Torque, moments and angular momentum
Center of Mass Introduction to the center of mass
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- 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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