WEBVTT

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

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where the center of mass was between

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this two kilogram mass and
this six kilogram mass,

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now they're separated by 10 centimeters,

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so it's somewhere in between them

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

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'cause the center of mass is always closer

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to the larger mass, but
exactly where is it gonna be?

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We need a formula to figure this out

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and the formula for the center
of mass looks like this,

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it says the location
of the center of mass,

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that's what this is,

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this Xcm is just the location
of the center of mass,

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

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you take all the masses
that you're trying to find

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the center of mass between,

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you take all those masses
times their positions

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and you add up all of these M times Xs,

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until you've accounted
for every single M times X

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there is in your system

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and then you just divide by all
of the masses added together

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and what you get out of this

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is the location of the center of mass.

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So let's use this, let's use
this for this example problem

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right here and let's see what we get,

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we'll have the center of mass,

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

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alright, so we'll take M1,

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which you could take either one as M1,

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but I already colored this one red,

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

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and we're gonna have to multiply by X1,

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the position of mass
one and at this point,

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you might be confused, you
might be like the position,

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I don't know what the position is,

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there's no coordinate system up here,

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well, you get to pick,
so you get to decide

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where you're measuring
these positions from

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and wherever you decide
to measure them from

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will also be the point,

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where the center of mass is measured from,

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in other words, you get to
choose where X equals zero.

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Let's just say for the sake of argument,

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

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

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

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

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

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

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because if this is X equals zero,

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the position of mass one is zero meters,

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

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

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which is six kilograms
times the position of M2,

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again we can choose
whatever point we want,

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but we have to be consistent,
we already chose this

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as X equals zero for mass one,

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so that still has to be X
equals zero for mass two,

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that means this has to
be 10 centimeters now

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and then those are our only
two masses, so we stop there

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and we just divide by all
the masses added together,

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which is gonna be two kilograms for M1

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

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is two times zero, zero plus six times 10

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is 60 kilogram centimeters
divided by two plus six

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is gonna be eight kilograms,

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which gives us 7.5 centimeters,

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so it's gonna be 7.5
centimeters from the point

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we called X equals zero,
which is right here,

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that's the location of the center of mass,

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so in other words, if you
connected these two spheres

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by a rod, a light rod and
you put a pivot right here,

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they would balance at
that point right there

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and just to show you, you might be like,

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"Wait, we can choose any
point as X equals zero,

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"won't we get a different number?"

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You will, so let's say you did this,

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instead of picking that as X equals zero,

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let's say we pick this
side as X equals zero,

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let's say we say X equals zero

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is this six kilogram mass's position,

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what are we gonna get then?

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

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for this calculation is gonna be,

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well, we'll have two kilograms,

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

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it's gonna be if this is zero

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and we're considering
this way is positive,

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it's gonna be negative 10 centimeters,

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'cause it's 10 centimeters to the left,

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so this is gonna be
negative 10 centimeters

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plus six kilograms times,

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now the location of the
six kilogram mass is zero,

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using this convention and we divide

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by both of the masses added up,

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so that's still two
kilograms plus six kilograms

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and what are we gonna get?

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We're gonna get two times negative 10

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

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so it's gonna be negative
20 kilogram centimeters

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divided by eight kilograms

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gives us negative 2.5 centimeters,

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

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"We got a different answer.

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"The location can't change,

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"based on where we're measuring from,"

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and it didn't change, it's still
in the exact same position,

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because now this negative 2.5 centimeters

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is measured relative
to this X equals zero,

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so what's negative 2.5
centimeters from here?

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It's 2.5 centimeters to the left,

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which lo and behold is
exactly at the same point,

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since this was 7.5 and
this is negative 2.5

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and the whole thing is 10 centimeters,

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it gives you the exact same
location for the center of mass,

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it has to, it can't change based on

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whether you're calling this
point zero or this point zero,

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but you have to be careful and
consistent with your choice,

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any choice will work, but you
have to be consistent with it

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and you have to know at the end

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where is this answer measured from,

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otherwise you won't be able to interpret

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what this number means at the end.

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So recapping, you can use
the center of mass formula

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to find the exact location
of the center of mass

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between a system of objects,

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you add all the masses
times their positions

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and divide by the total mass,

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the position can be measured

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relative to any point
you call X equals zero

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and the number you get
out of that calculation

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will be the distance from X equals zero

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to the center of mass of that system.