod with lambda=kx<sup>2</sup> result.

x_cm=3L/4.

A-Level 6. COM of semicircular ring from centre.

2R/π along symmetry axis.

A-Level 7. COM of semicircular lamina from centre.

4R/(3π) along symmetry axis.

A-Level 8. Negative mass formula.

r=(M_Fr_F-M_Rr_R)/(M_F-M_R).

A-Level 9. COM velocity relation.

V_cm=P_total/M.

A-Level 10. COM acceleration relation.

A_cm=F_ext/M.

A-Level 11. Explosion with no external force.

COM velocity remains constant.

A-Level 12. Projectile fragmentation.

COM follows projectile path.

A-Level 13. Composite lamina mass proportional to?

Area for uniform surface density.

A-Level 14. Composite wire mass proportional to?

Length for uniform linear density.

A-Level 15. Hollow sphere COM.

At centre by symmetry.

A-Level 16. Shift due to removed hole.

Away from removed mass.

A-Level 17. Same origin condition.

Coordinates must refer to one coordinate system.

A-Level 18. COM of two masses distance division.

Distances from COM are inversely proportional to masses.

A-Level 19. Internal forces and Newton's third law.

They cancel in total force equation.

A-Level 20. Rigid body motion decomposition.

Translation of COM plus rotation about COM.

A-Level 21. COM can be outside object example.

Ring, hoop or boomerang-like shape.

A-Level 22. Non-uniform density effect.

COM shifts toward denser region.

A-Level 23. Centre of gravity vs COM.

They coincide in uniform gravitational field.

A-Level 24. Total momentum of system.

P=Mv_cm.

A-Level 25. External impulse relation.

J_ext=ΔP_system=MΔv_cm.

Assertion Reason

30 assertion-reason questions for COM concepts and cut-out logic.

1. Assertion: COM of two equal masses lies at midpoint. Reason: Weighted average gives equal weight to both positions.

Answer: Both true; reason explains assertion.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

2. Assertion: COM always lies inside the body. Reason: Ring has COM at centre.

Answer: Assertion false; reason shows counterexample.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

3. Assertion: Removed part can be treated as negative mass. Reason: Remaining body equals full body minus removed body.

Answer: Both true; reason explains assertion.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

4. Assertion: Internal forces do not change COM motion. Reason: Internal forces cancel in action-reaction pairs.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

5. Assertion: COM of uniform rod is at L/2. Reason: Uniform rod has symmetry about midpoint.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

6. Assertion: COM and geometric centre are always same. Reason: Mass distribution may be non-uniform.

Answer: Assertion false; reason true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

7. Assertion: COM shifts toward heavier side. Reason: Weighted average gives more importance to larger mass.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

8. Assertion: Projectile fragments have COM following projectile path. Reason: Gravity is the external force on the system.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

9. Assertion: F_ext=Ma_cm. Reason: Total internal force of system is zero.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

10. Assertion: Using different origins in cut-out formula is allowed. Reason: Coordinates are relative to origin.

Answer: Assertion false; reason true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

11. Assertion: COM of two equal masses lies at midpoint. Reason: Weighted average gives equal weight to both positions.

Answer: Both true; reason explains assertion.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

12. Assertion: COM always lies inside the body. Reason: Ring has COM at centre.

Answer: Assertion false; reason shows counterexample.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

13. Assertion: Removed part can be treated as negative mass. Reason: Remaining body equals full body minus removed body.

Answer: Both true; reason explains assertion.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

14. Assertion: Internal forces do not change COM motion. Reason: Internal forces cancel in action-reaction pairs.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

15. Assertion: COM of uniform rod is at L/2. Reason: Uniform rod has symmetry about midpoint.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

16. Assertion: COM and geometric centre are always same. Reason: Mass distribution may be non-uniform.

Answer: Assertion false; reason true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

17. Assertion: COM shifts toward heavier side. Reason: Weighted average gives more importance to larger mass.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

18. Assertion: Projectile fragments have COM following projectile path. Reason: Gravity is the external force on the system.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

19. Assertion: F_ext=Ma_cm. Reason: Total internal force of system is zero.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

20. Assertion: Using different origins in cut-out formula is allowed. Reason: Coordinates are relative to origin.

Answer: Assertion false; reason true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

21. Assertion: COM of two equal masses lies at midpoint. Reason: Weighted average gives equal weight to both positions.

Answer: Both true; reason explains assertion.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

22. Assertion: COM always lies inside the body. Reason: Ring has COM at centre.

Answer: Assertion false; reason shows counterexample.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

23. Assertion: Removed part can be treated as negative mass. Reason: Remaining body equals full body minus removed body.

Answer: Both true; reason explains assertion.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

24. Assertion: Internal forces do not change COM motion. Reason: Internal forces cancel in action-reaction pairs.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

25. Assertion: COM of uniform rod is at L/2. Reason: Uniform rod has symmetry about midpoint.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

26. Assertion: COM and geometric centre are always same. Reason: Mass distribution may be non-uniform.

Answer: Assertion false; reason true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

27. Assertion: COM shifts toward heavier side. Reason: Weighted average gives more importance to larger mass.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

28. Assertion: Projectile fragments have COM following projectile path. Reason: Gravity is the external force on the system.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

29. Assertion: F_ext=Ma_cm. Reason: Total internal force of system is zero.

Answer: Both true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

30. Assertion: Using different origins in cut-out formula is allowed. Reason: Coordinates are relative to origin.

Answer: Assertion false; reason true.
Explanation: Check whether assertion and reason are individually true, then decide whether reason correctly explains assertion.

Case Study Questions

Case studies on disc with hole, composite plate, explosion, projectile system and two-particle system.

Case Study: Disc with hole

Passage: A uniform disc has a circular hole removed from one side. Questions: identify full body COM, removed part COM, remaining mass, shift direction and formula.
Answers and Explanations: Use negative mass. COM shifts away from hole. Same origin is compulsory.

Case Study: Composite plate

Passage: A rectangle and semicircle are joined to make a plate. Questions: choose components, use area masses, find x and y coordinates, use symmetry and locate final COM.
Answers and Explanations: Treat each shape as a mass at its own COM and use weighted average.

Case Study: Explosion into fragments

Passage: A shell explodes into fragments on a smooth surface. Questions: conserved quantity, COM velocity, fragment momentum, role of internal forces and external impulse.
Answers and Explanations: Total momentum and COM velocity stay constant if external impulse is zero.

Case Study: Projectile system

Passage: A projectile breaks into pieces in air. Questions: trajectory of COM, force on system, effect of explosion, horizontal motion and vertical acceleration.
Answers and Explanations: COM continues the original projectile trajectory under gravity.

Case Study: Two-particle system

Passage: Two particles of unequal masses move along a line. Questions: position of COM, velocity of COM, heavier side, internal force effect and total momentum.
Answers and Explanations: Use weighted averages; COM is closer to the heavier mass.

Common Student Mistakes

Avoid these errors in NEET, JEE, CBSE and international exam questions.

Different Origins

Use the same origin for full and removed bodies. Coordinates from different origins cannot be subtracted.

Forgetting Negative Mass

In a cut-out, the removed part is not ignored; it is subtracted as negative mass.

Geometric Centre Confusion

COM equals geometric centre only for uniform symmetric bodies.

COM Outside Body

Rings and cut-out bodies may have COM in empty space.

External Force Concept

Internal forces do not accelerate COM; only net external force does.

Wrong Signs

Coordinates left/below the origin must be negative if the axis choice says so.