Center of mass equation | Impacts and linear momentum | Physics | Khan Academy

- [Tutor] So let's say you wanted to know

where the center of mass was between

this two kilogram mass and
this six kilogram mass,

now they're separated by 10 centimeters,

so it's somewhere in between them

and we know it's gonna be
closer to the larger mass,

'cause the center of mass is always closer

to the larger mass, but
exactly where is it gonna 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 Xcm is just the location
of the center of mass,

it's the position of the
center of mass is gonna 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 Xs,

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 and let's see what we get,

we'll have the center of mass,

the position of the center
of mass is gonna be equal to,

alright, so we'll take M1,

which you could take either one as M1,

but I already colored this one red,

so we'll just say the
two kilogram mass is M1

and we're gonna have to multiply by X1,

the position of mass
one 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 zero.

Let's just say for the sake of argument,

the left-hand side over
here is X equals zero,

let's say right here is X
equals zero on our number line

and then it goes this way,
it's positive this way,

so if this is X equals zero,
halfway would be X equals five

and then over here, it
would be X equals 10,

we're free to choose that,
in fact, it's kind of cool,

because if this is X equals zero,

the position of mass one is zero meters,

so it's gonna be, this
term's just gonna go away,

which is okay, we're gonna
have to add to that M2,

which is six kilograms
times the position of M2,

again we can choose
whatever point we want,

but we have to be consistent,
we already chose this

as 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 gonna be two kilograms for M1

plus six kilograms for M2
and what we get out of this

is two times zero, zero plus six times 10

is 60 kilogram centimeters
divided by two plus six

is gonna be eight kilograms,

which gives us 7.5 centimeters,

so it's gonna be 7.5
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
connected 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,

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 mass's position,

what are we gonna get then?

We'll get that the location
of the center of mass

for this calculation is gonna be,

well, we'll have two kilograms,

but now the location of the
two kilogram mass is not zero,

it's gonna be if this is zero

and we're considering
this way is positive,

it's gonna be negative 10 centimeters,

'cause it's 10 centimeters to the left,

so this is gonna be
negative 10 centimeters

plus six 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 gonna get?

We're gonna get two times negative 10

plus six times zero,
well, that's just zero,

so it's gonna be negative
20 kilogram centimeters

divided by eight kilograms

gives us negative 2.5 centimeters,

so you might be worried,
you might be like, "What?

"We got a different answer.

"The location can't change,

"based on 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 zero,

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 7.5 and
this is negative 2.5

and the whole thing is 10 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.