so let's say you wanted to know where
the center of mass was between this two
kilogram mass and the six kilogram mass
now they're separated by 10 centimeters
so it's somewhere in between them and we
know it's going to be closer to the
larger mass because the center of mass
is always closer to the larger mass but
exactly where is it going to be we need
a formula to figure this out and the
formula for the center of mass looks
like this it says the location of the
center of mass that's what this is this
X cm is just the location of the center
of mass is the position of the center of
mass is going to equal you take all the
masses that you're trying to find the
center of mass between you take all
those masses times their positions and
you add up all of these M times X's
until you've accounted for every single
M times X there is in your system and
then you just divide by all of the
masses added together and what you get
out of this is the location of the
center of mass so let's use this let's
use this for this example problem right
here let's see what we get we'll have
that the center of mass the position of
the center of mass is going to be equal
to all right so I'll take M 1 which you
could take either one as M 1 but I
already colored this one red so we'll
just say the 2 kilogram mass is M 1 and
we're gonna have to multiply by X 1 the
position of mass 1 and at this point you
might be confused you might be like the
position I don't know what the position
is there's no coordinate system up here
well you get to pick so you get to
decide where you're measuring these
positions from and wherever you decide
to measure them from will also be the
point where the center of mass is
measured from in other words you get to
choose where x equals 0 let's just say
for the sake of argument the left-hand
side over here is x equals 0 let's say
right here is x equals 0 on our number
line and then it goes this way it's
positive this way so if this is x equals
0 half way would be x equals 5 and then
over here would be x equals 10 we're
free to choose that in fact it's kind of
cool because if this is x equals 0 the
position of mass 1 is 0 meters so it's
going to be this term is just going to
go away which is okay we're going to add
to that
- which is six kilograms times the
position of M - again we can choose
whatever point we want but we have to be
consistent we already chose this is x
equals zero for mass one so that still
has to be x equals zero for mass two
that means this has to be 10 centimeters
now and then those are our only two
masses so we stop there and we just
divide by all the masses added together
which is going to be two kilograms for
m1 plus six kilograms for m2 and what we
get out of this is two times zero zero
plus 6 times 10 is 60 kilograms
centimeters divided by 2 plus 6 is going
to be 8 kilograms which gives us seven
point five centimeters so it's going to
be seven point five centimeters from the
point we called x equals zero which is
right here that's the location of the
center of mass so in other words if you
connect these two spheres by a rod a
light rod and you put a pivot right here
they would balance at that point right
there and just to show you you might be
like wait we can choose any point as x
equals zero won't we get a different
number you will so let's say you did
this so instead of instead of picking
that as x equals zero let's say we pick
this side as x equals zero let's say we
say x equals zero is this six kilogram
masses position what are we going to get
then we'll get that the location of the
center of mass for this calculation is
going to be well we'll have two
kilograms but now the location of the
two kilogram mass is not zero it's going
to be if this is zero and we're
considering this way is positive it's
going to be negative 10 centimeters
because it's 10 centimeters to the left
so there's going to be negative 10
centimeters plus 6 kilograms times now
the location of the six kilogram mass is
zero using this convention and we divide
by both of the masses added up so that's
still two kilograms plus six kilograms
and what are we going to get we're going
to get two times negative 10 plus 6
times zero well that's just zero so
there's going to be negative 20 kilogram
centimeters divided by eight kilograms
gives us negative two point five
centimeters so you might be worried you
might be like what we got a different
answer the location can't change Bay
dawn where we're measuring from and it
didn't change it's still in the exact
same position because now this negative
2.5 centimeters is measured relative to
this x equals 0 so what's negative 2.5
centimeters from here it's 2.5
centimeters to the left which lo and
behold is exactly at the same point
since this was seven point five and this
is negative two point five and the whole
thing is ten centimeters it gives you
the exact same location for the center
of mass it has to it can't change based
on whether you're calling this point
zero or this point zero but you have to
be careful and consistent with your
choice any choice will work but you have
to be consistent with it and you have to
know at the end where is this answer
measured from otherwise you won't be
able to interpret what this number means
at the end so recapping you can use the
center of mass formula to find the exact
location of the center of mass between a
system of objects you add all the masses
times their positions and divide by the
total mass the position can be measured
relative to any point you call x equals
zero and the number you get out of that
calculation will be the distance from x
equals zero to the center of mass of
that system