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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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