Connecting period and frequency to angular velocity | AP Physics 1 | Khan Academy

- [Instructor] What we're going to do

in this video is continue talking

about uniform circular motion.

And in that context, we're gonna talk

about the idea of period,

which we denote with a capital T,

or we tend to denote with a capital T,

and a very related idea.

And that's of frequency,

which we typically denote
with a lower case f.

So you might have seen these
ideas in other contexts,

but we'll just make sure we get them.

And then we'll connect it to the idea

of angular velocity, in
particular the magnitude

of angular velocity,
which we've already seen

we can denote with a lower case omega.

Since I don't have a little arrow on top,

you could view it, just
the lower case omega,

as the magnitude of angular velocity.

But first, what is period
and what is frequency?

Well, period is how long does it take

to complete a cycle?

And if we're talking about
uniform circular motion,

a cycle is how long does it take,

if this is, say, some
type of a tennis ball

that's tethered to a nail right over here

and it's moving with some uniform speed,

a period is, well, how long does it take

to go all the way around once?

So, for example, if you
have a period of one second,

this ball would move like this,

one second, two seconds,

three seconds, four seconds.

That would be a period of one second.

If you had a period of two seconds,

well, it would go half the speed.

You would have one second, two seconds,

three seconds, four seconds,

five seconds, six seconds.

And if you went the other way,

if you had a period of half a second,

well, then it would be one second,

two seconds,

and so your period would be half a second.

It would take you half a
second to complete a cycle.

The unit of period is
going to be the second,

the unit of time and it's
typically given in seconds.

Now, what about frequency?

Well, frequency literally is
the reciprocal of the period.

So frequency is equal to one over,

let me write that one a little bit neater,

one over the period.

And one way to think about it is

well, how many cycles can
you complete in a second?

Period is how many seconds does it take

to complete a cycle, while frequency is

how many cycles can you do in a second?

So, for example, if I can
do two cycles in a second,

one second, two seconds,

three seconds,

then my frequency is
two cycles per second.

And the unit for frequency is,

sometimes you'll hear
people say just per second,

so the unit, sometimes
you'll see people just say

an inverse second like that,

or sometimes they'll use the shorthand Hz,

which stands for Hertz.

And Hertz is sometimes substituted
with cycles per second.

So this you could view as seconds or even

seconds per cycle.

And this is cycles per second.

Now with that out of the way,

let's see if we can connect these ideas

to the magnitude of angular velocity.

So let's just think about
a couple of scenarios.

Let's say that the magnitude
of our angular velocity,

let's say it is pi radians,

pi radians per second.

So if we knew that, what
is the period going to be?

Pause this video and see
if you can figure that out.

So let's work through it together.

So, this ball is going to move

through pi radians every second.

So how long is it going to take

for it to complete two pi radians?

'Cause remember, one complete
rotation is two pi radians.

Well, if it's going pi radians per second,

it's gonna take it two
seconds to go two pi radians.

And so, the period here, let me write it,

the period here is going
to be equal to two seconds.

Now, I kind of did that intuitively,

but how did I actually
manipulate the omega here?

Well, one way to think
about it, the period,

I said, look, in order to
complete one entire rotation,

I have to complete two pi radians.

So that is one entire cycle
is going to be two pi radians.

And then I'm gonna divide it by how fast,

what my angular velocity is going to be.

So I'm gonna divide it by,

in this case I'm gonna
divide it by pi radians,

pi, and I could write it
out pi radians per second.

I'm saying how far do I have to go

to complete a cycle and
that I'm dividing it

by how fast I am going through the angles.

And that's where I got
the two seconds from.

And so, already you can think of a formula

that connects period and angular velocity.

Period is equal to,
remember, two pi radians

is an entire cycle.

And so you just want to
divide that by how quickly

you're going through the angles.

And so that there will connect your period

and angular velocity.

Now if we know the period,

it's quite straightforward
to figure out the frequency.

So the frequency is just
one over the period.

So the frequency is,
we've already said it's

one over the period, and so the reciprocal

of two pi over omega is going to be

omega over two pi.

And in this situation where
the period was two seconds,

you don't even know what omega is,

and someone says the
period is two seconds,

then you know that the frequency

is going to be one over two seconds.

Or, you could view this
as being equal to 1/2.

You could sometimes see units like that,

which is kind of per second.

But I like to use Hertz, and in my brain

I say this means 1/2 cycles per second.

So one way to think about it,

it takes two seconds to complete.

If I'm doing pi radians per second,

my ball here is going to go one second,

two seconds, three seconds, four seconds.

And you see, just like that,

my period is indeed two seconds.

And you also see that in each second,

remember in each second
I cover pi radians.

Well, pi radians is half a cycle.

I complete half a cycle per second.