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### Course: Physics archive > Unit 6

Lesson 3: Center of mass# Equation for center of mass

The center of mass is a point in a system that responds to external forces as if the total mass of the system were concentrated at this point. The center of mass can be calculated by taking the masses you are trying to find the center of mass between and multiplying them by their positions. Then, you add these together and divide that by the sum of all the individual masses.

## Want to join the conversation?

- so i f you had more than 2 objects that are not in the same plane, would you just add their positions like vectors and determine the c of m from the results?(10 votes)
- Yes you would use the X,Y,Z coordinates......It is the unique position at which the weighted position vectors of all the parts of a system sum up to zero.....(8 votes)

- What about the motion of the centre of mass?

Suppose a box is moving in a projectile motion, it splits into parts somewhere in the trajectory. So, does the CoM follow the initial trajectory, or its trajectory changes?(2 votes)- The COM does follow the intial trajectory, no matter what internal force was applied to split it apart. The derivation is simple: the system of the box has no new external forces, so its COM will follow the same trajectory. This is only true if the box is still in the air.(3 votes)

- Can someone give me a link to an article referring to how the center of mass equation was derived?(6 votes)
- The open source CLP 2 textbook (http://www.math.ubc.ca/~CLP/CLP2/) used at UBC has a section dedicated to deriving center of mass from torque, if you're interested. The section is
*2.3.2 Optional - Torque*.(3 votes)

- So I understand how to find the center of mass between a system of objects, but how would you find it for one single object?(6 votes)
- For a single object,if it is a point object then its position vector itself gives the co-ordinates of the CM.If it is an extended body like a sphere or a cube,which is symmetrical then its geometric center gives the CM of the body.(1 vote)

- I don't get how a body's center of mass be outside it?(2 votes)
- try to imagine a doughnut(7 votes)

- If a ball strikes an object at a non-centered point, how do we find the speed of the block's rotation?(3 votes)
- A force applied through a point other than COM will cause a torque. Calculate the torque, and from that calculate the change in angular momentum and from that calculate the angular velocity.

You need to be a bit careful though to break up the force into components which are perpendicular and parallel to the radial vector from the COM to the point of force. The perpendicular component will generate the torque and rotational motion and the parallel component will generate a force causing linear translational motion.(5 votes)

- How's the center of mass gonna change if we account the weight of the string/rod connecting the two bodies?(3 votes)
- What is the mass of the string/rod in comparison to the mass of the two bodies? If you have a microscopically thin carbon nano-tube connecting two massed that have masses in the kilograms then there would be no measurable change in COM (Center of Mass). On the other hand if you had two styrofoam balls connected by a heavy iron chain it should make a big difference.

If you can determine the COM of the string/rod then you can calculate the the COM of the bodies and the string/rod by using both COMs.(4 votes)

- Why don't we take the mass of the pole that the two masses are connected by into account?(2 votes)
- We should probably not consider the mass of the pole as the mass is uniform throughout the pole and the centre of mass would be 5cm in our case(which is where it is supposed to be). So we need not consider the centre of mass of an object which is dimensionally stable.(2 votes)

- Does the equation change at all if the balance/plank has a given weight?(2 votes)
- Can we define center of mass as the point in a system at which a force may be applied to cause a linear acceleration without causing an angular acceleration in the system(2 votes)

## Video transcript

- [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.