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Center of mass

Interactions between parts of a system transfer momentum between the parts, but do not change the total momentum of the system. We can define apoint called the center of mass that serves as an average location of a system of parts.

The center of mass need not necessarily be at a location that is either in or on one of the parts. For example, the center of mass of a pair of heavy rods connected at oneend so as to form a "V" shape is somewhere in space between the two rods.

Having determined the center of mass for a system, we can treat the mass of the system as if it were all concentrated at the center of mass.

Location of the center of mass

For a system composed of two masses, the center of mass lies somewhere on a line between the two masses. The center of mass is a weighted average of the positions of the twomasses.

Facts worth remembering -- Center of mass for two objects

For a pair of masses located at two points along the x-axis, we can write

xcm = (m1*x1/M) + (m2*x2/M)


  • xcm is the x-coordinate of the center of mass
  • m1 and m2 are the values of the two masses
  • x1 and x2 are the locations of the two masses
  • M is the sum of m1 and m2

Multiple masses in three dimensions

When we have multiple masses in three dimensions, the definition of the center of mass is somewhat more complicated.

Facts worth remembering -- Center of mass for many objects

Vector form:

rcm = sum over all i(mi*ri / M)

Component form:

xcm = sum over all i(mi*xi / M)

ycm = sum over all i(mi*yi / M)

zcm = sum over all i(mi*zi / M)


  • Vector form
    • rcm is a position vector describing the location of the center of mass
    • ri are position vectors describing the locations of all the masses
    • mi are masses for i=1, i=2, etc.
  • Component form
    • xcm, ycm, and zcm are the locations of the center of mass along 3-dimensional axes.
    • mi are masses for i=1, i=2, etc.
    • xi, yi, and zi are the locations of the masses along 3-dimensional axes for i=1, i=2, etc.
    • M is the sum of all of the masses

Motion of the center of mass

It can be shown that in an isolated system, the center of mass must move with constant velocity regardless of the motions of the individual particles.

It can be shown that in a non-isolated system, if a net external force acts on a system, the center of mass does not movewith constant velocity. Instead, it moves as if all the mass were concentrated there into a fictitious point particle with all the external forces acting on that point.

Example scenarios

This section contains explanations and computations involving momentum, impulse, action and reaction, andthe conservation of momentum.

Momentum examples

This section contains several examples involving momentum

A sprinter

Use the Google calculator to compute the momentum of a 70-kg sprinter running 30 m/s at 0 degrees.

Answer: 2100 kg*m/s at 0 degrees

A truck

Use the Google calculator to compute the momentum in kg*m/s of a 2205-lb truck traveling 33.6 miles per hour at 0 degrees when the changes listed belowoccur:

  1. Initial momentum
  2. Momentum when velocity is doubled
  3. Momentum at initial velocity when mass is doubled
  4. Momentum when both velocity and mass are doubled

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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