Centre of MassJEE Main

Concept

The centre of mass is the mass-weighted average position of a system — the single point that moves as if all the mass were concentrated there. For two bodies it always lies closer to the heavier one, on the line joining them.

Key formula

Derivation

Placing $m_1$ at $x=0$ and $m_2$ at $x=d$, the weighted mean is $x_{cm} = m_2 d/(m_1+m_2)$ — measured from $m_1$.

Rearranging shows the lever rule: $m_1 r_1 = m_2 r_2$, where $r_1,r_2$ are the distances from the COM. Equal masses put the COM exactly midway; the heavier mass pulls it toward itself.

Scenarios to explore

• Centre of Mass — The mass-weighted balance point and lever rule.

Real-world applications

• Balancing seesaws and weighing scales.
• Projectile and rocket motion (the COM follows the trajectory).
• Stability of vehicles and structures.

JEE exam tips

• Lever rule $m_1r_1 = m_2r_2$ — the heavier mass sits the shorter distance from the COM.
• Internal forces never move the COM; only external forces do.
• In the absence of external force the COM moves with constant velocity.

Common mistakes

• Placing the COM midway regardless of the masses — it shifts toward the heavier body.
• Forgetting the COM lies on the line joining the masses.
• Confusing centre of mass with centre of gravity (identical only in a uniform field).

Exam traps to avoid

• Two equal masses: COM is exactly halfway, independent of the separation.
• When one mass is much larger, the COM is essentially at that mass.