Unit 5 · Topic 05

Center of mass and system motion

The center of mass summarizes mass distribution and collective motion. It may lie in empty space and responds only to net external force.

Unit 5Center of massOpen navigation

Concept 01

A mass-weighted average

Essential The minimum you should retain

Center-of-mass position is a weighted average: larger masses have more influence.

UnderstandInterpret and connect

For particles, r_cm=Σm_i r_i/M.

DeepenFormulation and conditions

Each component is averaged separately and M=Σm_i.

ExploreConnections for further study

The result may lie where there is no material, such as the center of a ring.

Worked example

Center of mass in the plane

Three particles: 1 kg at (0,0), 2 kg at (3,0), and 1 kg at (0,4) m.

Given
  • m_1=1 kg
  • m_2=2 kg
  • m_3=1 kg
Target

Calculate (x_cm,y_cm).

  1. Total mass

    M=4 kg.

  2. x component

    x_cm=[1(0)+2(3)+1(0)]/4=1.50 m.

  3. y component

    y_cm=[1(0)+2(0)+1(4)]/4=1.00 m.

  4. Interpretation

    The center of mass need not coincide with a particle.

Conclusion

(x_cm,y_cm)=(1.50 m,1.00 m).

Mathematical relation

Center-of-mass position

rcm=ΣmiriM \vec r_{cm}=\sum_i m_i\vec r_i/M
Represents

Mass-weighted average of positions.

Physical interpretation

It may lie where there is no material.

DeepenVariables, conditions, and checks

Variables

r_cm
center-of-mass position; usual unit: m
M
total mass; usual unit: kg

Conditions of application

  • M>0.

Dimensional check

m.

Two unequal masses and their center of mass on a line.2 kg5 kgx_cm

The center of mass lies closer to the larger mass and between the endpoints for positive masses.

Concept 02

Collective momentum

Essential The minimum you should retain

For constant total mass, P=Mv_cm P=Mv_{cm} .

UnderstandInterpret and connect

Center-of-mass velocity is the mass-weighted average of particle velocities.

DeepenFormulation and conditions

Internal motions can be complex while v_cm remains simple.

ExploreConnections for further study

A center-of-mass frame has zero total momentum and helps analyze collisions.

Concept 03

The resultant governs

Essential The minimum you should retain

For constant total mass, F_ext=M a_cm F_{ext}=Ma_{cm} .

UnderstandInterpret and connect

Internal forces change relative motion but cancel in the total-momentum balance.

DeepenFormulation and conditions

If F_ext=0, v_cm is constant even while particles collide or explode.

ExploreConnections for further study

The center-of-mass path separates collective motion from internal dynamics.

Mathematical relation

Center-of-mass motion

P→=Mvcm,F→=Macm \vec P=M\vec v_{cm},\ \vec F_{ext}=M\vec a_{cm}
Represents

Collective motion related to P and external force.

Physical interpretation

Only external force accelerates the center of mass.

DeepenVariables, conditions, and checks

Variables

v_cm
center-of-mass velocity; usual unit: m/s
a_cm
center-of-mass acceleration; usual unit: m/s²

Conditions of application

  • Constant total mass.

Dimensional check

kg·m/s and N.

Concept 04

A continuous trajectory

Essential The minimum you should retain

An explosion redistributes mass and momentum without itself making the center of mass jump.

UnderstandInterpret and connect

Ignoring air, gravity continues to govern the fragments' center of mass.

DeepenFormulation and conditions

Reconstructing r_cm before and after checks model consistency.

ExploreConnections for further study

This remains true even when no fragment follows the original path.

A reference parabola passes through several scattered fragments.r_cm123

Fragments separate, but their weighted average follows the path set by external force.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Requiring r_cm to lie in material.

It is a weighted average and may lie in empty space.

Attributing total-P change to internal forces.

For constant mass, F_ext=M a_cm F_{ext}=Ma_{cm} .