11. Consider a system of two point particles with particles having distinct masses m1 and m2. If the
first particle is pushed towards the centre of mass through a distance d, by what distance should the
second particle be moved, so as to keep the centre of mass at the same position?
a.(m1/m2)*d
b.d
c.((m1+m2)/m2)*d
d.(m2/m1)*d
Center of Mass Conservation Principle
For two particles with masses m₁ and m₂, center of mass position is x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂). To keep x_cm fixed when particle 1 moves towards CM by distance d (displacement -Δx₁ = -d), particle 2 must move by Δx₂.
Key Concept: Conservation requires m₁(-Δx₁) + m₂Δx₂ = 0, so m₁d = m₂|Δx₂|, yielding Δx₂ = (m₁/m₂)d away from CM.
Option Analysis
- a. (m1/m2)*d: Correct. CM balance equation m₁d = m₂x₂ gives exact ratio for position conservation.
- b. d: Incorrect. Equal displacement violates inverse mass proportionality of CM motion.
- c. ((m1+m2)/m2)*d: Incorrect. Overcomplicates with total mass; ignores simple lever rule balance.
- d. (m2/m1)*d: Incorrect. Inverse ratio would shift CM further, not maintain position.
Physical Insight: The lighter particle moves farther to balance the heavier particle’s smaller displacement, maintaining CM position.
Mathematical Derivation
Initial CM: x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂)
After displacement: x_cm’ = [m₁(x₁ – d) + m₂(x₂ + Δx₂)]/(m₁ + m₂)
Set equal: m₁d = m₂Δx₂ → Δx₂ = (m₁/m₂)d
