Let's learn a little bit about centripetal acceleration.
So let's say I have a dot.
It could be anything.
And we're in deep space, so we're not going to think about
gravity and all these things yet.
Let's say I have some object that's floating through space
with some velocity.
Let me draw the object.
It's this dot.
And it has some velocity.
And I will draw its velocity vector.
Let's say that this is-- I'll draw this really big.
That's not big.
Let me make a thicker line.
There you go.
Thicker line.
If you can see that.
So let's say that's the dot's velocity vector.
And I have a question for you.
So this is what the dot's doing at exactly this moment.
Let's say a little bit later, I want the dot to be doing
something like this.
I want its velocity vector to look like-- i want its
velocity factor to look like this.
A little bit later.
And its color changes to yellow.
That's out of the scope.
Color changing is out of the scope of this lecture.
But let's say a little bit later I want its velocity
vector to look something like that.
So I was a little careful, because I want it to have the
same magnitude, but just a little different direction.
I want it to tilt into the right a little bit.
Let me draw that.
And my question is, what had to be the acceleration acting,
when this was its velocity vector, in order to get its
velocity vector to look like this a little bit later?
Well, right here, its velocity vector is
completely in the x direction.
And there's no y component.
Let's break down this velocity vector
into the x and y component.
So its x component looks something like this.
And its y component looks something like this.
So in order to change its velocity, something would have
to-- Because clearly here its y component was 0.
There's no y component.
And now there is a y component.
So the acceleration that you would have to apply to this
dot would have to have some y component.
The acceleration would have to have some y component.
Let's say that.
This is acceleration, that's why it's not the same size.
Actually let me draw acceleration
in a different color.
And I'll try to do this in a bunch of different ways, so
that you get an intuition for it.
I'm doing it in a slightly different blue.
So you have to accelerate in the y direction a bit, and we
don't know how long, but long enough to get its velocity in
the y direction this big.
And then what would you have to
accelerate in the x direction?
Well, your x component shrinks a little bit.
This was its x component, and now since you're tilted down,
you get more y and less x.
So what direction does acceleration have to go in to
shrink the x component?
You have to go opposite.
So your acceleration is going to go that way a little bit.
And so if you were to add these two vectors, you would
have to accelerate something-- the acceleration vector would
look something like this.
So I'm-- And this might be a little confusing to you, but
all I'm showing is that, I want you to think about-- Say
an object is traveling through space at a constant velocity,
what does the acceleration vector have to look like in
order for that object to curve a little bit?
So for example, if this was the-- let me draw, let's say
this object was here.
And if we had no force on it, it would just keep moving in
this direction.
Its path would look like this.
I'll draw the path in brown.
The path would look like this.
It would be here, and then it would be here
a few seconds later.
It would just keep going in the direction that it's at.
And we know that from Newton's law, an object in motion tends
to stay in motion.
And the only way you can have a change in velocity is if you
have some net force and some net
acceleration on the object.
So in order for the object to curve, in order for the
object's path to look something like this-- so
here's the other path.
Let's say the object's path is like this.
And let's say it keeps curving.
The object's path is like that.
It keeps curving.
Goes there, to there.
It's going in a curved motion.
What has to keep happening?
Well, in order for the object's velocity to go from
this direction, in the x direction, in order for its
velocity to go from this, to go from this.
You have to accelerate a little bit inwards.
We drew that right here.
You have to have an inward acceleration.
And now its velocity-- and now if you did nothing else, and
the object will just keep going in this direction.
So in order to make it curve little more, you've got to
pull in a little bit again.
So you have to have-- and then if you don't pull in again,
it'll just keep going in this direction.
So you got to keep pulling inwards on it.
And I think you're starting to get a sense of what I'm
saying, so if you want the object to go in a circle, for
example, there has to be constant inward force always
pulling on the object.
Because if there isn't that constant inward-- so this is
this off white color, this is when there's a
constant inward force.
The object's going to travel in a circular path.
Well, what if that inward force doesn't exist?
Let's say that inward force is there, and then all of a
sudden when the object's here, the inward force disappears.
Then the object's just going to go in a straight line.
Tangent to the circular path.
And that makes more sense.
And I'll go into it in a little bit
more detail in a second.
But if you are spinning an object around in
a circle on a string.
And the string is providing the inward force, and as soon
as you let go of the string, the object goes off in a
straight line.
And so that that's what we're talking about.
So what does this inward acceleration has-- actually,
before I go into actually calculating what that inward
acceleration has to be, let's think about what happens when
that inwards acceleration isn't enough
or if it's too much?
So I just said, if I have an-- let's say this is like the
center of our rotation.
Let's say this is where the inward force is coming from.
Right there.
And let's say this is my object.
Let say it's a spaceship.
That's my spaceship.
Fire coming out of it.
And actually the fire would only go on an impulse.
Because once you have a little fire coming out in space,
there's no wind resistance, so it'll keep just going.
You don't have to keep-- so the science fiction movies,
where the fire just keeps going, that
doesn't make sense.
You would just have an impulse of fire that would accelerate
you, and then you wouldn't have to keep fueling the fire
in the back of the spaceship.
But anyway, maybe you realize it already, or that's out of
the scope of this lecture, but let's say that the object's
velocity is like that.
And I don't know, let's say this is a planet, or it's some
kind of weird forcefield, or something.
If this forcefield is a weak forcefield, if this object is
moving really fast, and this is a weak forcefield that's
always pulling in towards it, the object's path is going to
look something like this.
It's not going to be a circle.
It's going to kind of spiral outwards.
So that's kind of a weak inward force.
If it's a really strong inward force, the object's path is
going to look like this.
And I'll do that in yellow.
It's actually going to fall in.
It's going to spiral into the object.
And if it's just right, given how far the object is, and
I'll give you a formula for what just right implies.
But if the inward force is just right, the object's going
to essentially orbit around the subject.
It's going to go in a perfect circle.
Its path will be like that.
So let's think about what that inward force has to be.
And we're not going to-- well, this could have been gravity,
this inward force, but I won't work with gravity yet, because
gravity, actually the force changes depending on how close
you get to it.
So I won't deal with gravity yet.
But it could've been anything, it's just a tractor beam.
So let's try to get an intuition of how strong that
inward force is, and what variables it's dependent on.
So let's say I have a let's say this is my hand, and I
have a string.
And on that string, I have a rock.
And my question is, if I want this rock just spinning around
in a circle, and it'll spin around in a perfect circle.
We've all done this before.
If I want the rock to spin around in that circle, what
has to be the force that I pull on this string?
Or essentially, how much does the inward acceleration have
to be on that rock?
Let's get an intuition for maybe what has to happen.
So think about it, if the rock is moving, let's say that
right here, at this point, its velocity looks
something like that.
At this point, it's velocity looks something like this.
And then when the rock is here, its velocity looks
something like this.
So to move from this velocity to this velocity, if the
magnitude is really big, I'm going to have to apply more
acceleration to change direction.
Hopefully that makes a little sense.
To go from this velocity vector to
this velocity vector.
If this vector was even bigger, if to say the vector
looked like this, it had more magnitude.
I would have to pull inward with even more force and
accelerate inward at an even higher rate.
So definitely, the higher the velocity the more I'm going to
have to pull in.
And also the higher the velocity, the shorter the time
between this and this.
This object is moving around in a circle.
So there's kind of two components that are
affecting-- velocity affects the acceleration
I need in two ways.
Well, actually I'm running out of time, so I'll continue this
in the next video.