Lesson 6 – Centre of Mass & System of Particles
Stage 2 : Mathematical Applications
(Problem Practice – Page 7)

In this page, we apply the mathematical formulas of centre of mass learned in Stage 2 to solve exam-oriented problems. These problems are frequently asked in Intermediate exams and form the base for IIT-JEE numericals.

Problem 1: Two-Particle System (1D)

Question:
Two particles of masses 4 kg and 6 kg are placed at positions x = −2 m and x = 3 m respectively. Find the position of the centre of mass.

Solution:

x_cm = (4×(−2) + 6×3) / (4 + 6)

x_cm = (−8 + 18) / 10 = 10 / 10 = 1 m

Therefore, the centre of mass is at x = 1 m.

Problem 2: Three-Particle System

Question:
Three particles of masses 1 kg, 2 kg and 3 kg are located at x = 0 m, x = 2 m and x = 5 m respectively. Find the position of centre of mass.

Solution:

x_cm = (1×0 + 2×2 + 3×5) / (1+2+3)

x_cm = (0 + 4 + 15) / 6 = 19 / 6 ≈ 3.17 m

Problem 3: Centre of Mass in 2D

Question:
Two particles of masses 2 kg and 4 kg are located at (1, 2) and (5, 6) respectively. Find the coordinates of the centre of mass.

Solution:

x_cm = (2×1 + 4×5) / 6 = (2 + 20)/6 = 22/6 ≈ 3.67

y_cm = (2×2 + 4×6) / 6 = (4 + 24)/6 = 28/6 ≈ 4.67

Centre of mass = (3.67 , 4.67)

Problem 4: Uniform Rod

Question:
A uniform rod of length 1.6 m is placed along the x-axis with one end at x = 0. Find the position of its centre of mass.

Solution:

For a uniform rod, x_cm = L / 2

x_cm = 1.6 / 2 = 0.8 m

Problem 5: Exam-Type Concept Question

Question:
Why does the centre of mass of a ring lie at its centre even though there is no mass at that point?

Answer:

The centre of mass depends on mass distribution. Due to symmetry, mass is evenly distributed around the centre, so the weighted average position lies at the centre.

Common Mistakes (Very Important)

• Forgetting negative signs in coordinates
• Not dividing by total mass
• Mixing x and y calculations
• Skipping units in final answer

Stage 2 – Application Recap

✔ COM formula works for any number of particles
✔ Heavier masses dominate COM position
✔ 2D problems require separate x and y treatment
✔ These problems are guaranteed in exams

Next Transition

In the next stage, we will study the motion of centre of mass, its relation with external forces, and conservation of momentum. This is a very high-weightage IIT topic.