Treating systems (the hard way)

WEBVTT
Kind: captions
Language: en

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- [Voiceover] Alright,
this problem is a classic,

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you're gonna see this in basically

00:00:03.648 --> 00:00:05.617
every single physics text book.

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And the problem is this,
if you've got two masses

00:00:07.457 --> 00:00:10.978
tied together by a rope, and
that rope passes over a pulley,

00:00:10.978 --> 00:00:13.659
what's the acceleration of the masses?

00:00:13.659 --> 00:00:17.826
In other words, what's the
acceleration of the 3 kg mass?

00:00:18.890 --> 00:00:21.961
And then what's the
acceleration of the 5 kg mass?

00:00:21.961 --> 00:00:26.072
And if you're wondering
what the heck is a pulley?

00:00:26.072 --> 00:00:28.027
The pulley is this part right here.

00:00:28.027 --> 00:00:30.027
This right here is the pulley.

00:00:30.027 --> 00:00:33.498
So what a pulley does, a pulley
is a little piece of plastic

00:00:33.498 --> 00:00:35.788
or metal that can rotate.

00:00:35.788 --> 00:00:38.258
And it's usually got a
groove in it so that a string

00:00:38.258 --> 00:00:40.017
or a rope can pass over it.

00:00:40.017 --> 00:00:44.149
What it does, is it rotates
freely so that you can turn

00:00:44.149 --> 00:00:46.429
what's a horizontal tension on one side,

00:00:46.429 --> 00:00:48.380
into a vertical tension on the other.

00:00:48.380 --> 00:00:49.912
Or vice versa.

00:00:49.912 --> 00:00:52.461
It turns vertical forces
into horizontal forces.

00:00:52.461 --> 00:00:55.342
It allows you to transfer
a force from one direction

00:00:55.342 --> 00:00:56.741
to another direction.

00:00:56.741 --> 00:00:58.793
So that's what these
pulleys are useful for.

00:00:58.793 --> 00:01:01.263
And if they can spin
freely, and if this pulley

00:01:01.263 --> 00:01:03.954
has basically no mass,
if there's no resistance

00:01:03.954 --> 00:01:07.161
to motion at all, then
this tension on this side

00:01:07.161 --> 00:01:10.542
is gonna be equal to the
tension on this side.

00:01:10.542 --> 00:01:14.092
This vertical tension gets
transferred fully undiluted,

00:01:14.092 --> 00:01:17.459
into a horizontal tension
and these tension values

00:01:17.459 --> 00:01:20.898
will just be the same if
this pulley can spin freely.

00:01:20.898 --> 00:01:22.878
And if its mass is really small so that

00:01:22.878 --> 00:01:26.926
there's no inertial reason
why it doesn't wanna spin.

00:01:26.926 --> 00:01:28.398
So that's the problem.

00:01:28.398 --> 00:01:30.459
Let's say you wanted to figure this out.

00:01:30.459 --> 00:01:33.899
What is the acceleration of the 3 kg mass

00:01:33.899 --> 00:01:36.528
with the acceleration of the 5 kg mass?

00:01:36.528 --> 00:01:38.949
Now, I've gotta warn you,
there's an easy way to do this,

00:01:38.949 --> 00:01:40.616
and a hard way to do this.

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Now, I'm gonna show
you the hard way first.

00:01:43.376 --> 00:01:46.034
Sorry, no one ever wants to hear that.

00:01:46.034 --> 00:01:50.495
But, the reason is that, the
easy way won't make any sense

00:01:50.495 --> 00:01:51.975
unless I show you the hard way first.

00:01:51.975 --> 00:01:54.436
It won't make any sense why
the easy way works unless

00:01:54.436 --> 00:01:56.074
I show you the hard way.

00:01:56.074 --> 00:01:59.045
And for two, the hard way
isn't really all that hard.

00:01:59.045 --> 00:02:00.186
So I'm calling it the hard way,

00:02:00.186 --> 00:02:01.936
but it's not really that bad.

00:02:01.936 --> 00:02:04.185
And for three, sometimes teachers
and professors just wanna

00:02:04.185 --> 00:02:05.944
see you do it the hard
way, so you should know how

00:02:05.944 --> 00:02:07.186
to do this.

00:02:07.186 --> 00:02:08.019
So what do we do?

00:02:08.019 --> 00:02:09.946
We wanna find acceleration,
well you know how

00:02:09.946 --> 00:02:11.145
to find acceleration.

00:02:11.145 --> 00:02:13.142
We're gonna use Newton's second law.

00:02:13.142 --> 00:02:15.966
So we'll say that the
acceleration in a given direction

00:02:15.966 --> 00:02:18.166
is gonna equal the net
force in that direction,

00:02:18.166 --> 00:02:19.653
divided by the mass.

00:02:19.653 --> 00:02:20.965
Now what do we do?

00:02:20.965 --> 00:02:21.964
What mass are we gonna choose?

00:02:21.964 --> 00:02:23.824
We've got a couple masses here.

00:02:23.824 --> 00:02:26.123
One thing we could do, let's
just pick the 5 kg mass.

00:02:26.123 --> 00:02:27.761
Just pick one of them.

00:02:27.761 --> 00:02:30.703
So I'm gonna say that the
acceleration of the 5 kg mass

00:02:30.703 --> 00:02:34.613
is the net force on the 5
kg mass, divided by the mass

00:02:34.613 --> 00:02:36.313
of the 5 kg mass.

00:02:36.313 --> 00:02:38.845
And remember, we should always
pick a direction as well.

00:02:38.845 --> 00:02:40.694
So do we wanna pick the vertical direction

00:02:40.694 --> 00:02:42.403
or the horizontal direction?

00:02:42.403 --> 00:02:44.984
Well, since this box is gonna
be accelerating horizontally,

00:02:44.984 --> 00:02:47.361
and that's what we're interested
in, I'm gonna put one more

00:02:47.361 --> 00:02:49.843
sub-script up here, X, to
let us know we're picking

00:02:49.843 --> 00:02:52.025
the horizontal direction.

00:02:52.025 --> 00:02:54.814
So I can fill this out
now, I can plug stuff in.

00:02:54.814 --> 00:02:58.614
The acceleration of the 5
kg mass in the X direction

00:02:58.614 --> 00:03:00.474
is gonna be equal to.

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Alright what forces do we
have to figure out what goes

00:03:02.695 --> 00:03:03.923
up here?

00:03:03.923 --> 00:03:06.091
You always draw a force diagram.

00:03:06.091 --> 00:03:08.242
So what forces do I have on the 5 kg mass?

00:03:08.242 --> 00:03:10.375
I'm gonna have a force
of gravity, I'll draw

00:03:10.375 --> 00:03:11.933
that straight down.

00:03:11.933 --> 00:03:14.923
FG, and there's gonna be an equal force,

00:03:14.923 --> 00:03:16.821
normal force upward.

00:03:16.821 --> 00:03:19.532
So this normal force up
should be equal to the force

00:03:19.532 --> 00:03:23.113
of gravity and magnitude
because this box is probably not

00:03:23.113 --> 00:03:24.823
gonna be accelerating vertically.

00:03:24.823 --> 00:03:26.283
There's no real reason why it should be

00:03:26.283 --> 00:03:28.172
if this table is rigid.

00:03:28.172 --> 00:03:30.323
And there's one more force on this box.

00:03:30.323 --> 00:03:31.981
There's a force to the right.

00:03:31.981 --> 00:03:34.012
That's gonna be the force of tension.

00:03:34.012 --> 00:03:36.131
And if there's no friction on this table,

00:03:36.131 --> 00:03:38.259
then I have no leftward forces here.

00:03:38.259 --> 00:03:39.722
I'm ignoring air resistance since

00:03:39.722 --> 00:03:42.444
we usually ignore air resistance.

00:03:42.444 --> 00:03:43.695
So that's is, the only horizontal force

00:03:43.695 --> 00:03:45.675
I've got is T, tension.

00:03:45.675 --> 00:03:50.606
And I divide by the mass of
the 5 kg box, which is 5 kg.

00:03:50.606 --> 00:03:51.944
But we got a problem.

00:03:51.944 --> 00:03:54.994
Look it, we don't know the
acceleration of the 5 kg mass,

00:03:54.994 --> 00:03:56.115
and we don't know the tension.

00:03:56.115 --> 00:03:58.065
I can't solve this.

00:03:58.065 --> 00:03:59.865
Normally what you do in
this case, is you go to

00:03:59.865 --> 00:04:02.933
the vertical direction, the
other direction in other words.

00:04:02.933 --> 00:04:04.204
But that's not gonna help me either.

00:04:04.204 --> 00:04:06.214
That's just gonna tell me
that the normal force is gonna

00:04:06.214 --> 00:04:08.546
be equal to the force of gravity.

00:04:08.546 --> 00:04:09.594
And we kind of already knew that.

00:04:09.594 --> 00:04:10.894
So that doesn't help.

00:04:10.894 --> 00:04:12.035
So what do we do?

00:04:12.035 --> 00:04:14.492
Well, you might note, this
is only the equation for

00:04:14.492 --> 00:04:16.096
the 5 kg mass.

00:04:16.096 --> 00:04:18.613
And so now I have to do
this for the 3 kg mass.

00:04:18.613 --> 00:04:21.506
So let's come over here, let's
say that the acceleration

00:04:21.506 --> 00:04:25.575
of the 3 kg mass is gonna
be equal to the net force

00:04:25.575 --> 00:04:29.742
on the 3 kg mass, divided by
the mass of the 3 kg mass.

00:04:30.875 --> 00:04:33.104
And again, which direction should we pick?

00:04:33.104 --> 00:04:35.444
Well this acceleration over
here is gonna be vertical.

00:04:35.444 --> 00:04:37.813
So let's solve this for
the vertical direction.

00:04:37.813 --> 00:04:40.854
I'm gonna add one more
sub-script, Y, to remind myself.

00:04:40.854 --> 00:04:43.211
And you should do this too so
you remember which direction

00:04:43.211 --> 00:04:44.296
you're picking.

00:04:44.296 --> 00:04:46.894
So what forces do I plug in here?

00:04:46.894 --> 00:04:48.554
You figure that out with a force diagram.

00:04:48.554 --> 00:04:52.606
I'm gonna have a force of
gravity on this 3 kg mass,

00:04:52.606 --> 00:04:55.363
and then I'm gonna have
the same size of friction,

00:04:55.363 --> 00:04:57.906
or sorry, the same as
tension, that I had over here.

00:04:57.906 --> 00:04:59.784
So the tension on this side
of the rope, it's gonna

00:04:59.784 --> 00:05:01.554
be the same as the tension on this side.

00:05:01.554 --> 00:05:05.744
Assuming this pulley offers no
resistance either by its mass

00:05:05.744 --> 00:05:06.852
or friction.

00:05:06.852 --> 00:05:09.816
So assuming that its mass is
negligible, there's basically

00:05:09.816 --> 00:05:11.792
no friction, then I'm
gonna have a tension.

00:05:11.792 --> 00:05:14.681
That tension is gonna be the same size.

00:05:14.681 --> 00:05:16.733
So I'll draw that coming upward.

00:05:16.733 --> 00:05:19.592
But it's not gonna be as big
as the force of gravity is

00:05:19.592 --> 00:05:21.662
on this 3 kg mass.

00:05:21.662 --> 00:05:23.230
I've got the force of gravity here.

00:05:23.230 --> 00:05:26.163
This tension is gonna be
smaller, and the reason is,

00:05:26.163 --> 00:05:28.793
this 3 kg mass is accelerating downwards.

00:05:28.793 --> 00:05:30.691
So these forces can't be balanced.

00:05:30.691 --> 00:05:33.063
The upward force of tension
has gotta be smaller

00:05:33.063 --> 00:05:36.142
than the force of gravity
on this 3 kg mass.

00:05:36.142 --> 00:05:37.942
But this tension here should be the same

00:05:37.942 --> 00:05:39.780
as this tension here.

00:05:39.780 --> 00:05:40.934
So I'll plug those in.

00:05:40.934 --> 00:05:41.863
So let's plug this in.

00:05:41.863 --> 00:05:46.582
A of the 3 kg mass, in the Y
direction is gonna be equal to,

00:05:46.582 --> 00:05:48.434
I've got two vertical forces.

00:05:48.434 --> 00:05:50.324
I've got tension up, so
I'll make that positive,

00:05:50.324 --> 00:05:52.103
'cause we usually treat up as positive.

00:05:52.103 --> 00:05:55.525
I've got gravity down, and
so I'm gonna have negative,

00:05:55.525 --> 00:05:59.692
'cause it's downward force of
3 kg times the acceleration

00:06:01.200 --> 00:06:04.783
on Earth, is 9.8 meters
per second squared.

00:06:07.025 --> 00:06:07.858
Now what do we do?

00:06:07.858 --> 00:06:10.112
We divide by 3 kg, 'cause that's the mass.

00:06:10.112 --> 00:06:11.724
But I've still got a problem.

00:06:11.724 --> 00:06:14.974
I don't know this
acceleration or this tension.

00:06:14.974 --> 00:06:16.737
So what do I do?

00:06:16.737 --> 00:06:18.522
You might notice, if you're
clever you'll say wait,

00:06:18.522 --> 00:06:21.093
I've got my unknown on
this side is acceleration

00:06:21.093 --> 00:06:22.362
and tension.

00:06:22.362 --> 00:06:24.924
My unknown on this side is
acceleration and tension.

00:06:24.924 --> 00:06:27.223
It seems like I've got two
equations, two unknowns,

00:06:27.223 --> 00:06:28.841
maybe we should combine them.

00:06:28.841 --> 00:06:30.784
And that's exactly how you do these.

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So I've got tension in
both of these equations.

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Let me solve for tension over here,

00:06:35.962 --> 00:06:37.483
where it's kind of simple.

00:06:37.483 --> 00:06:41.564
And I'll just get the tension equals 5 kg,

00:06:41.564 --> 00:06:45.731
times the acceleration of the
5 kg mass in the X direction.

00:06:47.334 --> 00:06:48.402
So now I know what tension is.

00:06:48.402 --> 00:06:49.894
Tension is equal to this.

00:06:49.894 --> 00:06:51.714
And that tension over on
this side is the same as

00:06:51.714 --> 00:06:53.374
the tension on this side.

00:06:53.374 --> 00:06:56.405
So I can take this and
I can plug it in for

00:06:56.405 --> 00:06:57.842
this tension right here.

00:06:57.842 --> 00:06:59.383
And let's see what we get.

00:06:59.383 --> 00:07:02.443
We get that the acceleration
of the 3 kg mass vertically,

00:07:02.443 --> 00:07:06.313
is gonna equal, alright, I'm
gonna have a big mess on top,

00:07:06.313 --> 00:07:07.522
what am I gonna get?

00:07:07.522 --> 00:07:11.039
I'm gonna get, so T is the same as 5AX.

00:07:11.039 --> 00:07:14.622
So I'll plug in 5 kg
times the acceleration

00:07:16.340 --> 00:07:19.131
of the 5 kg mass in the X direction.

00:07:19.131 --> 00:07:21.253
And then I get all of
this stuff over here.

00:07:21.253 --> 00:07:24.586
So I'll get the rest of this right here.

00:07:27.383 --> 00:07:29.971
I'll just bring that down right there.

00:07:29.971 --> 00:07:31.322
Alright, now what do I have?

00:07:31.322 --> 00:07:33.533
I've got 3 kg on the bottom
still, so I have to put

00:07:33.533 --> 00:07:35.022
that here.

00:07:35.022 --> 00:07:36.511
Are we any better off?

00:07:36.511 --> 00:07:38.412
Yeah, we're better, because
now my only unknowns

00:07:38.412 --> 00:07:39.913
are acceleration.

00:07:39.913 --> 00:07:41.973
But these are not the same acceleration.

00:07:41.973 --> 00:07:43.831
Look, this acceleration
here is the acceleration of

00:07:43.831 --> 00:07:46.473
the 3 kg mass, vertically.

00:07:46.473 --> 00:07:48.733
This acceleration here is the acceleration

00:07:48.733 --> 00:07:51.622
of the 5kg mass horizontally.

00:07:51.622 --> 00:07:53.781
Now here's where I'm gonna
have to make an argument,

00:07:53.781 --> 00:07:55.100
and some people don't like this.

00:07:55.100 --> 00:07:58.213
But, it's crucial to
figuring out this problem.

00:07:58.213 --> 00:08:03.043
And the key idea is this, if
this 3 kg mass moves down,

00:08:03.043 --> 00:08:06.233
let's say one meter, let's say
it moves downward one meter.

00:08:06.233 --> 00:08:10.502
Well then this 5 kg mass had
better move forward one meter.

00:08:10.502 --> 00:08:14.281
Because if it doesn't, then it
didn't provide the one meter

00:08:14.281 --> 00:08:17.980
of rope that this 3 kg
mass needed to go downward.

00:08:17.980 --> 00:08:21.780
Which means either the rope
broke, or the rope stretched.

00:08:21.780 --> 00:08:23.654
And we're gonna assume that
our rope does not break

00:08:23.654 --> 00:08:25.114
or stretch.

00:08:25.114 --> 00:08:25.993
That's kind of a lie.

00:08:25.993 --> 00:08:29.023
All ropes are gonna stretch
a little bit under tension.

00:08:29.023 --> 00:08:31.911
We're gonna assume that
stretch is negligible.

00:08:31.911 --> 00:08:35.251
So the argument is that if
this 3 kg mass moves downward

00:08:35.251 --> 00:08:38.131
a certain amount, this 5
kg mass has to move forward

00:08:38.131 --> 00:08:41.010
by that same amount in order
to feed that amount of rope

00:08:41.010 --> 00:08:44.731
for this 3 kg mass to go
downward by that amount.

00:08:44.731 --> 00:08:45.741
Otherwise, think about it.

00:08:45.741 --> 00:08:49.630
If this 5 kg mass just sat
here and the 3 kg moved,

00:08:49.630 --> 00:08:53.070
or the 3 kg moved farther
than the 5 kg mass,

00:08:53.070 --> 00:08:54.880
then this rope is stretching or breaking.

00:08:54.880 --> 00:08:56.359
So if you believe that,
if you don't believe it,

00:08:56.359 --> 00:08:57.839
pause it and think about it.

00:08:57.839 --> 00:08:59.101
'Cause you've gotta
convince yourself of that.

00:08:59.101 --> 00:09:01.560
If you believe that then you
can also convince yourself

00:09:01.560 --> 00:09:04.369
that, well if the 3 kg
mass was moving downward

00:09:04.369 --> 00:09:07.260
at a certain speed, let's
say two meters per second.

00:09:07.260 --> 00:09:10.140
Then the 5 kg mass had
better also be moving forward

00:09:10.140 --> 00:09:13.300
two meters per second
because otherwise it wouldn't

00:09:13.300 --> 00:09:16.972
be feeding rope at a
rate that this 3 kg needs

00:09:16.972 --> 00:09:18.641
to move downward at that rate.

00:09:18.641 --> 00:09:20.412
And finally, if you believe all that,

00:09:20.412 --> 00:09:22.452
it's not too much harder
to convince yourself that

00:09:22.452 --> 00:09:25.198
this 3 kg mass, no matter what

00:09:25.198 --> 00:09:27.880
its acceleration downward must be,

00:09:27.880 --> 00:09:30.692
this 5 kg mass had better
have the same magnitude

00:09:30.692 --> 00:09:34.341
of acceleration forward
so that it's again,

00:09:34.341 --> 00:09:36.420
feeding the rope so
this rope doesn't break,

00:09:36.420 --> 00:09:38.357
or snap, or stretch.

00:09:38.357 --> 00:09:40.586
'Cause we're gonna assume
the rope doesn't do that.

00:09:40.586 --> 00:09:44.915
So what I'm saying is that the
acceleration of the 3 kg mass

00:09:44.915 --> 00:09:48.592
in the Y direction had
better equal the magnitude.

00:09:48.592 --> 00:09:50.327
So these magnitudes have to be the same.

00:09:50.327 --> 00:09:53.337
The sign doesn't have to be the same.

00:09:53.337 --> 00:09:56.707
So this 3 kg mass has
a negative acceleration

00:09:56.707 --> 00:09:58.797
just 'cause it points
down, and we're assuming up

00:09:58.797 --> 00:10:00.666
is positive, down is negative.

00:10:00.666 --> 00:10:03.897
This 5 kg mass has a
positive acceleration 'cause

00:10:03.897 --> 00:10:05.957
it's pointing to the right,
and we're assuming rightward

00:10:05.957 --> 00:10:08.477
is the positive horizontal direction.

00:10:08.477 --> 00:10:11.068
So, they can have different
signs, but the magnitudes

00:10:11.068 --> 00:10:13.167
had better be the same so
that your feeding this rope at

00:10:13.167 --> 00:10:16.217
a rate that the other one
needs in order to move.

00:10:16.217 --> 00:10:18.897
And so we can say that the
magnitudes are the same.

00:10:18.897 --> 00:10:21.175
In this case, since one
is negative of the other,

00:10:21.175 --> 00:10:23.755
I can say that the
acceleration of the 3 kg mass

00:10:23.755 --> 00:10:28.355
vertically downward is gonna
be equal to, let's say negative

00:10:28.355 --> 00:10:33.075
of the acceleration of the 5
kg mass in the X direction.

00:10:33.075 --> 00:10:34.596
I could have written it the other way.

00:10:34.596 --> 00:10:37.206
I could have wrote that A of the 5 kg mass

00:10:37.206 --> 00:10:40.477
in the X direction is a
negative A of the 3 kg mass

00:10:40.477 --> 00:10:41.365
in the Y direction.

00:10:41.365 --> 00:10:43.583
They're just different
by a negative sign is all

00:10:43.583 --> 00:10:44.647
that's important here.

00:10:44.647 --> 00:10:46.447
Okay, so this is the link we need.

00:10:46.447 --> 00:10:47.324
This is it.

00:10:47.324 --> 00:10:49.425
So this allows us to put
this final equation here

00:10:49.425 --> 00:10:51.358
in terms of only one variable.

00:10:51.358 --> 00:10:54.235
'Cause I know I've got A3Y
on this left hand side.

00:10:54.235 --> 00:10:56.985
I know A3Y should always be -A5X.

00:10:57.835 --> 00:11:02.002
If I take this and just plug
it in for A3Y right here,

00:11:02.986 --> 00:11:07.153
I'm gonna get -A5X =, well
all of this stuff, so I'll

00:11:10.014 --> 00:11:11.135
just copy this.

00:11:11.135 --> 00:11:12.766
Save some time.

00:11:12.766 --> 00:11:14.545
Copy, paste.

00:11:14.545 --> 00:11:15.663
Just equals all of that.

00:11:15.663 --> 00:11:19.055
All I did was plug in what I
know A3Y has to be equal to.

00:11:19.055 --> 00:11:22.094
'Cause now look, I've got one
equation with one unknown.

00:11:22.094 --> 00:11:24.077
I just need to solve for what A5X is.

00:11:24.077 --> 00:11:25.584
It's on both sides.

00:11:25.584 --> 00:11:28.097
So I'll need to combine
these and then isolate

00:11:28.097 --> 00:11:29.815
it on one side.

00:11:29.815 --> 00:11:31.566
So there's gonna be a
little bit of algebra here.

00:11:31.566 --> 00:11:35.107
Let's just take this, let's
give ourselves some room.

00:11:35.107 --> 00:11:36.816
Move this up just a little bit.

00:11:36.816 --> 00:11:38.534
Okay, so what do we do?

00:11:38.534 --> 00:11:39.783
We're gonna solve for A5X.

00:11:39.783 --> 00:11:41.315
Let me just get rid of this denominator.

00:11:41.315 --> 00:11:44.232
Let me multiply both sides by 3 kg.

00:11:45.387 --> 00:11:49.470
So I'm gonna get -3kg x
A5 in the X direction,

00:11:52.465 --> 00:11:55.385
if I multiply both sides by 3 kg,

00:11:55.385 --> 00:11:59.302
and then I get 5 kg x
A5 in the X direction.

00:12:01.753 --> 00:12:05.920
And I've still got minus,
alright 3 x 9.8 is 29.4 Newtons.

00:12:08.694 --> 00:12:10.655
So we'll just turn this into
what it's supposed to be.

00:12:10.655 --> 00:12:12.412
29.4 Newtons.

00:12:12.412 --> 00:12:14.533
So let's combine our A terms now.

00:12:14.533 --> 00:12:17.484
Let's move this negative
3A to the right hand side

00:12:17.484 --> 00:12:19.435
by adding it to both sides.

00:12:19.435 --> 00:12:22.184
And let's add this 29.4 to both sides.

00:12:22.184 --> 00:12:26.254
So I'll get the 29.4 Newtons
over here with a positive,

00:12:26.254 --> 00:12:27.812
if I add it to both sides.

00:12:27.812 --> 00:12:29.794
And it'll disappear on
the right hand side.

00:12:29.794 --> 00:12:31.965
And then I'll add this term to both sides.

00:12:31.965 --> 00:12:36.075
Add a positive 3 kg x A to both sides.

00:12:36.075 --> 00:12:40.242
It'll disappear on the
left, and I'll get 5 kg x A5

00:12:41.574 --> 00:12:45.741
in the X direction + 3 kg
x A5 in the X direction.

00:12:47.444 --> 00:12:50.411
Now we're close, look on the
right hand side I can combine

00:12:50.411 --> 00:12:54.578
these terms because 5A plus
3A is the same thing as 8A.

00:12:55.905 --> 00:13:00.072
So 29.4 Newtons, .4 = 8 kg x,
I'll put the parenthesis here,

00:13:05.115 --> 00:13:06.453
times five.

00:13:06.453 --> 00:13:08.165
85 in the X direction.

00:13:08.165 --> 00:13:12.180
Now I can divide both sides
by eight and usually we put

00:13:12.180 --> 00:13:14.034
the thing we're solving for
on the left, so I'm just gonna

00:13:14.034 --> 00:13:15.064
put that over here.

00:13:15.064 --> 00:13:18.647
I'll get 29.4 Newtons
over 8 kg is equal to

00:13:22.094 --> 00:13:24.474
the acceleration of mass five.

00:13:24.474 --> 00:13:27.225
The 5 kg mass in the X direction.

00:13:27.225 --> 00:13:29.435
And if we calculate that,

00:13:29.435 --> 00:13:30.964
I'll just put that into my calculator.

00:13:30.964 --> 00:13:32.797
29.4 divided by eight.

00:13:34.293 --> 00:13:35.293
I get 3.675.

00:13:36.464 --> 00:13:37.472
So we'll just round.

00:13:37.472 --> 00:13:39.722
We'll just say that's 3.68.

00:13:40.566 --> 00:13:44.566
3.6 whoops, 3.68 and it's
positive, that's good.

00:13:45.709 --> 00:13:49.154
We should get a positive
because the 5 kg mass has

00:13:49.154 --> 00:13:50.742
a positive acceleration.

00:13:50.742 --> 00:13:55.534
So we get positive 3.68
meters per second squared.

00:13:55.534 --> 00:13:57.992
But that's just of the 5 kg mass.

00:13:57.992 --> 00:14:00.082
How do we get the
acceleration of the 3 kg mass?

00:14:00.082 --> 00:14:00.978
Well that's easy.

00:14:00.978 --> 00:14:04.959
It's gotta have the same
magnitude of the 5 kg mass.

00:14:04.959 --> 00:14:06.658
All I have to do is take this number now.

00:14:06.658 --> 00:14:08.521
I know what A5X is.

00:14:08.521 --> 00:14:10.579
So if I just plug that in right here,

00:14:10.579 --> 00:14:14.579
well then I know that A3Y
is just gonna be equal

00:14:15.629 --> 00:14:18.546
to -3.68 meters per second squared.

00:14:22.258 --> 00:14:23.750
And I'm done, I did it.

00:14:23.750 --> 00:14:27.619
We figured out the
acceleration of the 3 kg mass,

00:14:27.619 --> 00:14:28.452
it's negative.

00:14:28.452 --> 00:14:30.760
No surprise, 'cause it's
accelerating downward.

00:14:30.760 --> 00:14:33.508
We figured out the
acceleration of the 5 kg mass,

00:14:33.508 --> 00:14:34.841
it's positive, not a surprise.

00:14:34.841 --> 00:14:36.600
It was accelerating to the right.

00:14:36.600 --> 00:14:38.861
The way we did it, recapping really quick,

00:14:38.861 --> 00:14:41.780
we did Newton's second
law for the 5 kg mass.

00:14:41.780 --> 00:14:43.104
That didn't let us solve.

00:14:43.104 --> 00:14:46.061
We did Newton's second
law for the 3 kg mass,

00:14:46.061 --> 00:14:47.722
that didn't let us solve.

00:14:47.722 --> 00:14:49.811
In fact it got really bleak,
because it seemed like we

00:14:49.811 --> 00:14:53.229
had three unknowns and only two equations,

00:14:53.229 --> 00:14:56.150
but the link that allowed us
to make it so that we only

00:14:56.150 --> 00:14:59.140
had one equation with one
unknown, is that we plugged

00:14:59.140 --> 00:15:01.401
one equation to the other first.

00:15:01.401 --> 00:15:03.799
We had to then write the
accelerations in terms

00:15:03.799 --> 00:15:07.180
of each other, that's
because these accelerations

00:15:07.180 --> 00:15:08.410
are not independent.

00:15:08.410 --> 00:15:10.851
The accelerations have to
have the same magnitude.

00:15:10.851 --> 00:15:13.578
And in this case one
had the opposite sign.

00:15:13.578 --> 00:15:15.490
So when we plugged that
in, we have one equation

00:15:15.490 --> 00:15:18.018
with one unknown, we
solve, we get the amount

00:15:18.018 --> 00:15:19.321
of acceleration.

00:15:19.321 --> 00:15:21.951
So that's the hard way
to do these problems.

00:15:21.951 --> 00:15:24.551
So in the next video I'll
show you the easy way

00:15:24.551 --> 00:15:26.170
to do these problems.


Derivation of the mirror equation

WEBVTT
Kind: captions
Language: en

00:00:00.375 --> 00:00:01.860
- [Voiceover] So imagine
you've got an object

00:00:01.860 --> 00:00:04.247
sitting in front of this concave mirror.

00:00:04.247 --> 00:00:07.035
If you wanted to figure out
where the image is formed,

00:00:07.035 --> 00:00:08.583
you can draw ray tracings.

00:00:08.583 --> 00:00:10.227
And so one ray you can draw

00:00:10.227 --> 00:00:12.747
is a parallel ray that goes
through the focal point.

00:00:12.747 --> 00:00:14.234
But these rays are reversible.

00:00:14.234 --> 00:00:15.811
I don't have to draw that one.

00:00:15.811 --> 00:00:19.149
Turns out, if you send rays
back along the way they came,

00:00:19.149 --> 00:00:22.036
they'll just retrace the path
they came along the other way.

00:00:22.036 --> 00:00:23.489
So these rays are reversible.

00:00:23.489 --> 00:00:26.306
I send a ray parallel, it gets
sent through the focal point.

00:00:26.306 --> 00:00:28.223
But if I send a ray
through the focal point,

00:00:28.223 --> 00:00:30.070
it will get sent parallel.

00:00:30.070 --> 00:00:31.904
In other words, I don't
have to draw this ray.

00:00:31.904 --> 00:00:33.603
I can draw this ray right here.

00:00:33.603 --> 00:00:34.561
The one that goes

00:00:34.561 --> 00:00:36.607
from the tip of the object
through the focal point.

00:00:36.607 --> 00:00:38.281
That's gonna get sent parallel.

00:00:38.281 --> 00:00:40.190
So I'm gonna draw this one just for fun

00:00:40.190 --> 00:00:41.213
and then I'm gonna draw another one.

00:00:41.213 --> 00:00:44.088
You need two in order to
find where the image is.

00:00:44.088 --> 00:00:45.248
I'll draw this one here.

00:00:45.248 --> 00:00:48.249
So this white line is
called the principal axis.

00:00:48.249 --> 00:00:51.148
It's drawn through the
center of this curved mirror.

00:00:51.148 --> 00:00:53.354
I'm gonna draw a ray that goes
from the tip of the object

00:00:53.354 --> 00:00:54.820
to that center point

00:00:54.820 --> 00:00:56.859
because I know the law of reflection

00:00:56.859 --> 00:01:00.270
says that the angle in has
to equal the angle out.

00:01:00.270 --> 00:01:02.319
And the angles are measured
form the normal line

00:01:02.319 --> 00:01:04.594
and it just so happens
that this principal axis

00:01:04.594 --> 00:01:06.445
that's usually drawn here anyways,

00:01:06.445 --> 00:01:08.755
is serving as a perfectly good normal line

00:01:08.755 --> 00:01:10.478
for this center of the mirror

00:01:10.478 --> 00:01:12.667
since it's passing through
that center of the mirror

00:01:12.667 --> 00:01:14.179
in a perpendicular way.

00:01:14.179 --> 00:01:15.422
I can just use that to my advantage.

00:01:15.422 --> 00:01:17.604
I know that the angle in's
gotta equal the angle out.

00:01:17.604 --> 00:01:19.599
I just have to make
sure that this angle out

00:01:19.599 --> 00:01:21.421
is about, looks like that,

00:01:21.421 --> 00:01:24.072
about equal to the angle
that it came in at.

00:01:24.072 --> 00:01:26.222
So these two have to be
the same angle theta.

00:01:26.222 --> 00:01:28.221
And now I can find where my image is.

00:01:28.221 --> 00:01:30.041
The image of this object is gonna be

00:01:30.041 --> 00:01:31.870
at the point where they cross.

00:01:31.870 --> 00:01:32.970
So the tip of this object

00:01:32.970 --> 00:01:35.364
gets mapped to this point right here

00:01:35.364 --> 00:01:37.270
and we get an image that's upside down

00:01:37.270 --> 00:01:38.283
and it looks like that.

00:01:38.283 --> 00:01:41.446
So ray tracing is cool; it lets
you find where the image is.

00:01:41.446 --> 00:01:44.216
But I mean I just kinda
eyeballed this angle here.

00:01:44.216 --> 00:01:45.938
This might have been off by a little bit.

00:01:45.938 --> 00:01:47.971
It might have been off by a degree or two.

00:01:47.971 --> 00:01:50.342
If I wanted to get exactly
where the image is,

00:01:50.342 --> 00:01:52.940
I'd want an equation that
I could just plug into.

00:01:52.940 --> 00:01:55.062
In other words, that I could plug into

00:01:55.062 --> 00:01:57.410
how far was the object from the mirror,

00:01:57.410 --> 00:01:59.442
and how long is this focal length,

00:01:59.442 --> 00:02:02.311
and it would spit out exactly
where the image is gonna be.

00:02:02.311 --> 00:02:04.068
So that's what we're gonna
derive in this video.

00:02:04.068 --> 00:02:06.344
It's called the mirror equation

00:02:06.344 --> 00:02:09.107
and it'll tell us how to
relate the object distance,

00:02:09.107 --> 00:02:11.189
the image distance, and the focal length.

00:02:11.189 --> 00:02:12.022
So let's do this.

00:02:12.022 --> 00:02:13.007
How do you derive it?

00:02:13.007 --> 00:02:15.330
If you look in the textbook,
it looks complicated.

00:02:15.330 --> 00:02:17.188
It's not actually as bad as it looks.

00:02:17.188 --> 00:02:19.001
When I used to look at the first time,

00:02:19.001 --> 00:02:21.653
I was like, "This is some sort
of mathematical witchcraft

00:02:21.653 --> 00:02:22.805
"I don't want to deal with."

00:02:22.805 --> 00:02:24.581
But it's not nearly as bad as it looks.

00:02:24.581 --> 00:02:26.096
We're gonna start with drawing triangles.

00:02:26.096 --> 00:02:28.915
So we'll notice that these
two angles are the same.

00:02:28.915 --> 00:02:31.156
We're gonna use this to our advantage.

00:02:31.156 --> 00:02:32.712
So we're gonna make two triangles

00:02:32.712 --> 00:02:35.712
that have these as one of their angles.

00:02:35.712 --> 00:02:37.632
So the first one, let's consider this one.

00:02:37.632 --> 00:02:39.098
Let's say one of the triangles will be

00:02:39.098 --> 00:02:41.963
from the base of this image
to the center of the mirror,

00:02:41.963 --> 00:02:45.089
and then from the mirror
to the tip of the image,

00:02:45.089 --> 00:02:46.388
and then from the tip of the image

00:02:46.388 --> 00:02:48.622
back down to the base of the image.

00:02:48.622 --> 00:02:50.503
So imagine this pink triangle right here,

00:02:50.503 --> 00:02:51.770
it's a right triangle

00:02:51.770 --> 00:02:54.155
'cause that angle right
here is a right angle,

00:02:54.155 --> 00:02:56.298
and it's got theta as one of its angles.

00:02:56.298 --> 00:02:57.714
But I could draw another triangle

00:02:57.714 --> 00:02:59.351
that has this angle theta.

00:02:59.351 --> 00:03:01.232
I can go from the tip of the object

00:03:01.232 --> 00:03:02.871
to the center of the mirror,

00:03:02.871 --> 00:03:03.956
and then from the center of the mirror

00:03:03.956 --> 00:03:05.385
to the base of the object,

00:03:05.385 --> 00:03:06.495
and then from the base of the object

00:03:06.495 --> 00:03:09.445
to the tip of the object and
I get this blue triangle.

00:03:09.445 --> 00:03:11.138
That's also a right triangle

00:03:11.138 --> 00:03:13.249
since this angle here is a right angle.

00:03:13.249 --> 00:03:15.670
So in other words, these
triangles are similar.

00:03:15.670 --> 00:03:17.264
They both have an angle theta,

00:03:17.264 --> 00:03:18.711
they both have a right angle.

00:03:18.711 --> 00:03:21.002
Or in other words, if you
don't like similar triangles,

00:03:21.002 --> 00:03:22.220
just think about it this way.

00:03:22.220 --> 00:03:23.361
You could use a trig function.

00:03:23.361 --> 00:03:24.816
Pick your favorite trig function.

00:03:24.816 --> 00:03:25.880
I'm gonna pick tangent.

00:03:25.880 --> 00:03:27.261
So let's take tangent theta.

00:03:27.261 --> 00:03:28.871
I know tangent theta by definition

00:03:28.871 --> 00:03:31.540
is always the opposite over the adjacent.

00:03:31.540 --> 00:03:33.056
The opposite to this angle,

00:03:33.056 --> 00:03:35.331
we'll take this angle down here first,

00:03:35.331 --> 00:03:38.049
the opposite to that theta is this side.

00:03:38.049 --> 00:03:39.406
What does that side mean?

00:03:39.406 --> 00:03:40.803
That's the height of the image.

00:03:40.803 --> 00:03:43.600
So I'm gonna call that
"hi" for image height.

00:03:43.600 --> 00:03:45.298
That's how tall the image is.

00:03:45.298 --> 00:03:47.623
So I know that this is gonna
equal height of the image

00:03:47.623 --> 00:03:51.109
divided by the adjacent
side to this angle here

00:03:51.109 --> 00:03:52.626
is this distance right here

00:03:52.626 --> 00:03:54.078
and we're gonna give that a name.

00:03:54.078 --> 00:03:56.694
That's just how far the
image is from the mirror.

00:03:56.694 --> 00:03:57.527
So we give this a name,

00:03:57.527 --> 00:04:00.610
we call this distance from
the mirror to the image,

00:04:00.610 --> 00:04:02.296
we call that the image distance.

00:04:02.296 --> 00:04:05.121
And it's measured from
the center of the mirror.

00:04:05.121 --> 00:04:08.084
So not from this end right
here, this little tip part,

00:04:08.084 --> 00:04:10.880
but from wherever the
center of the mirror is.

00:04:10.880 --> 00:04:12.973
It's measured from this point right here.

00:04:12.973 --> 00:04:15.608
And that's this adjacent
side since this is

00:04:15.608 --> 00:04:18.124
how far that image is from
the center of the mirror,

00:04:18.124 --> 00:04:21.529
so I'll call that "di" since
it's the image distance.

00:04:21.529 --> 00:04:23.301
But that was for this theta down here.

00:04:23.301 --> 00:04:26.458
I know that tangent of theta,
this is also a theta up here.

00:04:26.458 --> 00:04:28.651
I can use the same
relationship for this theta.

00:04:28.651 --> 00:04:30.338
And I know that tangent of this theta

00:04:30.338 --> 00:04:33.040
also has to be opposite over adjacent,

00:04:33.040 --> 00:04:36.224
but the opposite of this
theta is this side right here.

00:04:36.224 --> 00:04:37.422
And what is that side?

00:04:37.422 --> 00:04:39.907
That's just the height of the object.

00:04:39.907 --> 00:04:41.335
So I'll call that "ho".

00:04:41.335 --> 00:04:42.590
That's the object height,

00:04:42.590 --> 00:04:45.984
so the opposite side for this
theta is the object height.

00:04:45.984 --> 00:04:47.429
And then you divide by what?

00:04:47.429 --> 00:04:49.153
You divide by the adjacent side.

00:04:49.153 --> 00:04:50.397
That's just gonna be this distance

00:04:50.397 --> 00:04:51.933
from the mirror to the object.

00:04:51.933 --> 00:04:53.184
We'll give that a name,

00:04:53.184 --> 00:04:55.949
and if you guessed object
distance, then you guessed right.

00:04:55.949 --> 00:04:58.674
This is gonna be the object distance.

00:04:58.674 --> 00:05:00.923
And again we measure it from
the center of the mirror,

00:05:00.923 --> 00:05:03.713
not some curved portion
that sticks out over here,

00:05:03.713 --> 00:05:06.298
but from the center part of the mirror.

00:05:06.298 --> 00:05:09.418
So that's the adjacent side
to this theta down here,

00:05:09.418 --> 00:05:11.208
so I'm gonna write that as "do".

00:05:11.208 --> 00:05:13.007
And so this is an important relationship.

00:05:13.007 --> 00:05:15.920
This is actually given
a name just by itself.

00:05:15.920 --> 00:05:17.923
This isn't the equation we're hunting for,

00:05:17.923 --> 00:05:19.886
but it's so important
it gets its own name.

00:05:19.886 --> 00:05:21.967
Just part of this derivation
gets its own name.

00:05:21.967 --> 00:05:24.007
It's called the magnification equation

00:05:24.007 --> 00:05:25.504
but it's usually not written this way.

00:05:25.504 --> 00:05:29.099
People usually write it as hi
and then they divide by ho.

00:05:29.099 --> 00:05:32.371
So you get hi divided by ho equals,

00:05:32.371 --> 00:05:34.470
and then you multiply both sides by di,

00:05:34.470 --> 00:05:36.586
you get di over do.

00:05:36.586 --> 00:05:39.280
So a lot of times this is called
the magnification equation.

00:05:39.280 --> 00:05:41.798
It gives you a way to find what
the height of the image is.

00:05:41.798 --> 00:05:43.341
So we were lookin' for a way to find

00:05:43.341 --> 00:05:45.356
how far the image is from the mirror,

00:05:45.356 --> 00:05:46.838
but this lets you figure out, okay,

00:05:46.838 --> 00:05:48.869
you also need to know
how big is it gonna be?

00:05:48.869 --> 00:05:50.847
So if we solve this height of the image,

00:05:50.847 --> 00:05:52.130
we get the height of the image is

00:05:52.130 --> 00:05:54.772
the height of the object
times some factor,

00:05:54.772 --> 00:05:56.275
and that factor's just gonna be

00:05:56.275 --> 00:05:59.640
the image distance divided
by the object distance.

00:05:59.640 --> 00:06:00.473
Here's the thing, though.

00:06:00.473 --> 00:06:01.306
Look at.

00:06:01.306 --> 00:06:02.575
This image got flipped over.

00:06:02.575 --> 00:06:05.289
So if we define this
image distance as positive

00:06:05.289 --> 00:06:07.413
if it's on the same side as the object,

00:06:07.413 --> 00:06:09.341
we get a flipped over image,

00:06:09.341 --> 00:06:10.568
that'd be like a negative height.

00:06:10.568 --> 00:06:11.925
So since we want to represent

00:06:11.925 --> 00:06:14.209
flipped over images with a negative value,

00:06:14.209 --> 00:06:16.157
we actually write this equation down

00:06:16.157 --> 00:06:17.615
with a negative inside of here.

00:06:17.615 --> 00:06:18.817
We say that the height of the image

00:06:18.817 --> 00:06:22.215
is equal to negative ho, di over do,

00:06:22.215 --> 00:06:23.723
or you could put the
negative up here, too.

00:06:23.723 --> 00:06:25.418
So we stick a negative over here.

00:06:25.418 --> 00:06:26.886
That way we know that if you get

00:06:26.886 --> 00:06:28.318
a negative value for the image height,

00:06:28.318 --> 00:06:29.870
you know it's flipped over.

00:06:29.870 --> 00:06:30.703
So in other words,

00:06:30.703 --> 00:06:34.815
if the value I got for hi was
negative three centimeters,

00:06:34.815 --> 00:06:36.098
that means I'd get an image

00:06:36.098 --> 00:06:38.008
that had a height of three centimeters,

00:06:38.008 --> 00:06:39.185
but it'd be flipped over

00:06:39.185 --> 00:06:40.802
and that's what the negative represents.

00:06:40.802 --> 00:06:42.257
Okay, so that's kind of a side note.

00:06:42.257 --> 00:06:44.111
This is not what we're trying to derive.

00:06:44.111 --> 00:06:45.110
We're trying to derive a formula

00:06:45.110 --> 00:06:47.086
that would give us the image distance

00:06:47.086 --> 00:06:49.619
based on object distance and focal length.

00:06:49.619 --> 00:06:51.602
So we need to do another set of triangles.

00:06:51.602 --> 00:06:52.435
What we'll do now

00:06:52.435 --> 00:06:54.595
is instead of considering
these thetas here,

00:06:54.595 --> 00:06:56.332
we'll consider these angles here.

00:06:56.332 --> 00:06:57.692
Now I can't call them theta

00:06:57.692 --> 00:06:59.430
'cause we already called those theta,

00:06:59.430 --> 00:07:00.714
so I'll call these phi.

00:07:00.714 --> 00:07:02.850
So these angles also have to be the same

00:07:02.850 --> 00:07:04.519
because any time you have a line

00:07:04.519 --> 00:07:06.297
and then you cut another line through it,

00:07:06.297 --> 00:07:08.679
these angles here will always be the same.

00:07:08.679 --> 00:07:09.722
So we could do the same game.

00:07:09.722 --> 00:07:11.840
We could play the same
game we played for theta

00:07:11.840 --> 00:07:13.250
with these phis.

00:07:13.250 --> 00:07:14.783
We'll form two triangles,

00:07:14.783 --> 00:07:17.765
each triangle's gonna have
phi as one of the angles.

00:07:17.765 --> 00:07:18.723
So for the first one,

00:07:18.723 --> 00:07:21.361
we'll do this object height as one side

00:07:21.361 --> 00:07:23.099
over here to the base of phi

00:07:23.099 --> 00:07:24.727
and then back up over to here.

00:07:24.727 --> 00:07:25.560
And then for the other one

00:07:25.560 --> 00:07:27.744
we need to also have a
triangle that has phi in it,

00:07:27.744 --> 00:07:29.866
so we'll do this from here to here,

00:07:29.866 --> 00:07:32.786
down to there, and then back up to phi.

00:07:32.786 --> 00:07:34.065
So we've got two triangles.

00:07:34.065 --> 00:07:36.510
This triangle and this triangle,

00:07:36.510 --> 00:07:39.009
and they both have phi and
they both have a right angle.

00:07:39.009 --> 00:07:40.905
So these are also similar triangles.

00:07:40.905 --> 00:07:42.927
In other words, we'll play the same game.

00:07:42.927 --> 00:07:45.321
We'll say that tangent of this phi

00:07:45.321 --> 00:07:48.126
is gonna have to equal the
opposite over the adjacent.

00:07:48.126 --> 00:07:51.964
The opposite to this phi is
this side which is just ho.

00:07:51.964 --> 00:07:54.354
So we can write ho divided by

00:07:54.354 --> 00:07:56.710
the adjacent now is not do,

00:07:56.710 --> 00:07:58.595
'cause this side only goes to here.

00:07:58.595 --> 00:08:01.071
It doesn't go the whole way to the mirror.

00:08:01.071 --> 00:08:03.075
It only goes that far.

00:08:03.075 --> 00:08:05.223
So that's the entire object distance

00:08:05.223 --> 00:08:07.764
minus this piece right over here.

00:08:07.764 --> 00:08:11.577
So if I subtracted this much
from the object distance,

00:08:11.577 --> 00:08:13.156
I'd get the remaining amount

00:08:13.156 --> 00:08:15.576
which is the adjacent
part of this triangle.

00:08:15.576 --> 00:08:17.137
So this distance from the mirror

00:08:17.137 --> 00:08:19.351
to the focal point is given a name.

00:08:19.351 --> 00:08:22.305
It's called the focal length
and we represent it with an f.

00:08:22.305 --> 00:08:23.138
So a little confusing

00:08:23.138 --> 00:08:25.245
'cause f represents both a point

00:08:25.245 --> 00:08:28.394
and it represents the length
from the mirror to that point.

00:08:28.394 --> 00:08:30.842
So f is gonna represent
that length as well.

00:08:30.842 --> 00:08:33.371
So this adjacent side we could write as

00:08:33.371 --> 00:08:36.659
the object distance minus the focal length

00:08:36.659 --> 00:08:39.106
since this remaining part right here

00:08:39.106 --> 00:08:40.323
is the adjacent side

00:08:40.323 --> 00:08:42.638
which is object distance
minus the focal length.

00:08:42.638 --> 00:08:45.520
But we know that this phi
is also equal to this phi.

00:08:45.520 --> 00:08:47.654
So I can do tangent of theta for this phi,

00:08:47.654 --> 00:08:49.967
the opposite side would now be this side.

00:08:49.967 --> 00:08:50.848
What is that?

00:08:50.848 --> 00:08:52.557
What is this side of the triangle?

00:08:52.557 --> 00:08:54.710
That's just equal to
the height of the image.

00:08:54.710 --> 00:08:57.068
This side is the same as the image height

00:08:57.068 --> 00:08:59.510
so I can say that this
whole thing, tangent of phi,

00:08:59.510 --> 00:09:01.014
has to equal opposite over adjacent.

00:09:01.014 --> 00:09:03.881
This time the opposite of
phi is the image height.

00:09:03.881 --> 00:09:06.507
And we divide by this distance right here,

00:09:06.507 --> 00:09:08.152
which is just the focal length.

00:09:08.152 --> 00:09:11.046
So this adjacent side for this triangle

00:09:11.046 --> 00:09:13.331
is simply the focal length.

00:09:13.331 --> 00:09:14.860
So I'll just divide by f

00:09:14.860 --> 00:09:17.024
and so what I've got are two equations

00:09:17.024 --> 00:09:18.161
that I'm gonna put together

00:09:18.161 --> 00:09:20.860
and we will get the mirror
equation out of this.

00:09:20.860 --> 00:09:23.182
There's different ways
to proceed at this point.

00:09:23.182 --> 00:09:24.205
What I'm gonna do is

00:09:24.205 --> 00:09:27.041
I don't want ho or hi in either of these,

00:09:27.041 --> 00:09:30.795
so I'm just gonna solve this
one here for ho over hi.

00:09:30.795 --> 00:09:33.253
And I get ho over hi,

00:09:33.253 --> 00:09:35.811
so imagine dividing both sides by hi,

00:09:35.811 --> 00:09:38.623
and then multiplying
both sides by do minus f,

00:09:38.623 --> 00:09:42.983
and I'd get ho over hi equals do minus f,

00:09:42.983 --> 00:09:43.816
the focal length,

00:09:43.816 --> 00:09:45.441
divided by the focal length.

00:09:45.441 --> 00:09:46.694
But I could do the same thing up here.

00:09:46.694 --> 00:09:48.679
I can get ho over hi.

00:09:48.679 --> 00:09:51.422
It's just gonna equal do over di.

00:09:51.422 --> 00:09:54.381
So that's all you have to do
is divide both sides by hi

00:09:54.381 --> 00:09:56.894
and then multiply both sides by do.

00:09:56.894 --> 00:09:58.814
But this left-hand side right here

00:09:58.814 --> 00:10:01.322
is the same as this
left-hand side right here.

00:10:01.322 --> 00:10:03.053
So we know that ho over hi

00:10:03.053 --> 00:10:05.786
already equals do minus f over f,

00:10:05.786 --> 00:10:09.806
and up here we know that ho
over hi equals do over di,

00:10:09.806 --> 00:10:12.388
so that means that do minus f over f

00:10:12.388 --> 00:10:15.834
has to also equal do over di

00:10:15.834 --> 00:10:19.821
since both of these
expressions equal ho over hi.

00:10:19.821 --> 00:10:21.121
So we can set them all equal

00:10:21.121 --> 00:10:22.889
'cause they're all
equal to the same thing:

00:10:22.889 --> 00:10:24.440
ho over hi.

00:10:24.440 --> 00:10:26.135
And now we're gonna solve this.

00:10:26.135 --> 00:10:27.270
We're just gonna clean it up.

00:10:27.270 --> 00:10:29.008
The left-hand side I can write,

00:10:29.008 --> 00:10:30.481
I'll just stop using colors here,

00:10:30.481 --> 00:10:34.922
the left-hand side is gonna
be do over f minus one,

00:10:34.922 --> 00:10:37.701
since f over f just equals one.

00:10:37.701 --> 00:10:40.320
And that's gotta equal do over di.

00:10:40.320 --> 00:10:43.277
And now we can divide both sides by do.

00:10:43.277 --> 00:10:46.251
If I divide both sides
by do I get one over f,

00:10:46.251 --> 00:10:47.790
since the do cancels,

00:10:47.790 --> 00:10:50.755
minus one over do equals,

00:10:50.755 --> 00:10:53.576
and then the do will
cancel with this do on top,

00:10:53.576 --> 00:10:55.427
and I just get one over di.

00:10:55.427 --> 00:10:57.981
And now we made it except
it's usually written

00:10:57.981 --> 00:11:00.566
with this one over do added on the right.

00:11:00.566 --> 00:11:02.746
If we add one over do to both sides,

00:11:02.746 --> 00:11:04.656
we finally get the expression we wanted,

00:11:04.656 --> 00:11:07.636
which is one over the object distance,

00:11:07.636 --> 00:11:10.828
plus one over the image distance,

00:11:10.828 --> 00:11:13.786
equals one over the focal length.

00:11:13.786 --> 00:11:15.570
It's a pretty simple formula.

00:11:15.570 --> 00:11:17.925
It took a little bit of
effort to get to this,

00:11:17.925 --> 00:11:21.130
but this is called the mirror equation

00:11:21.130 --> 00:11:23.666
and it relates the focal
length of the mirror

00:11:23.666 --> 00:11:26.126
to the image distance
and the object distance.

00:11:26.126 --> 00:11:26.959
In other words,

00:11:26.959 --> 00:11:29.554
if you now how far you put
the object from the mirror,

00:11:29.554 --> 00:11:31.321
and you know the focal
length of the mirror,

00:11:31.321 --> 00:11:35.126
that lets you figure out exactly
where the image distance is

00:11:35.126 --> 00:11:38.583
instead of just kinda eyeballing
it and using a protractor.

00:11:38.583 --> 00:11:41.461
This'll let you solve for
where that image is exactly.

00:11:41.461 --> 00:11:42.438
And if you couple it

00:11:42.438 --> 00:11:44.800
with this magnification equation up here,

00:11:44.800 --> 00:11:47.620
you can also figure out the
exact height of the image.

00:11:47.620 --> 00:11:48.963
Now you might be feeling a little sketchy

00:11:48.963 --> 00:11:50.125
about this negative sign.

00:11:50.125 --> 00:11:51.811
I mean I kinda threw
it in quick over here.

00:11:51.811 --> 00:11:53.438
What's going on with this negative?

00:11:53.438 --> 00:11:54.952
In fact how do we know
if any of this stuff

00:11:54.952 --> 00:11:56.290
is negative or positive?

00:11:56.290 --> 00:11:58.188
Well the convention that I use,

00:11:58.188 --> 00:11:59.856
that a lot of textbooks use now,

00:11:59.856 --> 00:12:01.236
is the one where

00:12:01.236 --> 00:12:04.664
focal lengths will be
positive for concave mirrors,

00:12:04.664 --> 00:12:07.375
just like this mirror right
here, this is a concave mirror.

00:12:07.375 --> 00:12:08.878
But if it was bent the other way,

00:12:08.878 --> 00:12:12.377
if the mirror looked like
this, it'd be a convex mirror,

00:12:12.377 --> 00:12:14.232
and that would be a negative focal length.

00:12:14.232 --> 00:12:16.736
And it kinda makes sense if
the mirror was bent this way,

00:12:16.736 --> 00:12:18.727
the focal point is kinda back this way

00:12:18.727 --> 00:12:19.843
which is kinda behind it,

00:12:19.843 --> 00:12:21.573
so it makes sense that
it's sort of negative.

00:12:21.573 --> 00:12:23.337
And the object distance over here,

00:12:23.337 --> 00:12:24.991
if it's on the same side as your eye,

00:12:24.991 --> 00:12:26.276
if you're using this correctly,

00:12:26.276 --> 00:12:28.198
your eye should be over on this side,

00:12:28.198 --> 00:12:30.090
you would see an object right here

00:12:30.090 --> 00:12:32.642
but you'd also see an image
of that object right here.

00:12:32.642 --> 00:12:34.702
And if this object is on the same side

00:12:34.702 --> 00:12:36.502
as the mirror as your
eye, which it should be,

00:12:36.502 --> 00:12:38.496
then this object distance is positive.

00:12:38.496 --> 00:12:40.077
It's basically always positive

00:12:40.077 --> 00:12:42.237
unless you've got some
double set of mirrors

00:12:42.237 --> 00:12:43.509
or something weird happening.

00:12:43.509 --> 00:12:46.141
If you got a single mirror,
there's no craziness goin' on,

00:12:46.141 --> 00:12:49.605
this object distance is just
always defined to be positive

00:12:49.605 --> 00:12:51.651
using the convention
that we're using up here.

00:12:51.651 --> 00:12:54.047
And again, there are other
conventions you can use.

00:12:54.047 --> 00:12:56.864
But this is the one used in a
lot of textbooks these days.

00:12:56.864 --> 00:12:58.707
The di is a little trickier.

00:12:58.707 --> 00:13:01.384
The di is also positive if

00:13:01.384 --> 00:13:03.774
that image is on the
same side as your eye,

00:13:03.774 --> 00:13:04.791
like it is right here.

00:13:04.791 --> 00:13:05.932
So this image

00:13:05.932 --> 00:13:08.226
would be considered a
positive image distance

00:13:08.226 --> 00:13:09.708
since it's on this side.

00:13:09.708 --> 00:13:12.693
If the image got formed on
this side of the mirror,

00:13:12.693 --> 00:13:14.255
say you formed an image right here,

00:13:14.255 --> 00:13:15.970
that would be a negative image distance.

00:13:15.970 --> 00:13:18.374
If this was five centimeters
behind the mirror,

00:13:18.374 --> 00:13:19.237
we'd consider that

00:13:19.237 --> 00:13:22.035
a negative five centimeter image distance.

00:13:22.035 --> 00:13:25.498
And if you use that convention
with this mirror equation,

00:13:25.498 --> 00:13:27.122
you'll get a correct relationship

00:13:27.122 --> 00:13:28.416
between the object distance,

00:13:28.416 --> 00:13:30.119
image distance, and the focal length.

00:13:30.119 --> 00:13:32.187
And if you use that same sign convention

00:13:32.187 --> 00:13:34.379
with this magnification equation,

00:13:34.379 --> 00:13:36.613
you'll also get the
exact height of the image

00:13:36.613 --> 00:13:38.517
and if that image height
comes out negative,

00:13:38.517 --> 00:13:40.248
you'll know that it got
flipped upside down.

00:13:40.248 --> 00:13:41.980
And if the image height
comes out positive,

00:13:41.980 --> 00:13:44.277
you know that the image
will stay right-side up.

00:13:44.277 --> 00:13:47.285
So recapping, using a
bunch of similar triangles,

00:13:47.285 --> 00:13:49.452
we were able to derive a mirror equation

00:13:49.452 --> 00:13:51.218
that relates the object distance,

00:13:51.218 --> 00:13:53.015
image distance, and focal length,

00:13:53.015 --> 00:13:56.084
and along the way we derived
a magnification equation

00:13:56.084 --> 00:13:58.710
that relates the heights
of the image and object

00:13:58.710 --> 00:14:00.877
to the distances of the image and object.

00:14:00.877 --> 00:14:02.923
And you have to be very
careful with signs.

00:14:02.923 --> 00:14:04.415
Even though the object distance

00:14:04.415 --> 00:14:06.205
is basically always positive,

00:14:06.205 --> 00:14:08.710
focal length can be positive or negative.

00:14:08.710 --> 00:14:11.618
Focal length will be
positive for concave mirrors,

00:14:11.618 --> 00:14:14.094
and it'll be negative for convex mirrors.

00:14:14.094 --> 00:14:15.864
Image distances will be positive

00:14:15.864 --> 00:14:18.835
if they're on the same side
of the mirror as your eye,

00:14:18.835 --> 00:14:20.192
if they're in front of the mirror.

00:14:20.192 --> 00:14:22.579
But they'll be negative if
they're behind the mirror.

00:14:22.579 --> 00:14:23.469
And again this is not

00:14:23.469 --> 00:14:25.636
the only sign convention you could use,

00:14:25.636 --> 00:14:27.995
but it's as good as any
other sign convention

00:14:27.995 --> 00:14:31.371
and it's the one used in a
lot of textbooks these days.


Beat frequency

WEBVTT
Kind: captions
Language: en

00:00:00.366 --> 00:00:01.203
- [Voiceover] What's up everybody?

00:00:01.203 --> 00:00:03.667
I wanna talk to you about beat frequency,

00:00:03.667 --> 00:00:05.021
and to do so let me talk to you

00:00:05.021 --> 00:00:07.693
about this air displacement
versus time graph.

00:00:07.693 --> 00:00:09.583
So this is gonna give you the displacement

00:00:09.583 --> 00:00:14.121
of the air molecules for any
time at a particular location.

00:00:14.121 --> 00:00:16.590
So say you had some
speaker and it was playing

00:00:16.590 --> 00:00:20.296
a nice simple harmonic
tone and so it would sound

00:00:20.296 --> 00:00:21.852
something like this.

00:00:21.852 --> 00:00:23.636
(tone playing)

00:00:23.636 --> 00:00:27.212
That's 440 hertz, turns
out that's an A note.

00:00:27.212 --> 00:00:29.196
People use that a lot when
they're tuning instruments

00:00:29.196 --> 00:00:32.810
and whatnot so that's this
sound would sound like,

00:00:32.810 --> 00:00:35.036
and let's say it's sending this sound out

00:00:35.036 --> 00:00:38.296
and at a particular
point, one point in space,

00:00:38.296 --> 00:00:40.685
we measure what the
displacement of the air is

00:00:40.685 --> 00:00:42.057
as a function of time.

00:00:42.057 --> 00:00:43.763
Let's just say we're three meters

00:00:43.763 --> 00:00:45.397
to the right of this speaker.

00:00:45.397 --> 00:00:47.116
Just so we have a number to refer to,

00:00:47.116 --> 00:00:49.084
so there's air over
here, the air's chillin,

00:00:49.084 --> 00:00:52.297
just relaxin and then
the sound wave comes by

00:00:52.297 --> 00:00:54.688
and that causes this air to get displaced.

00:00:54.688 --> 00:00:56.155
It moves back and forth.

00:00:56.155 --> 00:00:58.580
A minuscule amount but some amount,

00:00:58.580 --> 00:01:00.325
and if we graphed that displacement

00:01:00.325 --> 00:01:02.859
as a function of time
we would get this graph.

00:01:02.859 --> 00:01:05.263
So in other words this entire graph

00:01:05.263 --> 00:01:08.497
is just personalized
for that point in space,

00:01:08.497 --> 00:01:10.782
three meters away from this speaker.

00:01:10.782 --> 00:01:12.029
So why am I telling you this?

00:01:12.029 --> 00:01:14.592
Well because we know if
you overlap two waves,

00:01:14.592 --> 00:01:16.359
if I take another wave and
let's just say this wave

00:01:16.359 --> 00:01:18.903
has the exact same
period as the first wave,

00:01:18.903 --> 00:01:20.152
right so I'll put these peak to peak

00:01:20.152 --> 00:01:21.693
so you can see, compare the peaks, yep.

00:01:21.693 --> 00:01:23.275
Takes the same amount of time

00:01:23.275 --> 00:01:25.380
for both of these to go through a cycle,

00:01:25.380 --> 00:01:26.783
that means they have the same period,

00:01:26.783 --> 00:01:28.363
so if I overlap these,
in other words if I took

00:01:28.363 --> 00:01:31.428
another speaker and I played
the same note next to it,

00:01:31.428 --> 00:01:34.080
if I played it like this I'd
hear constructive interference

00:01:34.080 --> 00:01:36.502
cause these are overlapping peak to peak,

00:01:36.502 --> 00:01:38.544
valley to valley perfectly.

00:01:38.544 --> 00:01:40.906
This note would get louder
if I was standing here

00:01:40.906 --> 00:01:43.362
and listening to it and it
would stay loud the whole time.

00:01:43.362 --> 00:01:45.739
It would just sound
louder the entire time,

00:01:45.739 --> 00:01:47.250
constructive interference,

00:01:47.250 --> 00:01:49.674
and if I moved that speaker
forward a little bit

00:01:49.674 --> 00:01:51.852
or I switched the leads,
if I found some way

00:01:51.852 --> 00:01:53.342
to get it out of phase so that

00:01:53.342 --> 00:01:54.930
it was destructive interference,

00:01:54.930 --> 00:01:57.033
I'd hear a softer note,
maybe it would be silent

00:01:57.033 --> 00:02:00.003
if I did this perfectly and
it would stay silent or soft

00:02:00.003 --> 00:02:03.084
the whole time, it would stay
destructive in other words.

00:02:03.084 --> 00:02:06.406
So if you overlap two waves
that have the same frequency,

00:02:06.406 --> 00:02:08.915
ie the same period, then
it's gonna be constructive

00:02:08.915 --> 00:02:11.111
and stay constructive, or be destructive

00:02:11.111 --> 00:02:14.404
and stay destructive, but
here's the crazy thing.

00:02:14.404 --> 00:02:15.623
Let me get rid of this.

00:02:15.623 --> 00:02:19.006
What if we overlapped two waves
that had different periods?

00:02:19.006 --> 00:02:19.994
What would happen then?

00:02:19.994 --> 00:02:21.048
Let's just try it out.

00:02:21.048 --> 00:02:23.608
So let me take this wave, this
wave has a different period.

00:02:23.608 --> 00:02:25.362
Look it, if I compare these two peaks,

00:02:25.362 --> 00:02:26.395
these two peeks don't line up,

00:02:26.395 --> 00:02:28.434
if I'm looking over here the distance

00:02:28.434 --> 00:02:31.206
between these two peaks is not the same

00:02:31.206 --> 00:02:33.086
as the distance between these two peaks.

00:02:33.086 --> 00:02:35.537
It's hard to see, it's almost the same,

00:02:35.537 --> 00:02:38.766
but this red wave has a
slightly longer period

00:02:38.766 --> 00:02:40.930
if you can see the time between peaks

00:02:40.930 --> 00:02:42.973
is a little longer than
the time between peaks

00:02:42.973 --> 00:02:44.465
for the blue wave and you might think,

00:02:44.465 --> 00:02:47.170
"Ah there's only a little difference here.

00:02:47.170 --> 00:02:48.799
"Can't be that big of a deal right?"

00:02:48.799 --> 00:02:49.991
It kind of is.

00:02:49.991 --> 00:02:53.231
It causes a new phenomenon
called beat frequency,

00:02:53.231 --> 00:02:54.805
and I'll show you why it happens here.

00:02:54.805 --> 00:02:57.071
So if I overlap these two.

00:02:57.071 --> 00:02:58.583
So now you take two speakers,

00:02:58.583 --> 00:03:00.502
but the second speaker you play it

00:03:00.502 --> 00:03:03.594
at a slightly different
frequency from the first.

00:03:03.594 --> 00:03:04.589
What would you get?

00:03:04.589 --> 00:03:05.880
Let's just look at what happens over here.

00:03:05.880 --> 00:03:09.220
They start out in phase
perfectly overlapping, right?

00:03:09.220 --> 00:03:11.443
Peak to peak, so this is constructive,

00:03:11.443 --> 00:03:13.701
this wave starts off constructively

00:03:13.701 --> 00:03:15.496
interfering with the other wave.

00:03:15.496 --> 00:03:16.993
So you hear constructive interference,

00:03:16.993 --> 00:03:18.364
that means if you were
standing at this point

00:03:18.364 --> 00:03:22.400
at that moment in time, notice
this axis is time not space,

00:03:22.400 --> 00:03:25.039
so at this moment in time right here,

00:03:25.039 --> 00:03:26.393
you would hear constructive interference

00:03:26.393 --> 00:03:29.135
which means that those
waves would sound loud.

00:03:29.135 --> 00:03:30.642
Sound really loud at that moment,

00:03:30.642 --> 00:03:33.981
but then you wait, this red
waves got a longer period.

00:03:33.981 --> 00:03:35.670
So it's taking longer for this red wave

00:03:35.670 --> 00:03:37.453
to go through a cycle, that means they're

00:03:37.453 --> 00:03:39.358
gonna start becoming out of phase, right?

00:03:39.358 --> 00:03:41.202
The peaks aren't gonna line up anymore.

00:03:41.202 --> 00:03:42.830
When this blue wave has displaced

00:03:42.830 --> 00:03:45.318
the air maximally to the right,

00:03:45.318 --> 00:03:47.481
this red wave is gonna
not have done that yet,

00:03:47.481 --> 00:03:49.903
it's gonna take a little longer
for it to try to do that.

00:03:49.903 --> 00:03:52.480
So these become out of phase,
now it's less constructive,

00:03:52.480 --> 00:03:54.444
less constructive, less constructive,

00:03:54.444 --> 00:03:56.925
over here look it, now the
peaks match the valleys.

00:03:56.925 --> 00:04:00.095
This is straight up
destructive, it's gonna be soft,

00:04:00.095 --> 00:04:01.727
and if you did this perfectly

00:04:01.727 --> 00:04:03.756
it might be silent at that point.

00:04:03.756 --> 00:04:05.781
You wait a little longer
and this blue wave

00:04:05.781 --> 00:04:08.493
has essentially lapped
the red wave, right?

00:04:08.493 --> 00:04:10.899
You waited so long the
blue wave has gone through

00:04:10.899 --> 00:04:14.390
an extra whole period
compared to the red wave,

00:04:14.390 --> 00:04:16.049
an so now the peaks line up again,

00:04:16.049 --> 00:04:17.438
and now it's constructive again

00:04:17.438 --> 00:04:19.086
because the peaks match the peaks

00:04:19.086 --> 00:04:20.469
and the valleys match the valleys.

00:04:20.469 --> 00:04:21.494
So at that point it's constructive

00:04:21.494 --> 00:04:23.595
and it's gonna be loud
again so what you would hear

00:04:23.595 --> 00:04:25.956
if you were standing at this
point three meters away,

00:04:25.956 --> 00:04:29.661
you'd first at this moment in
time hear the note be loud,

00:04:29.661 --> 00:04:31.287
then you'd hear it become soft

00:04:31.287 --> 00:04:33.257
and then you'd hear it become loud again.

00:04:33.257 --> 00:04:34.886
You'd hear this note wobble,

00:04:34.886 --> 00:04:36.671
and the name we have for this phenomenon

00:04:36.671 --> 00:04:40.838
is the beat frequency or
sometimes it's just called beats,

00:04:42.075 --> 00:04:44.777
and I don't mean you're
gonna hear Doctor Dre

00:04:44.777 --> 00:04:46.618
out of this thing that's
not the kind of beats

00:04:46.618 --> 00:04:48.173
I'm talking about, I'm just talking about

00:04:48.173 --> 00:04:51.479
that wobble from louder
to softer to louder.

00:04:51.479 --> 00:04:52.589
Actually let me just play it.

00:04:52.589 --> 00:04:53.842
Let me show you what this sounds like.

00:04:53.842 --> 00:04:56.082
So if we play the A note again.

00:04:56.082 --> 00:04:57.486
(tone playing)

00:04:57.486 --> 00:04:58.735
That's the A note.

00:04:58.735 --> 00:05:01.508
Let me play, that's 440 hertz, right?

00:05:01.508 --> 00:05:03.076
That's a particular frequency.

00:05:03.076 --> 00:05:05.075
Let me play just a slightly
different frequency.

00:05:05.075 --> 00:05:06.742
I'll play 443 hertz.

00:05:07.907 --> 00:05:09.737
(tone playing)

00:05:09.737 --> 00:05:10.952
And you're probably like that just sounds

00:05:10.952 --> 00:05:13.087
like the exact same thing,
I can't tell the difference

00:05:13.087 --> 00:05:15.570
between the two, but if I play them both

00:05:15.570 --> 00:05:17.661
you'll definitely be able
to tell the difference.

00:05:17.661 --> 00:05:18.861
So I'm gonna play them both now.

00:05:18.861 --> 00:05:22.489
Here's the 443 hertz, and here's the 440.

00:05:22.489 --> 00:05:24.915
(two tones playing)

00:05:24.915 --> 00:05:25.933
And you hear a wobble.

00:05:25.933 --> 00:05:27.322
This thing starts to wobble.

00:05:27.322 --> 00:05:28.845
So let me stop this.

00:05:28.845 --> 00:05:30.291
So that's what physicists
are talking about

00:05:30.291 --> 00:05:32.638
when they say beat frequency or beats,

00:05:32.638 --> 00:05:33.875
they're referring to that wobble

00:05:33.875 --> 00:05:36.261
and sound loudness that you hear

00:05:36.261 --> 00:05:38.825
when you overlap two waves
that different frequencies.

00:05:38.825 --> 00:05:40.244
This is important, it only works

00:05:40.244 --> 00:05:42.620
when you have waves of
different frequency.

00:05:42.620 --> 00:05:45.212
So what if you wanted to know
the actual beat frequency?

00:05:45.212 --> 00:05:46.702
What if you wanted to know how many

00:05:46.702 --> 00:05:48.912
wobbles you get per second?

00:05:48.912 --> 00:05:51.318
So how often is it going from constructive

00:05:51.318 --> 00:05:53.330
to destructive back to constructive?

00:05:53.330 --> 00:05:56.255
If that takes a long time the
frequency is gonna be small,

00:05:56.255 --> 00:05:58.586
cause there aren't gonna
be many wobbles per second,

00:05:58.586 --> 00:06:00.387
but if this takes a short amount of time,

00:06:00.387 --> 00:06:02.185
if there's not much time
between constructive

00:06:02.185 --> 00:06:04.258
back to constructive
then the beat frequency's

00:06:04.258 --> 00:06:07.410
gonna be large, there will
be many wobbles per second.

00:06:07.410 --> 00:06:10.688
How would you figure
out this beat frequency,

00:06:10.688 --> 00:06:13.461
I'll call it FB, this
would be how many times

00:06:13.461 --> 00:06:17.741
this goes from constructive
back to constructive per second.

00:06:17.741 --> 00:06:19.661
So if it does that 20 times per second,

00:06:19.661 --> 00:06:22.008
this thing would be
wobbling 20 times per second

00:06:22.008 --> 00:06:23.732
and the frequency would be 20 hertz.

00:06:23.732 --> 00:06:24.951
So how do you find this if you know

00:06:24.951 --> 00:06:26.686
the frequency of each wave,

00:06:26.686 --> 00:06:28.286
and it turns out it's very very easy.

00:06:28.286 --> 00:06:30.232
I'm just gonna show you
the formula in this video,

00:06:30.232 --> 00:06:31.699
in the next video we'll derive it

00:06:31.699 --> 00:06:33.192
for those that are interested,

00:06:33.192 --> 00:06:34.714
but in this one I'll
just show you what it is,

00:06:34.714 --> 00:06:35.953
show you how to use it.

00:06:35.953 --> 00:06:37.703
So the beat frequency
if you wanna find it,

00:06:37.703 --> 00:06:40.098
if I know the frequency of the first wave,

00:06:40.098 --> 00:06:43.158
so if wave one has a frequency, f1.

00:06:43.158 --> 00:06:45.430
So say that blue wave has a frequency f1,

00:06:45.430 --> 00:06:48.235
and wave two has a frequency f2,

00:06:48.235 --> 00:06:50.168
then I can find the beat frequency

00:06:50.168 --> 00:06:51.587
by just taking the difference.

00:06:51.587 --> 00:06:55.092
I can just take f1 and then subtract f2,

00:06:55.092 --> 00:06:56.189
and it's as simple as that.

00:06:56.189 --> 00:06:57.713
That gives you the beat frequency.

00:06:57.713 --> 00:06:59.023
Now you might wonder like wait a minute,

00:06:59.023 --> 00:07:02.417
what if f1 has a smaller
frequency than f2?

00:07:02.417 --> 00:07:04.692
That would give me a
negative beat frequency?

00:07:04.692 --> 00:07:06.595
That doesn't make sense we
can't have a negative frequency

00:07:06.595 --> 00:07:09.786
so we typically put an absolute
value sign around this.

00:07:09.786 --> 00:07:12.764
You should take the higher
frequency minus the lower,

00:07:12.764 --> 00:07:14.353
but just in case you don't just stick

00:07:14.353 --> 00:07:16.255
an absolute value and that gives you

00:07:16.255 --> 00:07:18.132
the size of this beat frequency,

00:07:18.132 --> 00:07:22.002
which is basically the
number of wobbles per second,

00:07:22.002 --> 00:07:24.636
ie the number of times
it goes from constructive

00:07:24.636 --> 00:07:27.833
all the way back to
constructive per second.

00:07:27.833 --> 00:07:29.954
That's what this beat frequency means

00:07:29.954 --> 00:07:32.042
and this formula is how you can find it.

00:07:32.042 --> 00:07:33.505
Now I should say to be clear,

00:07:33.505 --> 00:07:35.329
we're playing two different sound waves,

00:07:35.329 --> 00:07:38.777
our ears really just sort of
gonna hear one total wave.

00:07:38.777 --> 00:07:40.500
So these waves overlap.

00:07:40.500 --> 00:07:43.501
You can do this whole analysis
using wave interference.

00:07:43.501 --> 00:07:45.208
You write down the equation of one wave,

00:07:45.208 --> 00:07:46.989
you write down the
equation of the other wave,

00:07:46.989 --> 00:07:48.345
you add up the two, right?

00:07:48.345 --> 00:07:51.105
We know that the total wave is gonna equal

00:07:51.105 --> 00:07:54.973
the summation of each wave at
a particular point in time.

00:07:54.973 --> 00:07:56.271
So at one point in time if we take

00:07:56.271 --> 00:07:58.495
the value of each wave and add them up,

00:07:58.495 --> 00:08:00.521
we'd get the total wave,
what would that look like?

00:08:00.521 --> 00:08:02.594
What would the total wave look like?

00:08:02.594 --> 00:08:03.555
It would look like this.

00:08:03.555 --> 00:08:05.228
If we just add it up
you'd get a total wave

00:08:05.228 --> 00:08:07.608
that looks like this
green dashed wave here.

00:08:07.608 --> 00:08:10.216
Right over here, they
add up to twice the wave,

00:08:10.216 --> 00:08:12.684
and then in the middle they
cancel to almost nothing,

00:08:12.684 --> 00:08:15.212
and then back over here they add up again,

00:08:15.212 --> 00:08:17.481
and so if you just
looked at the total wave,

00:08:17.481 --> 00:08:18.701
it would look something like this.

00:08:18.701 --> 00:08:21.155
So the total wave would
start with a large amplitude,

00:08:21.155 --> 00:08:24.035
and then it would die out because
they'd become destructive,

00:08:24.035 --> 00:08:26.094
and then it would become
a large amplitude again.

00:08:26.094 --> 00:08:27.801
So you see this picture a lot

00:08:27.801 --> 00:08:29.446
when you're talking about beat frequency

00:08:29.446 --> 00:08:30.922
because it's showing what the total wave

00:08:30.922 --> 00:08:32.890
looks like as a function of time

00:08:32.890 --> 00:08:35.311
when you add up those two individual waves

00:08:35.311 --> 00:08:36.700
since this is going from constructive

00:08:36.700 --> 00:08:39.029
to destructive to constructive again,

00:08:39.029 --> 00:08:40.874
and this is why it sounds loud

00:08:40.874 --> 00:08:43.509
and then soft and then
loud again to our ear.

00:08:43.509 --> 00:08:46.187
So what would an example
problem look like for beats?

00:08:46.187 --> 00:08:49.299
Let's say you were told
that there's a flute,

00:08:49.299 --> 00:08:51.494
and let's say this flute
is playing a frequency

00:08:51.494 --> 00:08:54.677
of 440 hertz like that
note we heard earlier,

00:08:54.677 --> 00:08:56.857
and let's say there's also a clarinet.

00:08:56.857 --> 00:08:58.429
They play it, they wanna
make sure they're in tune,

00:08:58.429 --> 00:09:00.928
they wanna make sure
they're jam sounds good

00:09:00.928 --> 00:09:02.404
for everyone in the audience,

00:09:02.404 --> 00:09:04.598
but when they both try to play the A note,

00:09:04.598 --> 00:09:07.298
this flute plays 440, this
clarinet plays a note,

00:09:07.298 --> 00:09:09.581
and let's say we hear a beat frequency,

00:09:09.581 --> 00:09:12.293
I'll write it in this color,
we hear a beat frequency

00:09:12.293 --> 00:09:16.075
of five hertz so we hear
five wobbles per second.

00:09:16.075 --> 00:09:18.617
In fact if you've ever
tried to tune an instrument

00:09:18.617 --> 00:09:20.781
you know that one way to tune it is to try

00:09:20.781 --> 00:09:23.189
to check two notes that are
supposed to be the same.

00:09:23.189 --> 00:09:25.033
You can tell immediately
if they're not the same

00:09:25.033 --> 00:09:26.741
cause you'll hear these wobbles,

00:09:26.741 --> 00:09:27.991
and so you keep tuning it until

00:09:27.991 --> 00:09:29.881
you don't hear the wobble anymore.

00:09:29.881 --> 00:09:31.875
As those notes get closer and closer,

00:09:31.875 --> 00:09:34.160
there'll be less wobbles per second,

00:09:34.160 --> 00:09:36.263
and once you hear no wobble at all,

00:09:36.263 --> 00:09:38.441
you know you're at the
exact same frequency,

00:09:38.441 --> 00:09:39.999
but these aren't, these are off,

00:09:39.999 --> 00:09:41.504
and so the question might ask,

00:09:41.504 --> 00:09:46.000
what are the two possible
frequencies of the clarinet?

00:09:46.000 --> 00:09:47.979
Well we know that the beat frequency

00:09:47.979 --> 00:09:49.368
is equal to the absolute value

00:09:49.368 --> 00:09:51.699
of the difference in the two frequencies.

00:09:51.699 --> 00:09:54.182
So if there's a beat
frequency of five hertz

00:09:54.182 --> 00:09:57.156
and the flutes playing 440,
that means the clarinet

00:09:57.156 --> 00:09:59.595
is five hertz off from the flute.

00:09:59.595 --> 00:10:01.972
So the clarinet might
be a little too high,

00:10:01.972 --> 00:10:05.172
it might be 445 hertz,
playing a little sharp,

00:10:05.172 --> 00:10:09.347
or it might be 435 hertz,
might be playing a little flat.

00:10:09.347 --> 00:10:11.128
So we'd have to tune to figure out

00:10:11.128 --> 00:10:12.608
how it can get to the point where

00:10:12.608 --> 00:10:14.557
there'd be zero beat frequency,

00:10:14.557 --> 00:10:16.475
cause when there's zero
beat frequencies you know

00:10:16.475 --> 00:10:18.307
both of these frequencies are the same,

00:10:18.307 --> 00:10:19.435
but what do you do?

00:10:19.435 --> 00:10:20.271
How does it know?

00:10:20.271 --> 00:10:22.208
How does the clarinet
player know which one to do?

00:10:22.208 --> 00:10:23.485
You kind of don't sometimes.

00:10:23.485 --> 00:10:25.025
Sometimes you just have to test it out.

00:10:25.025 --> 00:10:26.962
Let's say the clarinet player assumed,

00:10:26.962 --> 00:10:29.627
all right maybe they were
a little too sharp 445,

00:10:29.627 --> 00:10:31.455
so they're gonna lower their note.

00:10:31.455 --> 00:10:34.198
So they start to tune down,
what will they listen for?

00:10:34.198 --> 00:10:36.667
They'll listen for less
wobbles per second.

00:10:36.667 --> 00:10:39.576
So if you become more in tune in stead of,

00:10:39.576 --> 00:10:41.134
(imitates wobbling tone)

00:10:41.134 --> 00:10:41.967
you would hear,

00:10:41.967 --> 00:10:44.425
(imitates slowing wobble)

00:10:44.425 --> 00:10:46.437
right, and then once
you're perfectly in tune,

00:10:46.437 --> 00:10:47.561
(hums tone)

00:10:47.561 --> 00:10:49.711
and it would be perfect,
there'd be no wobbles.

00:10:49.711 --> 00:10:51.684
If this person tried it and there were

00:10:51.684 --> 00:10:54.182
more wobbles per second
then this person would know,

00:10:54.182 --> 00:10:55.990
"Oh, I was probably at this lower note.

00:10:55.990 --> 00:11:00.260
"cause if I'm at 435, and
I go to say 430 hertz,

00:11:00.260 --> 00:11:02.150
"that's gonna be more out of tune."

00:11:02.150 --> 00:11:03.996
Now the beat frequency would be 10 hertz,

00:11:03.996 --> 00:11:05.897
you'd hear 10 wobbles per second,

00:11:05.897 --> 00:11:07.252
and the person would know immediately,

00:11:07.252 --> 00:11:08.824
"Whoa, that was a bad idea.

00:11:08.824 --> 00:11:11.506
"I must not have been too sharp.

00:11:11.506 --> 00:11:12.834
"I must've been too flat."

00:11:12.834 --> 00:11:14.357
So now that you know
you're a little too flat

00:11:14.357 --> 00:11:16.156
you start tuning the other way,

00:11:16.156 --> 00:11:19.478
so you can raise this up to 440 hertz

00:11:19.478 --> 00:11:21.743
and then you would hear
zero beat frequency,

00:11:21.743 --> 00:11:24.232
zero wobbles per second, a nice tune,

00:11:24.232 --> 00:11:26.224
and you would be playing in harmony.

00:11:26.224 --> 00:11:28.573
So recapping beats or
beat frequency occurs

00:11:28.573 --> 00:11:31.864
when you overlap two waves that
have different frequencies.

00:11:31.864 --> 00:11:33.679
This causes the waves to go from being

00:11:33.679 --> 00:11:37.042
constructive to destructive
to constructive over and over,

00:11:37.042 --> 00:11:40.780
which we perceive as a wobble
in the loudness of the sound,

00:11:40.780 --> 00:11:42.394
and the way you can
find the beat frequency

00:11:42.394 --> 00:11:45.139
is by taking the difference
of the two frequencies

00:11:45.139 --> 00:11:47.160
of the waves that are overlapping.