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- [Voiceover] The figure
above shows a string

with one end attached to an oscillator

and the other end
attached to a block there.

There's our block.

The string passes over a
massless pulley that turns

with negligible friction.

There's our massless pulley that turns

with negligible friction.

Four such strings, A, B, C,
and D are set up side-by-side

as shown in the figure
in the diagram below.

So this is a top view,
you can see oscillator.

This is a top view of the
oscillator string pulley

mass system, and we have four of them.

Each oscillator is adjusted
to vibrate the string

at its fundamental frequency f.

So let's think about what
it, I'll keep reading

then we'll think about what
fundamental frequency means.

The distance between each
oscillator and the pulley L

is the same.

So the length between the
oscillator and the pulley

is the same, and the mass
of each block is the same.

So the mass is what's providing
the tension in the string.

However, the fundamental
frequency of each string

is different.

So let's just first of all think about

what the fundamental frequency is,

and then let's think about
what makes them different.

So the fundamental frequency,

one way you could think about it is

it's the lowest frequency
that is going to produce

a standing wave in your string.

So it's the frequency that
produces a standing wave

that looks like this.

It's a standing wave where the
string is half a wavelength.

I guess there's two
ways to think about it.

It's the lowest frequency we could produce

the standing wave, or it's
the frequency at which

you're producing the
wave, the standing wave,

with the longest wavelength.

So the string at the fundamental frequency

is just going to go to, is gonna vibrate

between those two positions.

And you see here that the wavelength here

is twice the length of the string.

And if you want to see
that a little bit clearer,

if I were to continue with this wave,

I would have to go another
length of the string

in order to complete one wavelength.

Or another way to think about it,

you're going up, down, and
then you're going down back.

It's reflecting back off of this end here.

And as we mentioned, essentially
what the mass is doing

is providing the tension, the
force of gravity on this mass

is providing the tension in this string.

So the oscillators is vibrating
at the right frequency

to produce this, the lowest
frequency where you can produce

this standing wave.

So let's answer the questions
now, and we have four

of this set up and they all
have different frequencies.

The equation for the velocity
v of a wave on a string

is v is equal to the
square root of the tension

of the string divided by
the mass of the string

divided by the length of the string,

where F sub T is tension of the string

and m divided by L is
the mass per unit length,

linear mass density of the string.

And hopefully this makes sense.

It makes sense that it's
going to be that if,

if the tension increases, that
your velocity will increase,

the tension, you could think
of the atoms of the string

of how much they're pulling on each other.

And so if there's higher tension, well,

they're going to be able
to move each other better,

or I guess you could say
accelerate each other better

as the wave goes through the string.

And also make sense that
the larger the mass,

if you hold everything else equal,

that you're gonna have a slower velocity.

Mass, you could view it

as a measure of inertia,
it's how hard it is

to accelerate something.

So if the string itself,
especially the mass

per unit length, if there's a
lot of mass per unit length,

actually, let me circle
that because that's actually

the more interesting thing.

If there's a lot of mass per unit length,

it makes sense that for a
given amount of the string,

it's gonna be harder to
accelerate it back and forth

as you vibrate it.

And so this part right over
here would be inversely related

to the velocity.

It's not proportional though.

You have this square root here,

but they're definitely,

if the linear mass density increases,

then you're gonna have a slower velocity.

And if your tension increases,
you're going to have

a higher velocity.

So hopefully this makes
some intuitive sense.

And they ask, what is the
difference about the four string

shown above that would
result in having different

fundamental frequencies?

Explain how you arrive at your answer.

And then part b is
student graphs frequency

is a function of the inverse
of the linear mass density.

Will the graph be linear?

Explain how you arrived at your answer.

All right, let's answer each of these.

So a, part a, what is different
about the four strings

because they all have different
fundamental frequencies?

So the fundamental frequency,

let's just go back to
what we know about waves,

that the velocity of a wave
is equal to the frequency

times the wavelength of the wave.

Or you could say, you could
divide both sides by lambda,

you could say that the frequency of a wave

is equal to the velocity
over the velocity over

the wavelength.

So if we're talking about
the fundamental frequency,

if we're talking about
the fundamental frequency,

funda, let me just,

let me write frequency and
then let me write fundamental

real small here.

Fundamental, the fundamental frequency,

is going to be the velocity of our waves

divided by the wavelength is
going to be twice the length

of our string, divided by two L.

And if you look at the
expression that they gave us

for the velocity of
the wave on the string,

well, this is going to be equal to

the square root of the
tension in the string

divided by, divided by
the linear mass density.

Divided by the linear mass density.

And all of that is going to be over two L.

Now all of them have different
fundamental frequencies,

but let's think about
what's different over here.

They all have the same tension.

They all have the same tension.

How do I know that?

Well, what's causing the
tension is the masses

hanging over the pulleys.

So the weight of those masses.

So that's all going to be
the same for all of them,

and they all have the same length.

They all have the same length.

So these are all the same.

So the only way that you're
going to have different

fundamental frequencies is
if you have different masses.

So different masses.

So that has to be different.

Different.

So to answer their question,

the strings must have different mass,

well, they would have different
linear mass densities,

but since they're all the same length,

they would also have different masses.

So let me write this down.

Strings, strings must

have different,

different masses

and mass densities,

and mass densities,

densities, since

all other variables

driving, driving

fundamental frequency are the same,

fundamental frequency

are the same,

are the same.

All right, let's tackle part b now.

A student graphs frequency as a function

of the inverse of linear mass density.

Will the graph be linear?

Explain how you arrived at your answer.

So student graphs frequency.

Let me underline this.

They're graphing frequency as a function

of linear mass density.

So we actually can write this down.

If we wanted to write frequency

as a function of linear mass densities,

so we could write as a
function of m over L,

well, this is going to be equal to,

is going to be equal to,

we could rewrite this expression,

actually, we could just
leave it like this.

This is the same thing as one over two L

times the square root of the tension

divided by the linear mass density.

Or you can do this as being equal to,

you could say this is a square root of

the tension over two L,

and I'm putting all of
this separate because

we're doing it as a function
of our linear mass density.

So we can assume that all of
this is going to be a constant.

And then times the square root

of one over the linear mass density.

So if you're plotting, if
you're plotting frequency

as a function of this,

it's not going to be,
the graph is definitely

not going to be linear.

You see here, one, I have
the reciprocal of it,

and then I take the square root of it.

So let me write this down.

This f or graph of f of two L,

definitely not linear.

Let me write it that way.

F of m over L graph,

definitely, definitely

not linear.

And you could point out
that it has a reciprocal

and it's a square root.

Involves, involves square root

and inverse,

or I could say and
reciprocal of the variable.

And reciprocal, reciprocal

of the linear mass density

of m over L.

So definitely not linear.

All right, let's tackle part c.

The frequency of the oscillator
connected to string D

is changed so that the string vibrates

in its second harmonic.

On the side view of string D below,

mark and label the points
on the string that have

the greatest average vertical speed.

So one way to think about
the fundamental frequency,

that's our first harmonic.

So if they're talking
about the first harmonic

and the string is what I showed before,

that's the lowest frequency
that produces a standing wave

or it's the frequency
that produces the largest

standing wave, I guess
you could say the one

with the largest wavelength.

And if it was a first harmonic,

the part of the string that moves the most

is going to be that center.

But our second harmonic is
the next highest frequency

that produces a standing wave,

and that's going to be
a situation in which

the wave, instead of having
a wavelength twice the length

of the string, it's
gonna have a wavelength

equal to the length of the string.

So now, it's going to look like this.

So it's going to vibrate between this,

actually, let me draw it a
little bit neater than that.

It's going to vibrate between,

between this.

And so

it's gonna vibrate between that and this.

And this right over here.

So when you see this, this version of it,

where now that we have
half the wavelength,

the wavelength is exactly
the wavelength of,

it's actually L this time,

the part that moves the most is here.

So that's going to move the most.

And here.

I didn't draw it perfectly.

But one way to think
about it, it's exactly 1/4

and 3/4 of the way.

Exactly halfway is not
going to move as much,

is not gonna move because
it's a standing wave now

or it's gonna move very imperceptibly.

These two places are where
you're gonna move the most

at the first, or sorry,
at the second harmonic.

The second harmonic being
the next highest frequency

that produces a standing wave again.