[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.02,0:00:01.19,Default,,0000,0000,0000,,- [Tutor] So let's say you wanted to know
Dialogue: 0,0:00:01.19,0:00:03.53,Default,,0000,0000,0000,,where the center of mass was between
Dialogue: 0,0:00:03.53,0:00:06.48,Default,,0000,0000,0000,,this two kilogram mass and\Nthis six kilogram mass,
Dialogue: 0,0:00:06.48,0:00:08.69,Default,,0000,0000,0000,,now they're separated by 10 centimeters,
Dialogue: 0,0:00:08.69,0:00:10.67,Default,,0000,0000,0000,,so it's somewhere in between them
Dialogue: 0,0:00:10.67,0:00:14.03,Default,,0000,0000,0000,,and we know it's gonna be\Ncloser to the larger mass,
Dialogue: 0,0:00:14.03,0:00:16.25,Default,,0000,0000,0000,,'cause the center of mass is always closer
Dialogue: 0,0:00:16.25,0:00:19.82,Default,,0000,0000,0000,,to the larger mass, but\Nexactly where is it gonna be?
Dialogue: 0,0:00:19.82,0:00:22.41,Default,,0000,0000,0000,,We need a formula to figure this out
Dialogue: 0,0:00:22.41,0:00:25.60,Default,,0000,0000,0000,,and the formula for the center\Nof mass looks like this,
Dialogue: 0,0:00:25.60,0:00:27.96,Default,,0000,0000,0000,,it says the location\Nof the center of mass,
Dialogue: 0,0:00:27.96,0:00:29.81,Default,,0000,0000,0000,,that's what this is,
Dialogue: 0,0:00:29.81,0:00:33.73,Default,,0000,0000,0000,,this Xcm is just the location\Nof the center of mass,
Dialogue: 0,0:00:33.73,0:00:37.79,Default,,0000,0000,0000,,it's the position of the\Ncenter of mass is gonna equal,
Dialogue: 0,0:00:37.79,0:00:40.27,Default,,0000,0000,0000,,you take all the masses\Nthat you're trying to find
Dialogue: 0,0:00:40.27,0:00:42.05,Default,,0000,0000,0000,,the center of mass between,
Dialogue: 0,0:00:42.05,0:00:44.63,Default,,0000,0000,0000,,you take all those masses\Ntimes their positions
Dialogue: 0,0:00:44.63,0:00:47.69,Default,,0000,0000,0000,,and you add up all of these M times Xs,
Dialogue: 0,0:00:47.69,0:00:50.44,Default,,0000,0000,0000,,until you've accounted\Nfor every single M times X
Dialogue: 0,0:00:50.44,0:00:51.27,Default,,0000,0000,0000,,there is in your system
Dialogue: 0,0:00:51.27,0:00:54.84,Default,,0000,0000,0000,,and then you just divide by all\Nof the masses added together
Dialogue: 0,0:00:54.84,0:00:56.59,Default,,0000,0000,0000,,and what you get out of this
Dialogue: 0,0:00:56.59,0:00:59.21,Default,,0000,0000,0000,,is the location of the center of mass.
Dialogue: 0,0:00:59.21,0:01:01.69,Default,,0000,0000,0000,,So let's use this, let's use\Nthis for this example problem
Dialogue: 0,0:01:01.69,0:01:03.47,Default,,0000,0000,0000,,right here and let's see what we get,
Dialogue: 0,0:01:03.47,0:01:05.05,Default,,0000,0000,0000,,we'll have the center of mass,
Dialogue: 0,0:01:05.05,0:01:08.67,Default,,0000,0000,0000,,the position of the center\Nof mass is gonna be equal to,
Dialogue: 0,0:01:08.67,0:01:10.85,Default,,0000,0000,0000,,alright, so we'll take M1,
Dialogue: 0,0:01:10.85,0:01:13.09,Default,,0000,0000,0000,,which you could take either one as M1,
Dialogue: 0,0:01:13.09,0:01:14.27,Default,,0000,0000,0000,,but I already colored this one red,
Dialogue: 0,0:01:14.27,0:01:17.04,Default,,0000,0000,0000,,so we'll just say the\Ntwo kilogram mass is M1
Dialogue: 0,0:01:17.04,0:01:20.34,Default,,0000,0000,0000,,and we're gonna have to multiply by X1,
Dialogue: 0,0:01:20.34,0:01:22.97,Default,,0000,0000,0000,,the position of mass\None and at this point,
Dialogue: 0,0:01:22.97,0:01:25.73,Default,,0000,0000,0000,,you might be confused, you\Nmight be like the position,
Dialogue: 0,0:01:25.73,0:01:27.37,Default,,0000,0000,0000,,I don't know what the position is,
Dialogue: 0,0:01:27.37,0:01:29.67,Default,,0000,0000,0000,,there's no coordinate system up here,
Dialogue: 0,0:01:29.67,0:01:31.98,Default,,0000,0000,0000,,well, you get to pick,\Nso you get to decide
Dialogue: 0,0:01:31.98,0:01:34.87,Default,,0000,0000,0000,,where you're measuring\Nthese positions from
Dialogue: 0,0:01:34.87,0:01:37.15,Default,,0000,0000,0000,,and wherever you decide\Nto measure them from
Dialogue: 0,0:01:37.15,0:01:38.02,Default,,0000,0000,0000,,will also be the point,
Dialogue: 0,0:01:38.02,0:01:40.44,Default,,0000,0000,0000,,where the center of mass is measured from,
Dialogue: 0,0:01:40.44,0:01:44.05,Default,,0000,0000,0000,,in other words, you get to\Nchoose where X equals zero.
Dialogue: 0,0:01:44.05,0:01:45.52,Default,,0000,0000,0000,,Let's just say for the sake of argument,
Dialogue: 0,0:01:45.52,0:01:48.72,Default,,0000,0000,0000,,the left-hand side over\Nhere is X equals zero,
Dialogue: 0,0:01:48.72,0:01:52.01,Default,,0000,0000,0000,,let's say right here is X\Nequals zero on our number line
Dialogue: 0,0:01:52.01,0:01:54.53,Default,,0000,0000,0000,,and then it goes this way,\Nit's positive this way,
Dialogue: 0,0:01:54.53,0:01:58.15,Default,,0000,0000,0000,,so if this is X equals zero,\Nhalfway would be X equals five
Dialogue: 0,0:01:58.15,0:02:00.94,Default,,0000,0000,0000,,and then over here, it\Nwould be X equals 10,
Dialogue: 0,0:02:00.94,0:02:03.54,Default,,0000,0000,0000,,we're free to choose that,\Nin fact, it's kind of cool,
Dialogue: 0,0:02:03.54,0:02:05.65,Default,,0000,0000,0000,,because if this is X equals zero,
Dialogue: 0,0:02:05.65,0:02:08.71,Default,,0000,0000,0000,,the position of mass one is zero meters,
Dialogue: 0,0:02:08.71,0:02:11.44,Default,,0000,0000,0000,,so it's gonna be, this\Nterm's just gonna go away,
Dialogue: 0,0:02:11.44,0:02:14.38,Default,,0000,0000,0000,,which is okay, we're gonna\Nhave to add to that M2,
Dialogue: 0,0:02:14.38,0:02:18.22,Default,,0000,0000,0000,,which is six kilograms\Ntimes the position of M2,
Dialogue: 0,0:02:18.22,0:02:19.94,Default,,0000,0000,0000,,again we can choose\Nwhatever point we want,
Dialogue: 0,0:02:19.94,0:02:22.71,Default,,0000,0000,0000,,but we have to be consistent,\Nwe already chose this
Dialogue: 0,0:02:22.71,0:02:24.48,Default,,0000,0000,0000,,as X equals zero for mass one,
Dialogue: 0,0:02:24.48,0:02:27.46,Default,,0000,0000,0000,,so that still has to be X\Nequals zero for mass two,
Dialogue: 0,0:02:27.46,0:02:30.32,Default,,0000,0000,0000,,that means this has to\Nbe 10 centimeters now
Dialogue: 0,0:02:30.32,0:02:32.59,Default,,0000,0000,0000,,and then those are our only\Ntwo masses, so we stop there
Dialogue: 0,0:02:32.59,0:02:35.44,Default,,0000,0000,0000,,and we just divide by all\Nthe masses added together,
Dialogue: 0,0:02:35.44,0:02:38.50,Default,,0000,0000,0000,,which is gonna be two kilograms for M1
Dialogue: 0,0:02:38.50,0:02:42.97,Default,,0000,0000,0000,,plus six kilograms for M2\Nand what we get out of this
Dialogue: 0,0:02:42.97,0:02:46.56,Default,,0000,0000,0000,,is two times zero, zero plus six times 10
Dialogue: 0,0:02:46.56,0:02:50.95,Default,,0000,0000,0000,,is 60 kilogram centimeters\Ndivided by two plus six
Dialogue: 0,0:02:50.95,0:02:53.38,Default,,0000,0000,0000,,is gonna be eight kilograms,
Dialogue: 0,0:02:53.38,0:02:56.74,Default,,0000,0000,0000,,which gives us 7.5 centimeters,
Dialogue: 0,0:02:56.74,0:03:00.57,Default,,0000,0000,0000,,so it's gonna be 7.5\Ncentimeters from the point
Dialogue: 0,0:03:00.57,0:03:03.49,Default,,0000,0000,0000,,we called X equals zero,\Nwhich is right here,
Dialogue: 0,0:03:03.49,0:03:05.78,Default,,0000,0000,0000,,that's the location of the center of mass,
Dialogue: 0,0:03:05.78,0:03:08.94,Default,,0000,0000,0000,,so in other words, if you\Nconnected these two spheres
Dialogue: 0,0:03:08.94,0:03:12.56,Default,,0000,0000,0000,,by a rod, a light rod and\Nyou put a pivot right here,
Dialogue: 0,0:03:12.56,0:03:15.09,Default,,0000,0000,0000,,they would balance at\Nthat point right there
Dialogue: 0,0:03:15.09,0:03:16.40,Default,,0000,0000,0000,,and just to show you, you might be like,
Dialogue: 0,0:03:16.40,0:03:19.35,Default,,0000,0000,0000,,"Wait, we can choose any\Npoint as X equals zero,
Dialogue: 0,0:03:19.35,0:03:21.05,Default,,0000,0000,0000,,"won't we get a different number?"
Dialogue: 0,0:03:21.05,0:03:23.07,Default,,0000,0000,0000,,You will, so let's say you did this,
Dialogue: 0,0:03:23.07,0:03:26.00,Default,,0000,0000,0000,,instead of picking that as X equals zero,
Dialogue: 0,0:03:26.00,0:03:28.16,Default,,0000,0000,0000,,let's say we pick this\Nside as X equals zero,
Dialogue: 0,0:03:28.16,0:03:30.73,Default,,0000,0000,0000,,let's say we say X equals zero
Dialogue: 0,0:03:30.73,0:03:33.24,Default,,0000,0000,0000,,is this six kilogram mass's position,
Dialogue: 0,0:03:33.24,0:03:34.98,Default,,0000,0000,0000,,what are we gonna get then?
Dialogue: 0,0:03:34.98,0:03:37.71,Default,,0000,0000,0000,,We'll get that the location\Nof the center of mass
Dialogue: 0,0:03:37.71,0:03:38.95,Default,,0000,0000,0000,,for this calculation is gonna be,
Dialogue: 0,0:03:38.95,0:03:40.96,Default,,0000,0000,0000,,well, we'll have two kilograms,
Dialogue: 0,0:03:40.96,0:03:44.77,Default,,0000,0000,0000,,but now the location of the\Ntwo kilogram mass is not zero,
Dialogue: 0,0:03:44.77,0:03:45.92,Default,,0000,0000,0000,,it's gonna be if this is zero
Dialogue: 0,0:03:45.92,0:03:48.41,Default,,0000,0000,0000,,and we're considering\Nthis way is positive,
Dialogue: 0,0:03:48.41,0:03:50.10,Default,,0000,0000,0000,,it's gonna be negative 10 centimeters,
Dialogue: 0,0:03:50.10,0:03:51.93,Default,,0000,0000,0000,,'cause it's 10 centimeters to the left,
Dialogue: 0,0:03:51.93,0:03:54.50,Default,,0000,0000,0000,,so this is gonna be\Nnegative 10 centimeters
Dialogue: 0,0:03:54.50,0:03:57.38,Default,,0000,0000,0000,,plus six kilograms times,
Dialogue: 0,0:03:57.38,0:04:00.37,Default,,0000,0000,0000,,now the location of the\Nsix kilogram mass is zero,
Dialogue: 0,0:04:00.37,0:04:02.13,Default,,0000,0000,0000,,using this convention and we divide
Dialogue: 0,0:04:02.13,0:04:03.69,Default,,0000,0000,0000,,by both of the masses added up,
Dialogue: 0,0:04:03.69,0:04:06.84,Default,,0000,0000,0000,,so that's still two\Nkilograms plus six kilograms
Dialogue: 0,0:04:06.84,0:04:07.73,Default,,0000,0000,0000,,and what are we gonna get?
Dialogue: 0,0:04:07.73,0:04:10.43,Default,,0000,0000,0000,,We're gonna get two times negative 10
Dialogue: 0,0:04:10.43,0:04:12.16,Default,,0000,0000,0000,,plus six times zero,\Nwell, that's just zero,
Dialogue: 0,0:04:12.16,0:04:16.14,Default,,0000,0000,0000,,so it's gonna be negative\N20 kilogram centimeters
Dialogue: 0,0:04:16.14,0:04:18.36,Default,,0000,0000,0000,,divided by eight kilograms
Dialogue: 0,0:04:18.36,0:04:21.06,Default,,0000,0000,0000,,gives us negative 2.5 centimeters,
Dialogue: 0,0:04:21.06,0:04:23.25,Default,,0000,0000,0000,,so you might be worried,\Nyou might be like, "What?
Dialogue: 0,0:04:23.25,0:04:24.67,Default,,0000,0000,0000,,"We got a different answer.
Dialogue: 0,0:04:24.67,0:04:26.18,Default,,0000,0000,0000,,"The location can't change,
Dialogue: 0,0:04:26.18,0:04:28.40,Default,,0000,0000,0000,,"based on where we're measuring from,"
Dialogue: 0,0:04:28.40,0:04:31.16,Default,,0000,0000,0000,,and it didn't change, it's still\Nin the exact same position,
Dialogue: 0,0:04:31.16,0:04:34.20,Default,,0000,0000,0000,,because now this negative 2.5 centimeters
Dialogue: 0,0:04:34.20,0:04:37.13,Default,,0000,0000,0000,,is measured relative\Nto this X equals zero,
Dialogue: 0,0:04:37.13,0:04:40.05,Default,,0000,0000,0000,,so what's negative 2.5\Ncentimeters from here?
Dialogue: 0,0:04:40.05,0:04:42.68,Default,,0000,0000,0000,,It's 2.5 centimeters to the left,
Dialogue: 0,0:04:42.68,0:04:46.24,Default,,0000,0000,0000,,which lo and behold is\Nexactly at the same point,
Dialogue: 0,0:04:46.24,0:04:49.58,Default,,0000,0000,0000,,since this was 7.5 and\Nthis is negative 2.5
Dialogue: 0,0:04:49.58,0:04:51.96,Default,,0000,0000,0000,,and the whole thing is 10 centimeters,
Dialogue: 0,0:04:51.96,0:04:54.98,Default,,0000,0000,0000,,it gives you the exact same\Nlocation for the center of mass,
Dialogue: 0,0:04:54.98,0:04:57.57,Default,,0000,0000,0000,,it has to, it can't change based on
Dialogue: 0,0:04:57.57,0:05:00.59,Default,,0000,0000,0000,,whether you're calling this\Npoint zero or this point zero,
Dialogue: 0,0:05:00.59,0:05:03.85,Default,,0000,0000,0000,,but you have to be careful and\Nconsistent with your choice,
Dialogue: 0,0:05:03.85,0:05:06.44,Default,,0000,0000,0000,,any choice will work, but you\Nhave to be consistent with it
Dialogue: 0,0:05:06.44,0:05:08.30,Default,,0000,0000,0000,,and you have to know at the end
Dialogue: 0,0:05:08.30,0:05:10.47,Default,,0000,0000,0000,,where is this answer measured from,
Dialogue: 0,0:05:10.47,0:05:12.52,Default,,0000,0000,0000,,otherwise you won't be able to interpret
Dialogue: 0,0:05:12.52,0:05:14.31,Default,,0000,0000,0000,,what this number means at the end.
Dialogue: 0,0:05:14.31,0:05:17.00,Default,,0000,0000,0000,,So recapping, you can use\Nthe center of mass formula
Dialogue: 0,0:05:17.00,0:05:19.64,Default,,0000,0000,0000,,to find the exact location\Nof the center of mass
Dialogue: 0,0:05:19.64,0:05:21.10,Default,,0000,0000,0000,,between a system of objects,
Dialogue: 0,0:05:21.10,0:05:24.02,Default,,0000,0000,0000,,you add all the masses\Ntimes their positions
Dialogue: 0,0:05:24.02,0:05:26.01,Default,,0000,0000,0000,,and divide by the total mass,
Dialogue: 0,0:05:26.01,0:05:27.44,Default,,0000,0000,0000,,the position can be measured
Dialogue: 0,0:05:27.44,0:05:29.75,Default,,0000,0000,0000,,relative to any point\Nyou call X equals zero
Dialogue: 0,0:05:29.75,0:05:32.55,Default,,0000,0000,0000,,and the number you get\Nout of that calculation
Dialogue: 0,0:05:32.55,0:05:35.37,Default,,0000,0000,0000,,will be the distance from X equals zero
Dialogue: 0,0:05:35.37,0:05:38.16,Default,,0000,0000,0000,,to the center of mass of that system.