- [Instructor] Just for kicks,
let's imagine someone spinning a flaming tennis ball
attached to some type of a string or chain
that they're spinning it above their head like this
and let's say they're spinning it at a constant speed.
We've already described situations like this,
maybe not with as much drama as this one
but we can visualize the velocity vectors
at different points for the ball.
So, at this point, let's say the velocity vector
will look like this,
the linear velocity vector just to be super clear.
So, the linear velocity vector
might look something like that
and it's going to have magnitude V,
the magnitude of the velocity vector
you could also view as its linear speed.
Now, a few moments later,
what is the ball going to be doing?
Well, a few moments later
the ball might be let's say right over here,
we don't wanna lose the drama,
it's still flaming we're assuming,
it's still attached to our chain right over here
but what would its velocity vector be?
Well, we're assuming it has a constant speed,
a constant linear speed,
so the magnitude is going to be the same
but now the direction is going to be tangential
to the circular path at that point.
So, our direction has changed.
Now, one way to think about this change
in direction of velocity,
it's a little counterintuitive at first
because when we first think about acceleration,
we tend to think in terms of changing the magnitude
of velocity but keeping the magnitude the same,
but changing the direction
still involves an acceleration
and at first, it's a little counterintuitive
the direction of that acceleration
but if I were to take this second velocity vector
and if I were to shift it over here,
and if I were to start it at the exact same point,
it would look something like this,
actually, let me do it in a slightly different color
so it's a little bit more visible,
so it would look something like this.
This and this, they have the same length
and they're parallel, so they are the same vector
and so, in some amount of time,
if you wanna go from this velocity vector
and actually this should be a little bit longer,
this should look like this,
it should have the same magnitude as this one,
so it should look like this.
So, if in some amount of time,
this and this should have the same magnitude.
If in some amount of time
you go from this velocity vector
to this velocity vector,
your net change in velocity
is going radially inward.
This right over here is your net change in velocity
and so, in other videos
we talk about this notion of centripetal acceleration.
In order to keep something going
in this uniform circular motion,
in order to keep changing the direction
of our velocity vector,
you are accelerating it radially inward,
centripetal acceleration, inward acceleration
and so, at all points in time,
you have an inward acceleration
which we denote, the magnitude,
we usually say is A with a C subscript for centripetal.
Sometimes you'll see an A with an R subscript for radial
but in this context we will use centripetal.
Now, one question that you might have been wondering
this whole time that we talked
about centripetal acceleration
is Newton's First Law might be nagging.
Newton's First Law tells us that the velocity
of an object both it's magnitude and its direction
will not change unless there's some net force
acting on the object
and we clearly see here
that the direction of our velocity vector is changing,
so Newton's First Law tells us
that there must be some net force acting on it
and that net force is going to be acting
in the same direction as our acceleration
and so, what we're gonna do here
is introduce an idea of centripetal force.
So, centripetal force,
if it's accelerating the object inwards
in the inward direction,
so we have a centripetal force
that is causing our centripetal acceleration.
F sub C right over here,
you can view that as the magnitude of our centripetal force
and the way that they would be connected
comes straight out for Newton's Second Law.
This isn't some type of new, different type of force,
this is the same type of forces
that we talk about throughout physics.
We know that the magnitude of our centripetal force
is going to be equal to the mass of our object
times the magnitude of our centripetal acceleration.
If you want, you could put vectors on top of this.
You could say something like this
but we know the direction of the centripetal force
and the centripetal acceleration, it is inward.
Now, what inward means,
the exact arrow's going to be different
at different points but for any position for the ball
we know at least conceptually
what inward is going to be.
So, this is just to appreciate the idea.
Centripetal acceleration in classical mechanics
isn't just going to show up out of nowhere.
Newton's First Law tells us
that if something is being accelerated,
there must be a net force acting on it
and if it's being accelerated inward
in the centripetal direction I guess you could say,
then the force must also be acting inward
and they would just be related by F equals MA
which we learn from Newton's Second Law.
And to appreciate the intuition for this,
just remember the last time
that you were spinning or rotating a flaming tennis ball
attached to a chain above your head.
In order to do that, in order to keep the ball spinning
and not just going and veering off in a straight line,
you can keep pulling inward on your chain
so that the flaming tennis ball doesn't go hit a wall
and set things onto fire.
And so, what you are providing is that centripetal force
to keep that flaming tennis ball
in its uniform circular motion.