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Formulas/physics/Centre Of Mass/Acceleration of Centre of Mass

Acceleration of Centre of Mass

The CM of a system accelerates as if all external forces act on total mass M at the CM.
Class 11Class JEE
Derivation

Derivation

Differentiate the CM velocity with respect to time:

v⃗cm=∑miv⃗iM\vec{v}_{cm} = \frac{\sum m_i \vec{v}_i}{M}

a⃗cm=dv⃗cmdt=∑mia⃗iM\vec{a}_{cm} = \frac{d\vec{v}_{cm}}{dt} = \frac{\sum m_i \vec{a}_i}{M}

From Newton's Second Law for each particle:

mia⃗i=F⃗iext+F⃗iintm_i \vec{a}_i = \vec{F}_i^{ext} + \vec{F}_i^{int}

Summing over all particles:

∑mia⃗i=∑F⃗iext+∑F⃗iint\sum m_i \vec{a}_i = \sum \vec{F}_i^{ext} + \sum \vec{F}_i^{int}

Internal forces cancel in pairs by Newton's Third Law: ∑F⃗iint=0\sum \vec{F}_i^{int} = 0

Ma⃗cm=∑F⃗iext=F⃗extM\vec{a}_{cm} = \sum \vec{F}_i^{ext} = \vec{F}_{ext}

F⃗ext=Ma⃗cm\boxed{\vec{F}_{ext} = M\vec{a}_{cm}}

What this means

The CM of any system — however complex its internal structure — obeys Newton's Second Law as if it were a single particle of mass MM subject to the total external force.

Internal forces (between parts of the system) have absolutely no effect on the CM motion. Only external forces matter.

Examples

Exploding shell: A shell fired at angle θ\theta with speed uu explodes in mid-air. The only external force is gravity (−Mgj^-Mg\hat{j}). So a⃗cm=−gj^\vec{a}_{cm} = -g\hat{j} — the CM continues on the original parabolic path. The fragments fly in all directions, but their CM follows the undisturbed trajectory.

Person jumping on a cart: A person and cart form a system. If no external horizontal force acts, the CM moves at constant velocity horizontally. When the person walks forward, the cart moves back to keep the CM fixed.

Binary star system: Two stars orbit their common CM. External gravitational forces (from other stars) cause the CM to accelerate, but the internal gravitational force between the two stars does not affect the CM.

CM and Newton's laws

This result elevates the CM concept beyond a geometric point. It shows that the CM is the natural "particle" representation of any extended body or system:

  • F⃗ext=Ma⃗cm\vec{F}_{ext} = M\vec{a}_{cm} — Newton's Second Law
  • v⃗cm=const\vec{v}_{cm} = \text{const} when F⃗ext=0\vec{F}_{ext} = 0 — Newton's First Law
  • Momentum p⃗=Mv⃗cm\vec{p} = M\vec{v}_{cm}

All of classical mechanics for a point particle applies to the CM of any system.

Key Idea
$\vec{F}_{ext} = M\vec{a}_{cm}$ applies to the whole system. Do not use it for individual parts. Each part obeys its own Newton's Second Law with its own forces — including internal forces. Only the whole system's CM obeys this clean equation.