1
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so let's say you wanted to know where
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the center of mass was between this two
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kilogram mass and the 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 and we
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know it's going to be closer to the
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larger mass because the center of mass
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is always closer to the larger mass but
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exactly where is it going to be we need
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a formula to figure this out and the
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formula for the center of mass looks
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like this it says the location of the
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center of mass that's what this is this
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X cm is just the location of the center
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of mass is the position of the center of
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mass is going to equal you take all the
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masses that you're trying to find the
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center of mass between you take all
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those masses times their positions and
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you add up all of these M times X's
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until you've accounted for every single
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M times X there is in your system and
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then you just divide by all of the
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masses added together and what you get
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out of this is the location of the
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center of mass so let's use this let's
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use this for this example problem right
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here let's see what we get we'll have
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that the center of mass the position of
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the center of mass is going to be equal
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to all right so I'll take M 1 which you
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could take either one as M 1 but I
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already colored this one red so we'll
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just say the 2 kilogram mass is M 1 and
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we're gonna have to multiply by X 1 the
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position of mass 1 and at this point you
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might be confused you might be like the
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position I don't know what the position
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is there's no coordinate system up here
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well you get to pick so you get to
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decide where you're measuring these
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positions from and wherever you decide
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to measure them from will also be the
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point where the center of mass is
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measured from in other words you get to
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choose where x equals 0 let's just say
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for the sake of argument the left-hand
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side over here is x equals 0 let's say
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right here is x equals 0 on our number
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line and then it goes this way it's
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positive this way so if this is x equals
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0 half way would be x equals 5 and then
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over here would be x equals 10 we're
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free to choose that in fact it's kind of
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cool because if this is x equals 0 the
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position of mass 1 is 0 meters so it's
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going to be this term is just going to
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go away which is okay we're going to add
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to that
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- which is six kilograms times the
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position of M - again we can choose
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whatever point we want but we have to be
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consistent we already chose this is x
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equals zero for mass one so that still
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has to be x equals zero for mass two
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that means this has to be 10 centimeters
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now and then those are our only two
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masses so we stop there and we just
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divide by all the masses added together
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which is going to be two kilograms for
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m1 plus six kilograms for m2 and what we
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get out of this is two times zero zero
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plus 6 times 10 is 60 kilograms
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centimeters divided by 2 plus 6 is going
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to be 8 kilograms which gives us seven
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point five centimeters so it's going to
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be seven point five centimeters from the
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point we called x equals zero which is
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right here that's the location of the
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center of mass so in other words if you
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connect these two spheres by a rod a
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light rod and you put a pivot right here
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they would balance at that point right
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there and just to show you you might be
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like wait we can choose any point as x
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equals zero won't we get a different
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number you will so let's say you did
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this so instead of instead of picking
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that as x equals zero let's say we pick
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this side as x equals zero let's say we
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say x equals zero is this six kilogram
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masses position what are we going to get
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then we'll get that the location of the
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center of mass for this calculation is
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going to be well we'll have two
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kilograms but now the location of the
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two kilogram mass is not zero it's going
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to be if this is zero and we're
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considering this way is positive it's
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going to be negative 10 centimeters
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because it's 10 centimeters to the left
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so there's going to be negative 10
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centimeters plus 6 kilograms times now
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the location of the six kilogram mass is
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zero using this convention and we divide
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by both of the masses added up so that's
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still two kilograms plus six kilograms
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and what are we going to get we're going
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to get two times negative 10 plus 6
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times zero well that's just zero so
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there's going to be negative 20 kilogram
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centimeters divided by eight kilograms
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gives us negative two point five
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centimeters so you might be worried you
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might be like what we got a different
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answer the location can't change Bay
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dawn where we're measuring from and it
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didn't change it's still in the exact
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same position because now this negative
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2.5 centimeters is measured relative to
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this x equals 0 so what's negative 2.5
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centimeters from here it's 2.5
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centimeters to the left which lo and
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behold is exactly at the same point
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since this was seven point five and this
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is negative two point five and the whole
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thing is ten centimeters it gives you
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the exact same location for the center
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00:04:53,930 --> 00:04:57,230
of mass it has to it can't change based
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on whether you're calling this point
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zero or this point zero but you have to
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be careful and consistent with your
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choice any choice will work but you have
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to be consistent with it and you have to
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know at the end where is this answer
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measured from otherwise you won't be
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able to interpret what this number means
138
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at the end so recapping you can use the
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center of mass formula to find the exact
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location of the center of mass between a
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system of objects you add all the masses
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00:05:22,400 --> 00:05:24,740
times their positions and divide by the
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total mass the position can be measured
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relative to any point you call x equals
145
00:05:29,630 --> 00:05:31,670
zero and the number you get out of that
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00:05:31,670 --> 00:05:34,160
calculation will be the distance from x
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00:05:34,160 --> 00:05:37,070
equals zero to the center of mass of
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that system