Main content

## MCAT

### Unit 8: Lesson 4

Newton's laws and equilibrium# Center of mass

Sal gives a conceptual understanding of what center of mass means. Created by Sal Khan.

## Want to join the conversation?

- But how do you really calculate the centre of mass in physics problems?(162 votes)
- @ Mat Hildebrand: This method can in fact be generalized to an integral.

In which case it is Xcom=Integral(x*dm)/Integral(dm)

Where dm is the mass of an infinitesimal (meaning really really small) piece of your object, x is the position of that infinitesimal piece, and Xcom is the center of mass. This can then be repeated for the y dimension and the z dimension by simply changing all of the x's to y's or z's respectively.(Note: to actually evaluate the integral, you'll want to convert your dm into terms of dx. Specifically, dm=rho*dx , where rho is the density (mass per length) of your object.(8 votes)

- Is the centre of mass and centre of gravity the same?

Also are they always at the same place?

If not then could you please give and example?(24 votes)- If the gravitational field is constant, then the centre of mass and centre of gravity will be in the same place. For almost all problems you will ever come across, you can treat them as the same. In reality, they are not quite. For example, because your head is further from the centre of the Earth than your feet are, each kilo of your head experiences less of a gravitational pull than each kilo of your feet. So your centre of gravity will be very slightly lower (more towards the Earth) than your centre of mass is. But the difference is so very, very slight that you can almost always ignore it.(57 votes)

- Wait... how does their center of mass go below the bar? I'm not understanding why it doesn't remain where it originally was?(17 votes)
- If you bend your body, more of your body mass will be in behind you. In the diagram you can see that the guys legs and arms are behind him. So the center of mass is actually outside the persons body. This may seem weird, but think of a donut, its center is hollow, but its center of mass is in its center, even though its empty in the middle(54 votes)

- is it possible the center of mass of an object is outside the body??(11 votes)
- Yes, it is. As an example, think of a doughnut (or any other type of ring) -- the center of mass is located at the center of the doughnut (where the hole is) and, hence, outside its doughnut body.

I should note, however, that I am assuming we're talking about a regular doughnut that contains a hole in the center (thus excluding all jelly/cream-filled doughnuts that do not contain said holes). This answer is making me hungry... Hope this helps!(45 votes)

- I don't understand how the Center of Mass is coming out of the body when the man is jumping in an arched manner over the pole at9:02. Please help!(6 votes)
- centre of mass is a theoretical concept... its major advantage is that it simplifies motion ... like if we are looking for motion of a elephant .. then we can study elephants motion just by seeing motion of its "centre of mass" ...

centre of mass need not always be inside body ... it is generally inside body when body has some regular shape or uniform density ... when jumper arches he tries to form a kind of semi circular ring like structure .. which is ir-regular figure ... making a circle(imaginary) by taking arc (jumper's body) as its part .. gives us a regular figure ie .. a circle but .. its density distribution would be non-uniform ... the body arc part will be high density one .. while other part would be low density (since other part of circle is of air).. so .. centre of mass would be near body & centre of circle .. (but not inside body).. i hope you got it ..(22 votes)

- How to exemplify the center of mass if it is outside the object? For example, if there is a hole in the center of the ruler, how to apply the force 'F'?(7 votes)
- That question can be expanded: if the center of Mass is inside an object, how can you apply a force to it (e.g. inside a 3D horse)?

Since the center of mass is the integral (sum) of the objects moments divided by the total mass, you can do the same with applied forces. It's hard to apply a force at a singular point. So sum up all the impact vectors of forces applied to the object and you get a "Center of force". If that matches with the center of mass, you get a rotation free accelleration(7 votes)

- how does centre of mass change?(2 votes)
- centre of mass can only be changed if the shape , size and mass of the body changes(6 votes)

- i would really appreciate if someone would give an accurate answer to this question.

what is the advantage of being able to move the counterweight along the arm?(4 votes)- Moving the counterweight makes the center of mass adjustable.(2 votes)

- is the center of mass in one place always(3 votes)
- Within a rigid object the center of mass is in a specific location the it doesn't change but if you have an object that shape changes or its mass distribution changes then the center of mass can change.(2 votes)

- sir, you said if force is applied on the centre of mass the object will move in the direction of the force , BUT what if centre of mass is present outside the body ?? and can u please provide me a definition of centre of mass?? ... thanks ..(2 votes)
- It does not matter if the center of mass is physically within an object or not - the equations of motion still apply. The behavior of the object will appear the same as if a force had acted at that point (even if the force cannot contact the object at that point). Think of a hollow billiard ball - it will behave like a billiard ball in some respects - and have predictable behavior in collisions even though the center of mass will be in a void.

Definition: Wikipedia's definition is (correctly) mathematical in nature (the center of mass is the point where the sum of all elements of mass (you can think of it as each particle of mass) times the distance of that particle from the center of mass will be zero (sums to zero). This distance must of course be expressed in a coordinate system (for example x-y-z) so that particles on one side contribute to the sum with an opposite sign to those diametrically opposite. Think of it as the point where the object would 'balance' if you could put a point fulcrum at that spot (you usually can not - especially when the center of mass is outside the body, as in a horseshoe).(4 votes)

## Video transcript

I will now do a presentation
on the center of mass. And the center mass, hopefully,
is something that will be a little bit intuitive
to you, and it actually has some very neat applications. So in very simple terms, the
center of mass is a point. Let me draw an object. Let's say that this
is my object. Let's say it's a ruler. This ruler, it exists,
so it has some mass. And my question to you is
what is the center mass? And you say, Sal, well, in order
to know figure out the center mass, you have to tell me
what the center of mass is. And what I tell you is the
center mass is a point, and it actually doesn't have to even
be a point in the object. I'll do an example soon
where it won't be. But it's a point. And at that point, for dealing
with this object as a whole or the mass of the object as a
whole, we can pretend that the entire mass exists
at that point. And what do I mean
by saying that? Well, let's say that the
center of mass is here. And I'll tell you why
I picked this point. Because that is pretty close
to where the center of mass will be. If the center of mass is there,
and let's say the mass of this entire ruler is, I don't
know, 10 kilograms. This ruler, if a force is applied at
the center of mass, let's say 10 Newtons, so the mass
of the whole ruler is 10 kilograms. If a force is applied
at the center of mass, this ruler will accelerate the
same exact way as would a point mass. Let's say that we just had a
little dot, but that little dot had the same mass, 10
kilograms, and we were to push on that dot with 10 Newtons. In either case, in the case of
the ruler, we would accelerate upwards at what? Force divided by mass, so we
would accelerate upwards at 1 meter per second squared. And in this case of this
point mass, we would accelerate that point. When I say point mass, I'm just
saying something really, really small, but it has a mass
of 10 kilograms, so it's much smaller, but it has the
same mass as this ruler. This would also accelerate
upwards with a magnitude of 1 meters per second squared. So why is this useful to us? Well, sometimes we have some
really crazy objects and we want to figure out exactly
what it does. If we know its center of mass
first, we can know how that object will behave without
having to worry about the shape of that object. And I'll give you a really easy
way of realizing where the center of mass is. If the object has a uniform
distribution-- when I say that, it means, for simple
purposes, if it's made out of the same thing and that thing
that it's made out of, its density, doesn't really change
throughout the object, the center of mass will be the
object's geometric center. So in this case, this
ruler's almost a one-dimensional object. We just went halfway. The distance from here to here
and the distance from here to here are the same. This is the center of mass. If we had a two-dimensional
object, let's say we had this triangle and we want to figure
out its center of mass, it'll be the center in
two dimensions. So it'll be something
like that. Now, if I had another situation,
let's say I have this square. I don't know if that's big
enough for you to see. I need to draw it a
little thicker. Let's say I have this square,
but let's say that half of this square is made from lead. And let's say the other half
of the square is made from something lighter than lead. It's made of styrofoam. That is lighter than lead. So in this situation, the center
of mass isn't going to be the geographic center. I don't know how much denser
lead is than styrofoam, but the center of mass is going to
be someplace closer to the right because this object does
not have a uniform density. It'll actually depend on how
much denser the lead is than the styrofoam, which
I don't know. But hopefully, that gives you a
little intuition of what the center of mass is. And now I'll tell you something
a little more interesting. Every problem we have done so
far, we actually made the simplifying assumption that
the force acts on the center of mass. So if I have an object, let's
say the object that looks like a horse. Let's say that object. If this is the object's center
of mass, I don't know where the horse's center of mass
normally is, but let's say a horse's center of
mass is here. If I apply a force directly on
that center of mass, then the object will move in the
direction of that force with the appropriate acceleration. We could divide the force by the
mass of the entire horse and we would figure out the acceleration in that direction. But now I will throw in a twist.
And actually, every problem we did, all of these
Newton's Law's problems, we assumed that the force acted
at the center of mass. But something more interesting
happens if the force acts away from the center of mass. Let me actually take
that ruler example. I don't know why I even
drew the horse. If I have this ruler again and
this is the center of mass, as we said, any force that we act
on the center of mass, the whole object will move in the
direction of the force. It'll be shifted by the
force, essentially. Now, this is what's
interesting. If that's the center of mass and
if I were to apply a force someplace else away from the
center of mass, let' say I apply a force here, I want you
to think about for a second what will probably happen
to the object. Well, it turns out that the
object will rotate. And so think about if we're on
the space shuttle or we're in deep space or something, and
if I have a ruler, and if I just push at one end of the
ruler, what's going to happen? Am I just going to push the
whole ruler or is the whole ruler is going to rotate? And hopefully, your intuition
is correct. The whole ruler will rotate
around the center of mass. And in general, if you were to
throw a monkey wrench at someone, and I don't recommend
that you do, but if you did, and while the monkey wrench is
spinning in the air, it's spinning around its
center of mass. Same for a knife. If you're a knife catcher,
that's something you should think about, that the object,
when it's free, when it's not fixed to any point, it rotates
around its center of mass, and that's very interesting. So you can actually throw random
objects, and that point at which it rotates
around, that's the object's center of mass. That's an experiment that you
should do in an open field around no one else. Now, with all of this, and
I'll actually in the next video tell you what this is. When you have a force that
causes rotational motion as opposed to a shifting motion,
that's torque, but we'll do that in the next video. But now I'll show you just a
cool example of how the center of mass is relevant in everyday
applications, like high jumping. So in general, let's say
that this is a bar. This is a side view of a
bar, and this is the thing holding the bar. And a guy wants to jump
over the bar. His center of mass is-- most
people's center of mass is around their gut. I think evolutionarily that's
why our gut is there, because it's close to our
center of mass. So there's two ways to jump. You could just jump straight
over the bar, like a hurdle jump, in which case your center
of mass would have to cross over the bar. And we could figure out this
mass, and we can figure out how much energy and how much
force is required to propel a mass that high because we know
projectile motion and we know all of Newton's laws. But what you see a lot in the
Olympics is people doing a very strange type of jump,
where, when they're going over the bar, they look something
like this. Their backs are arched
over the bar. Not a good picture. But what happens when someone
arches their back over the bar like this? I hope you get the point. This is the bar right here. Well, it's interesting. If you took the average of
this person's density and figured out his geometric center
and all of that, the center of mass in this
situation, if someone jumps like that, actually travels
below the bar. Because the person arches their
back so much, if you took the average of the total
mass of where the person is, their center of mass actually
goes below the bar. And because of that, you can
clear a bar without having your center of mass go as high
as the bar and so you need less force to do it. Or another way to say it, with
the same force, you could clear a higher bar. , Hopefully, I didn't confuse you,
but that's exactly why these high jumpers arch their
back, so that their center of mass is actually below the bar
and they don't have to exert as much force. Anyway, hopefully you found that
to be a vaguely useful introduction to the center of
mass, and I'll see you in the next video on torque.