Change in centripetal acceleration from change in linear velocity and radius: Worked examples

- [Instructor] We are told
that a van drives around

a circular curve of radius
r with linear speed v.

On a second curve of the same radius,

the van has linear speed 1/3 v.

And you could view linear
speed as the magnitude

of your linear velocity.

How does the magnitude of the
van's centripetal acceleration

change after the linear speed decreases?

So pause this video and see
if you can figure it out

on your own, and I'll give
you a little bit of a hint.

We know that the magnitude
of centripetal acceleration

in general is equal to
linear speed squared

divided by radius.

The radius of the curve.

Alright, now let's work
through this together.

So let's first think
about the first curve.

So the first curve the
magnitude of our centripetal

acceleration for curve one,
I have another subscript

one here, this is around the first curve.

They tell us that our linear speed is v.

So we have v squared over and
the radius of that curve is r.

This is going to be a
straight up v squared over r

for that first curve.

The magnitude of our
centripetal acceleration.

Now what about the second curve?

So the magnitude of our
centripetal acceleration

around second curve,
that's what that two is,

is going to be equal to,

they tell us we now have
a linear speed of 1/3 v.

So in our numerator
we're gonna square that,

1/3 v squared all of that over

the curve of the same radius.

So our radius is still r.

So let's just do a little
algebraic simplification.

1/3 v times 1/3 v is just going to be

1/9 v squared.

So it's going to be 1/9 v squared over r.

All I did is square this numerator here.

Or I could write this as
1/9 times v squared over r.

The reason why I wrote
this in green is because

this is the exact same thing as this.

And so this is going to be equal to,

this is equal to 1/9
times, instead of writing

v squared over r, I could say hey that's r

the magnitude of our
centripetal acceleration

around the first curve.

The magnitude of our
centripetal acceleration

around the first curve.

So how does the magnitude
of the van's centripetal

acceleration change after
the linear speed decreases?

Well around the second curve,

we have 1/9 the magnitude
of centripetal acceleration.

So we could say the magnitude,

or I could just say, they already asked us

how does the magnitude change

so we could say decreases,

decreases by a factor

factor of nine.

And I wrote it in this language.

You could say it got
multiplied by a factor of 1/9

or you could say decreases
by a factor of nine

because on the Khan Academy exercises

that deal with this, they
use language like that.

Let's do another example.

Here we are told a
father spins his daughter

in a circle of radius r
at angular speed omega.

Then the father extends
his arms and spins her

in a circle of radius two r
with the same angular speed.

How does the magnitude of
the child's centripetal

acceleration change when
the father extends his arms?

Once again pause this
video and see if you can

figure it out.

Well the key realization
here and we derived this

at a previous video, is to
realize that the magnitude

of centripetal acceleration
is equal to r times

our angular speed squared.

And so initially so the
magnitude of our centripetal

acceleration initially,
I'll do that with a sub i.

That is going to be equal
to, well they're using

the same notation.

We have omega as our angular speed.

And our radius is r.

So it's just going to be r omega squared.

And then when we think about the father,

he extends his arms.

So then you have the
magnitude of your centripetal

acceleration, I could
say final or extended or

I'll just say final sub f.

What does that going to be equal to?

Well now our radius, the
radius of our circle is two r.

So it's going to be two r

and they say the same angular speed.

So our angular speed is still omega.

Two r omega squared.

Well this part right over
here r omega squared,

that was just the magnitude of our initial

centripetal acceleration.

That was the magnitude of our initial

centripetal acceleration.

And so you see that the magnitude of our

centripetal acceleration, has
increased by a factor of two.

Increased, increased by a factor

of two.

And we are done.