Harmonic motion part 3 (no calculus)

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:01.810
Welcome back.

00:00:01.810 --> 00:00:03.860
And if you were covering your
eyes because you didn't want

00:00:03.860 --> 00:00:06.250
to see calculus, I think you
can open your eyes again.

00:00:06.250 --> 00:00:08.830
There shouldn't be any
significant displays of

00:00:08.830 --> 00:00:10.330
calculus in this video.

00:00:10.330 --> 00:00:12.550
But just to review what we went
over, we just said, OK if

00:00:12.550 --> 00:00:14.760
we have a spring-- and I drew it
vertically this time-- but

00:00:14.760 --> 00:00:16.750
pretend like there's no gravity,
or maybe pretend like

00:00:16.750 --> 00:00:19.440
we're viewing-- we're looking at
the top of a table, because

00:00:19.440 --> 00:00:21.080
we don't want to look
at the effect of

00:00:21.080 --> 00:00:21.990
a spring and gravity.

00:00:21.990 --> 00:00:23.450
We just want to look at
a spring by itself.

00:00:23.450 --> 00:00:25.760
So this could be in deep space,
or something else.

00:00:25.760 --> 00:00:26.780
But we're not thinking
about gravity.

00:00:26.780 --> 00:00:28.670
But I drew it vertically just
so that we can get more

00:00:28.670 --> 00:00:30.080
intuition for this curve.

00:00:30.080 --> 00:00:33.520
Well, we started off saying is
if I have a spring and 0-- x

00:00:33.520 --> 00:00:35.740
equals 0 is kind of the natural
resting point of the

00:00:35.740 --> 00:00:38.430
spring, if I just let this
mass-- if I didn't pull on the

00:00:38.430 --> 00:00:39.380
spring at all.

00:00:39.380 --> 00:00:41.340
But I have a mass attached to
the spring, and if I were to

00:00:41.340 --> 00:00:46.060
stretch the spring to point A,
we said, well what happens?

00:00:46.060 --> 00:00:49.990
Well, it starts with very little
velocity, but there's a

00:00:49.990 --> 00:00:52.100
restorative force, that's going
to be pulling it back

00:00:52.100 --> 00:00:53.430
towards this position.

00:00:53.430 --> 00:00:56.170
So that force will accelerate
the mass, accelerate the mass,

00:00:56.170 --> 00:00:59.620
accelerate the mass, until
it gets right here.

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And then it'll have a lot of
velocity here, but then it'll

00:01:03.040 --> 00:01:04.360
start decelerating.

00:01:04.360 --> 00:01:06.680
And then it'll decelerate,
decelerate, decelerate.

00:01:06.680 --> 00:01:08.560
Its velocity will stop, and
it'll come back up.

00:01:08.560 --> 00:01:10.625
And if we drew this as
a function of time,

00:01:10.625 --> 00:01:12.060
this is what happens.

00:01:12.060 --> 00:01:15.010
It starts moving very
slowly, accelerates.

00:01:15.010 --> 00:01:18.300
At this point, at x equals 0,
it has its maximum speed.

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So the rate of change of
velocity-- or the rate of

00:01:21.010 --> 00:01:25.230
change of position is fastest.
And we can see the slope is

00:01:25.230 --> 00:01:26.770
very fast right here.

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And then, we start slowing
down again, slowing down,

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until we get back to
the spot of A.

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And then we keep going up and
down, up and down, like that.

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And we showed that actually,
the equation for the mass's

00:01:40.280 --> 00:01:45.320
position as a function of time
is x of t-- and we used a

00:01:45.320 --> 00:01:47.620
little bit of differential
equations to prove it.

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But this equation-- not that I
recommend that you memorize

00:01:50.310 --> 00:01:51.950
anything-- but this is a pretty

00:01:51.950 --> 00:01:53.640
useful equation to memorize.

00:01:53.640 --> 00:01:58.390
Because you can use it to
pretty much figure out

00:01:58.390 --> 00:02:05.210
anything-- about the position,
or of the mass at any given

00:02:05.210 --> 00:02:09.490
time, or the frequency of this
oscillatory motion, or

00:02:09.490 --> 00:02:10.120
anything else.

00:02:10.120 --> 00:02:12.005
Even the velocity, if you know
a little bit of calculus, you

00:02:12.005 --> 00:02:13.720
can figure out the velocity
at anytime, of the object.

00:02:13.720 --> 00:02:16.030
And that's pretty neat.

00:02:16.030 --> 00:02:18.740
So what can we do now?

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Well, let's try to figure
out the period of

00:02:20.690 --> 00:02:26.220
this oscillating system.

00:02:26.220 --> 00:02:28.450
And just so you know-- I know
I put the label harmonic

00:02:28.450 --> 00:02:30.980
motion on all of these-- this
is simple harmonic motion.

00:02:30.980 --> 00:02:34.470
Simple harmonic motion is
something that can be

00:02:34.470 --> 00:02:36.690
described by a trigonometric
function like this.

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And it just oscillates back
and forth, back and forth.

00:02:39.720 --> 00:02:41.850
And so, what we're doing
is harmonic motion.

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And now, let's figure out
what this period is.

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Remember we said that after T
seconds, it gets back to its

00:02:47.670 --> 00:02:50.210
original position, and then
after another T seconds, it

00:02:50.210 --> 00:02:51.860
gets back to its original
position.

00:02:51.860 --> 00:02:53.540
Let's figure out
with this T is.

00:02:53.540 --> 00:02:55.320
And that's essentially
its period, right?

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What's the period
of a function?

00:02:57.830 --> 00:03:00.650
It's how long it takes to get
back to your starting point.

00:03:00.650 --> 00:03:06.560
Or how long it takes for the
whole cycle to happen once.

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So what is this T?

00:03:08.140 --> 00:03:09.140
So let me ask you a question.

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What are all the points--
that if this is a

00:03:11.320 --> 00:03:12.950
cosine function, right?

00:03:12.950 --> 00:03:18.290
What are all of the points at
which cosine is equal to 1?

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Or this function would
be equal to A, right?

00:03:20.470 --> 00:03:22.860
Because whenever cosine is
equal to 1, this whole

00:03:22.860 --> 00:03:24.415
function is equal to A.

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And it's these points.

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Well cosine is equal to 1 when--
so, theta-- let's say,

00:03:31.450 --> 00:03:36.430
when is cosine of theta
equal to 1?

00:03:36.430 --> 00:03:38.890
So, at what angles
is this true?

00:03:38.890 --> 00:03:42.150
Well it's true at theta
is equal to 0, right?

00:03:42.150 --> 00:03:44.200
Cosine of 0 is 1.

00:03:44.200 --> 00:03:46.430
Cosine of 2 pi is
also 1, right?

00:03:46.430 --> 00:03:48.400
We could just keep going around
that unit circle.

00:03:48.400 --> 00:03:50.770
You should watch the unit circle
video if this makes no

00:03:50.770 --> 00:03:51.260
sense to you.

00:03:51.260 --> 00:03:53.600
Or the graphing trig
functions.

00:03:53.600 --> 00:03:55.850
It's also true at 4 pi.

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Really, any multiple of
2 pi, this is true.

00:03:59.790 --> 00:04:00.320
Right?

00:04:00.320 --> 00:04:04.310
Cosine of that angle
is equal to 1.

00:04:04.310 --> 00:04:05.900
So the same thing is true.

00:04:05.900 --> 00:04:14.400
This function, x of t, is equal
to A at what points?

00:04:14.400 --> 00:04:18.980
x of t is equal to A whenever
this expression-- within the

00:04:18.980 --> 00:04:24.540
cosines-- whenever this
expression is equal to 0, 2

00:04:24.540 --> 00:04:27.990
pi, 4 pi, et cetera.

00:04:27.990 --> 00:04:30.610
And this first time that it
cycles, right, from 0 to 2

00:04:30.610 --> 00:04:36.650
pi-- from 0 to T, that'll
be at 2 pi, right?

00:04:36.650 --> 00:04:41.730
So this whole expression will
equal A, when k-- and that's

00:04:41.730 --> 00:04:42.970
these points, right?

00:04:42.970 --> 00:04:44.760
That's when this function
is equal to A.

00:04:44.760 --> 00:04:46.930
It'll happen again over
here someplace.

00:04:46.930 --> 00:04:50.260
When this little internal
expression is equal to 2 pi,

00:04:50.260 --> 00:04:52.180
or really any multiple
of 2 pi.

00:04:52.180 --> 00:04:56.010
So we could say, so x of t is
equal to A when the square

00:04:56.010 --> 00:05:03.460
root of k over m times
t, is equal to 2 pi.

00:05:03.460 --> 00:05:07.330
Or another way of thinking about
it, is let's multiply

00:05:07.330 --> 00:05:10.510
both sides of this equation
times the inverse of the

00:05:10.510 --> 00:05:12.450
square root of k over m.

00:05:12.450 --> 00:05:20.410
And you get, t is equal to 2 pi
times the square root-- and

00:05:20.410 --> 00:05:21.840
it's going to be the inverse
of this, right?

00:05:21.840 --> 00:05:25.550
Of m over k.

00:05:25.550 --> 00:05:28.440
And there we have the period
of this function.

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This is going to be equal
to 2 pi times the square

00:05:31.460 --> 00:05:33.860
root of m over k.

00:05:33.860 --> 00:05:41.475
So if someone tells you, well I
have a spring that I'm going

00:05:41.475 --> 00:05:43.925
to pull from some-- I'm going to
stretch it, or compress it

00:05:43.925 --> 00:05:46.080
a little bit, then I let go--
what is the period?

00:05:46.080 --> 00:05:49.180
How long does it take for the
spring to go back to its

00:05:49.180 --> 00:05:49.990
original position?

00:05:49.990 --> 00:05:52.480
It'll keep doing that, as we
have no friction, or no

00:05:52.480 --> 00:05:54.060
gravity, or any air
resistance, or

00:05:54.060 --> 00:05:54.790
anything like that.

00:05:54.790 --> 00:05:56.700
Air resistance really is just
a form of friction.

00:05:56.700 --> 00:05:58.940
You could immediately-- if you
memorize this formula,

00:05:58.940 --> 00:06:01.360
although you should know where
it comes from-- you could

00:06:01.360 --> 00:06:03.840
immediately say, well I know
how long the period is.

00:06:03.840 --> 00:06:06.390
It's 2 pi times m over k.

00:06:06.390 --> 00:06:09.090
That's how long it's going to
take the spring to get back--

00:06:09.090 --> 00:06:11.560
to complete one cycle.

00:06:11.560 --> 00:06:13.520
And then what about
the frequency?

00:06:13.520 --> 00:06:16.320
If you wanted to know cycles per
second, well that's just

00:06:16.320 --> 00:06:19.200
the inverse of the
period, right?

00:06:19.200 --> 00:06:22.190
So if I wanted to know the
frequency, that equals 1 over

00:06:22.190 --> 00:06:23.820
the period, right?

00:06:23.820 --> 00:06:26.860
Period is given in seconds
per cycle.

00:06:26.860 --> 00:06:33.340
So frequency is cycles
per second, and this

00:06:33.340 --> 00:06:35.880
is seconds per cycle.

00:06:35.880 --> 00:06:38.970
So frequency is just going
to be 1 over this.

00:06:38.970 --> 00:06:44.500
Which is 1 over 2 pi times the
square root of k over m.

00:06:44.500 --> 00:06:46.180
That's the frequency.

00:06:46.180 --> 00:06:50.110
But I have always had trouble
memorizing this, and this.

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You always [UNINTELLIGIBLE]

00:06:50.620 --> 00:06:52.240
k over m, and m over
k, and all of that.

00:06:52.240 --> 00:06:54.650
All you have to really
memorize is this.

00:06:54.650 --> 00:06:56.840
And even that, you might
even have an intuition

00:06:56.840 --> 00:06:57.660
as to why it's true.

00:06:57.660 --> 00:06:59.240
You can even go to the
differential equations if you

00:06:59.240 --> 00:07:00.680
want to reprove it
to yourself.

00:07:00.680 --> 00:07:04.020
Because if you have this, you
really can answer any question

00:07:04.020 --> 00:07:07.990
about the position of the
mass, at any time.

00:07:07.990 --> 00:07:10.150
The velocity of the mass, at any
time, just by taking the

00:07:10.150 --> 00:07:10.980
derivative.

00:07:10.980 --> 00:07:13.140
Or the period, or the frequency
of the function.

00:07:13.140 --> 00:07:14.810
As long as you know how to take
the period and frequency

00:07:14.810 --> 00:07:16.180
of trig functions.

00:07:16.180 --> 00:07:19.010
You can watch my videos, and
watch my trig videos, to get a

00:07:19.010 --> 00:07:20.340
refresher on that.

00:07:20.340 --> 00:07:22.850
One thing that's pretty
interesting about this, is

00:07:22.850 --> 00:07:26.930
notice that the frequency
and the period, right?

00:07:26.930 --> 00:07:28.900
This is the period of the
function, that's how long it

00:07:28.900 --> 00:07:30.470
takes do one cycle.

00:07:30.470 --> 00:07:33.090
This is how many cycles it does
in one second-- both of

00:07:33.090 --> 00:07:35.010
them are independent of A.

00:07:35.010 --> 00:07:37.830
So it doesn't matter, I could
stretch it only a little bit,

00:07:37.830 --> 00:07:40.750
like there, and it'll take the
same amount of time to go

00:07:40.750 --> 00:07:43.560
back, and come back like
that, as it would if I

00:07:43.560 --> 00:07:44.470
stretch it a lot.

00:07:44.470 --> 00:07:45.290
It would just do that.

00:07:45.290 --> 00:07:48.800
If I stretched it just a little
bit, the function would

00:07:48.800 --> 00:07:51.450
look like this.

00:07:51.450 --> 00:07:53.340
Make sure I do this right.

00:07:53.340 --> 00:07:55.030
I'm not doing that right.

00:07:55.030 --> 00:07:56.280
Edit, undo.

00:07:58.680 --> 00:08:00.800
If I just do it a little bit,
the amplitude is going to be

00:08:00.800 --> 00:08:03.180
less, but the function is going
to essentially do the

00:08:03.180 --> 00:08:04.890
same thing.

00:08:04.890 --> 00:08:08.190
It's just going to do that.

00:08:08.190 --> 00:08:10.090
So it's going to take the same
amount of time to complete the

00:08:10.090 --> 00:08:11.450
cycle, it'll just have
a lower amplitude.

00:08:11.450 --> 00:08:14.890
So that's interesting to me,
that how much I stretch it,

00:08:14.890 --> 00:08:18.000
it's not going to make it take
longer or less time to

00:08:18.000 --> 00:08:19.850
complete one cycle.

00:08:19.850 --> 00:08:21.500
That's interesting.

00:08:21.500 --> 00:08:26.360
And so if I just told you, that
I actually start having

00:08:26.360 --> 00:08:27.930
objects compressed, right?

00:08:27.930 --> 00:08:33.419
So in that case, let's
say my A is minus 3.

00:08:33.419 --> 00:08:37.490
I have a spring constant
of-- let's say k is,

00:08:37.490 --> 00:08:39.280
I don't know, 10.

00:08:39.280 --> 00:08:43.830
And I have a mass of 2
kilograms. Then I could

00:08:43.830 --> 00:08:47.100
immediately tell you what the
equation of the position as a

00:08:47.100 --> 00:08:48.630
function of time at
any point is.

00:08:48.630 --> 00:08:53.120
It's going to be x of t will
equal-- I'm running out of

00:08:53.120 --> 00:08:57.310
space-- so x of t would equal--
this is just basic

00:08:57.310 --> 00:09:03.450
subsitution-- minus 3 cosine of
10 divided by 2, right? k

00:09:03.450 --> 00:09:04.970
over m, is 5.

00:09:04.970 --> 00:09:07.600
So square root of 5t.

00:09:07.600 --> 00:09:10.370
I know that's hard to read,
but you get the point.

00:09:10.370 --> 00:09:11.860
I just substituted that.

00:09:11.860 --> 00:09:14.550
But the important thing to
know is this-- this is, I

00:09:14.550 --> 00:09:17.360
think, the most important
thing-- and then if given a

00:09:17.360 --> 00:09:19.690
trig function, you have trouble
remembering how to

00:09:19.690 --> 00:09:21.690
figure out the period or
frequency-- although I always

00:09:21.690 --> 00:09:26.120
just think about, when does
this expression equal 1?

00:09:26.120 --> 00:09:29.952
And then you can figure out--
when does it equal 1, or when

00:09:29.952 --> 00:09:32.416
does it equal 0-- and you can
figure out its period.

00:09:32.416 --> 00:09:33.360
If you don't have it,

00:09:33.360 --> 00:09:36.170
you can memorize this formula
for period, and this formula

00:09:36.170 --> 00:09:38.830
for frequency, but I think that
might be a waste of your

00:09:38.830 --> 00:09:39.830
brain space.

00:09:39.830 --> 00:09:43.090
Anyway, I'll see you
in the next video.


Harmonic motion part 2 (calculus)

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:04.310
So where I left off in the last
video, I'd just rewritten

00:00:04.310 --> 00:00:05.440
the spring equation.

00:00:05.440 --> 00:00:08.900
And I just wrote force is
mass times acceleration.

00:00:08.900 --> 00:00:11.980
And I was in the process of
saying, well if x is a

00:00:11.980 --> 00:00:14.000
function of t, what's
acceleration?

00:00:14.000 --> 00:00:17.170
Well, velocity is this
derivative of x with respect

00:00:17.170 --> 00:00:17.790
to time, right?

00:00:17.790 --> 00:00:20.060
Your change in position
over change of time.

00:00:20.060 --> 00:00:23.350
And acceleration is the
derivative of velocity, or the

00:00:23.350 --> 00:00:25.635
second derivative of position.

00:00:25.635 --> 00:00:29.350
So you take the derivative
twice of x of t, right?

00:00:29.350 --> 00:00:35.930
So let's rewrite this equation
in those terms. Let me erase

00:00:35.930 --> 00:00:37.570
all this--I actually want to
keep all of this, just so we

00:00:37.570 --> 00:00:40.710
remember what we're talking
about this whole time.

00:00:40.710 --> 00:00:43.800
Let me see if I can
erase it cleanly.

00:00:43.800 --> 00:00:45.050
That's pretty good.

00:00:48.260 --> 00:00:50.270
Let me erase all of this.

00:00:56.600 --> 00:00:57.850
All of this.

00:00:57.850 --> 00:00:59.100
I'll even erase this.

00:01:02.430 --> 00:01:05.280
That's pretty good, all right.

00:01:05.280 --> 00:01:08.350
Now back to work.

00:01:08.350 --> 00:01:11.070
So, we know that-- or hopefully
we know-- that

00:01:11.070 --> 00:01:13.390
acceleration is the second
derivative of x as

00:01:13.390 --> 00:01:14.060
a function of t.

00:01:14.060 --> 00:01:18.910
So we can rewrite this as
mass times the second

00:01:18.910 --> 00:01:21.290
derivative of x.

00:01:21.290 --> 00:01:23.870
So I'll write that as-- well,
I think the easiest notation

00:01:23.870 --> 00:01:26.940
would just be x prime prime.

00:01:26.940 --> 00:01:29.640
That's just the second
derivative of x as

00:01:29.640 --> 00:01:31.180
a function of t.

00:01:31.180 --> 00:01:33.530
I'll write the function
notation, just so you remember

00:01:33.530 --> 00:01:36.190
this is a function of time.

00:01:36.190 --> 00:01:43.520
Is equal to minus
k times x of t.

00:01:43.520 --> 00:01:46.080
And what you see here, what
I've just written, this is

00:01:46.080 --> 00:01:50.270
actually a differential
equation.

00:01:50.270 --> 00:01:51.650
And so what is a differential
equation?

00:01:51.650 --> 00:01:55.030
Well, it's an equation where,
in one expression, or in one

00:01:55.030 --> 00:01:57.560
equation, on both sides of
this, you not only have a

00:01:57.560 --> 00:02:01.100
function, but you have
derivatives of that function.

00:02:01.100 --> 00:02:05.670
And the solution to a
differential equation isn't

00:02:05.670 --> 00:02:06.880
just a number, right?

00:02:06.880 --> 00:02:10.250
A solution to equations that
we've done in the past are

00:02:10.250 --> 00:02:14.420
numbers, essentially, or a set
of numbers, or maybe a line.

00:02:14.420 --> 00:02:17.660
But the solution to differential
equations is

00:02:17.660 --> 00:02:20.250
actually going to be a function,
or a class of

00:02:20.250 --> 00:02:22.470
functions, or a set
of functions.

00:02:22.470 --> 00:02:25.700
So it'll take a little time to
get your head around it, but

00:02:25.700 --> 00:02:28.100
this is as good an example as
ever to be exposed to it.

00:02:28.100 --> 00:02:31.390
And we're not going to solve
this differential equation

00:02:31.390 --> 00:02:32.130
analytically.

00:02:32.130 --> 00:02:35.550
We're going to use our intuition
behind what we did

00:02:35.550 --> 00:02:37.710
earlier in the previous video.

00:02:37.710 --> 00:02:41.040
We're going to use that to guess
at what a solution to

00:02:41.040 --> 00:02:42.710
this differential equation is.

00:02:42.710 --> 00:02:46.070
And then, if it works out, then
we'll have a little bit

00:02:46.070 --> 00:02:46.640
more intuition.

00:02:46.640 --> 00:02:49.140
And then we'll actually know
what the position is, at any

00:02:49.140 --> 00:02:52.870
given time, of this mass
attached to the spring.

00:02:52.870 --> 00:02:53.700
So this is exciting.

00:02:53.700 --> 00:02:56.140
This is a differential
equation.

00:02:56.140 --> 00:02:58.580
When we drew the position-- our
intuition for the position

00:02:58.580 --> 00:03:01.070
over time-- our intuition tells
us that it's a cosine

00:03:01.070 --> 00:03:02.790
function, with amplitude A.

00:03:02.790 --> 00:03:08.130
So we said it's A cosine omega
t, where this is the angular

00:03:08.130 --> 00:03:11.400
velocity of-- well, I don't want
to go into that just yet,

00:03:11.400 --> 00:03:13.080
we'll get a little bit more
intuition in a second.

00:03:13.080 --> 00:03:17.490
And now, what we can do is,
let's test this expression--

00:03:17.490 --> 00:03:23.980
this function-- to see if it
satisfies this equation.

00:03:23.980 --> 00:03:25.230
Right?

00:03:28.260 --> 00:03:39.880
If we say that x of t is equal
to A cosine of wt, what is the

00:03:39.880 --> 00:03:42.530
derivative of this?
x prime of t.

00:03:42.530 --> 00:03:45.450
And you could review
the derivative

00:03:45.450 --> 00:03:47.560
videos to remember this.

00:03:47.560 --> 00:03:50.100
Well, it's the derivative of the
inside, so it'll be that

00:03:50.100 --> 00:03:53.680
omega, times the
outside scalar.

00:03:53.680 --> 00:03:56.650
A omega.

00:03:56.650 --> 00:03:58.915
And then the derivative-- I'm
just doing the chain rule--

00:03:58.915 --> 00:04:01.700
the derivative of cosine of t is
minus sine of whatever's in

00:04:01.700 --> 00:04:03.070
the inside.

00:04:03.070 --> 00:04:04.780
I'll put the minus outside.

00:04:04.780 --> 00:04:11.040
So it's minus sine of wt.

00:04:11.040 --> 00:04:15.510
And then, if we want the second
derivative-- so that's

00:04:15.510 --> 00:04:17.120
x prime prime of t.

00:04:20.899 --> 00:04:22.490
Let me do this in a different
color, just so it doesn't get

00:04:22.490 --> 00:04:23.430
monotonous.

00:04:23.430 --> 00:04:25.540
That's the derivative
of this, right?

00:04:25.540 --> 00:04:28.270
So what's the derivative
of-- these are just

00:04:28.270 --> 00:04:29.260
scalar values, right?

00:04:29.260 --> 00:04:30.570
These are just constants.

00:04:30.570 --> 00:04:32.870
So the derivative of the
inside is an omega.

00:04:32.870 --> 00:04:35.270
I multiply the omega times
the scalar constant.

00:04:35.270 --> 00:04:43.190
I get minus A omega squared.

00:04:43.190 --> 00:04:45.350
And then the derivative of
sine is just cosine.

00:04:45.350 --> 00:04:46.910
But the minus is still there,
because I had the minus to

00:04:46.910 --> 00:04:47.950
begin with.

00:04:47.950 --> 00:04:54.422
Minus cosine of omega t.

00:04:54.422 --> 00:04:56.560
Now let's see if this is true.

00:04:56.560 --> 00:05:07.840
So if this is true, I should
be able to say that m times

00:05:07.840 --> 00:05:10.740
the second derivative of x of
t, which is in this case is

00:05:10.740 --> 00:05:21.750
this, times minus Aw
squared cosine wt.

00:05:21.750 --> 00:05:31.430
That should be equal to minus k
times my original function--

00:05:31.430 --> 00:05:32.380
times x of t.

00:05:32.380 --> 00:05:34.020
And x of t is a cosine wt.

00:05:37.210 --> 00:05:39.530
I'm running out of space.

00:05:39.530 --> 00:05:41.590
Hopefully you understand
what I'm saying.

00:05:41.590 --> 00:05:44.870
I just substituted x prime
prime, the second derivative,

00:05:44.870 --> 00:05:51.310
into this, and I just
substituted x of t, which I

00:05:51.310 --> 00:05:53.800
guess that's that, in here.

00:05:53.800 --> 00:05:55.070
And now I got this.

00:05:55.070 --> 00:05:56.570
And let me see if
I can rewrite.

00:05:56.570 --> 00:05:58.670
Maybe I can get rid of
the spring up here.

00:05:58.670 --> 00:05:59.510
I'm trying to look for space.

00:05:59.510 --> 00:06:00.930
I don't want to get rid of this,
because I think this

00:06:00.930 --> 00:06:04.200
gives us some intuition
of what we're doing.

00:06:04.200 --> 00:06:06.250
One of those days that I wish
I had a larger blackboard.

00:06:10.680 --> 00:06:13.160
Erase the spring.

00:06:13.160 --> 00:06:16.490
Hopefully you can remember
that image in your mind.

00:06:16.490 --> 00:06:19.190
And actually, I can
erase that.

00:06:19.190 --> 00:06:21.945
I can erase that.

00:06:21.945 --> 00:06:24.600
I can erase all of this, just so
I have some space, without

00:06:24.600 --> 00:06:27.150
getting rid of that nice curve I
took the time to draw in the

00:06:27.150 --> 00:06:28.650
last video.

00:06:28.650 --> 00:06:29.750
Almost there.

00:06:29.750 --> 00:06:31.860
OK.

00:06:31.860 --> 00:06:33.110
Back to work.

00:06:35.470 --> 00:06:37.420
Make sure my pen feels
right, OK.

00:06:37.420 --> 00:06:42.960
So all I did is I took-- we
said that by the spring

00:06:42.960 --> 00:06:46.900
constant, if you rewrite force
as mass times acceleration,

00:06:46.900 --> 00:06:47.580
you get this.

00:06:47.580 --> 00:06:49.340
Which is essentially a
differential equation, I just

00:06:49.340 --> 00:06:52.220
rewrote acceleration as
the second derivative.

00:06:52.220 --> 00:06:56.450
Then I took a guess, that this
is x of t, just based on our

00:06:56.450 --> 00:06:58.500
intuition of the drawing.

00:06:58.500 --> 00:06:59.680
I took a guess.

00:06:59.680 --> 00:07:01.430
And then I took the second
derivative of it.

00:07:01.430 --> 00:07:01.490
Right?

00:07:01.490 --> 00:07:04.160
This is the first derivative,
this is the second derivative.

00:07:04.160 --> 00:07:06.300
And then I substituted the
second derivative here, and I

00:07:06.300 --> 00:07:07.450
substituted the function here.

00:07:07.450 --> 00:07:08.770
And this is what I got.

00:07:08.770 --> 00:07:12.500
And so let me see if I can
simplify that a little bit.

00:07:12.500 --> 00:07:28.550
So if I rewrite there, I get
minus mAw squared cosine of wt

00:07:28.550 --> 00:07:37.720
is equal to minus
kA cosine of wt.

00:07:37.720 --> 00:07:38.900
Well it looks good so far.

00:07:38.900 --> 00:07:41.685
Let's see, we can get rid of the
minus signs on both sides.

00:07:46.450 --> 00:07:47.680
Get rid of the A's
on both sides.

00:07:47.680 --> 00:07:47.750
Right?

00:07:47.750 --> 00:07:50.335
We can divide both sides by A.

00:07:50.335 --> 00:07:54.090
Let me do this in black, just
so it really erases it.

00:07:54.090 --> 00:07:57.780
So if we get rid of A on both
sides, we're left with that.

00:08:01.210 --> 00:08:04.690
And then-- so let's see, we
have mw squared cosine of

00:08:04.690 --> 00:08:08.620
omega t is equal to k
cosine of omega t.

00:08:08.620 --> 00:08:12.660
So this equation holds
true if what is true?

00:08:12.660 --> 00:08:21.120
This equation holds true if mw
squared-- or omega squared, I

00:08:21.120 --> 00:08:21.520
think that's omega.

00:08:21.520 --> 00:08:24.796
I always forget my--
is equal to k.

00:08:24.796 --> 00:08:29.110
Or another way of saying
it, if omega squared is

00:08:29.110 --> 00:08:32.909
equal to k over m.

00:08:32.909 --> 00:08:40.309
Or, omega is equal to the
square root of k over m.

00:08:40.309 --> 00:08:41.220
So there we have it.

00:08:41.220 --> 00:08:44.080
We have figured out what
x of t has to be.

00:08:44.080 --> 00:08:47.650
We said that this differential
equation is true, if this is x

00:08:47.650 --> 00:08:49.840
of t, and omega is
equal to this.

00:08:49.840 --> 00:08:55.520
So now we've figured out the
actual function that describes

00:08:55.520 --> 00:08:58.350
the position of that spring
as a function of time.

00:08:58.350 --> 00:09:04.886
x of t is going to be equal to--
we were right about the

00:09:04.886 --> 00:09:07.590
A, and that's just intuition,
right, because the amplitude

00:09:07.590 --> 00:09:13.820
of this cosine function is A--
A cosine-- and instead of

00:09:13.820 --> 00:09:18.880
writing w, we can now write the
square root of k over m.

00:09:18.880 --> 00:09:26.060
The square root of k over m t.

00:09:26.060 --> 00:09:27.490
That to me is amazing.

00:09:27.490 --> 00:09:34.140
We have now, using not too
sophisticated calculus, solved

00:09:34.140 --> 00:09:34.980
a differential equation.

00:09:34.980 --> 00:09:38.860
And now can-- if you tell me at
5.8 seconds, where is x, I

00:09:38.860 --> 00:09:39.920
can tell you.

00:09:39.920 --> 00:09:42.180
And I just realized that I am
now running out of time, so I

00:09:42.180 --> 00:09:44.210
will see you in the
next video.


Introduction to harmonic motion

WEBVTT
Kind: captions
Language: en

00:00:01.140 --> 00:00:04.080
Let's see if we can use what we
know about springs now to

00:00:04.080 --> 00:00:05.640
get a little intuition
about how the

00:00:05.640 --> 00:00:06.810
spring moves over time.

00:00:06.810 --> 00:00:07.600
And hopefully we'll
learn a little bit

00:00:07.600 --> 00:00:08.660
about harmonic motion.

00:00:08.660 --> 00:00:11.010
We'll actually even step into
the world of differential

00:00:11.010 --> 00:00:11.930
equations a little bit.

00:00:11.930 --> 00:00:14.200
And don't get daunted
when we get there.

00:00:14.200 --> 00:00:15.950
Or just close your eyes
when it happens.

00:00:15.950 --> 00:00:18.340
Anyway, so I've drawn a spring,
like I've done in the

00:00:18.340 --> 00:00:19.410
last couple of videos.

00:00:19.410 --> 00:00:23.150
And 0, this point in the x-axis,
that's where the

00:00:23.150 --> 00:00:25.500
spring's natural resting
state is.

00:00:25.500 --> 00:00:28.890
And in this example I
have a mass, mass m,

00:00:28.890 --> 00:00:30.220
attached to the spring.

00:00:30.220 --> 00:00:31.480
And I've stretched the string.

00:00:31.480 --> 00:00:32.680
I've essentially pulled it.

00:00:32.680 --> 00:00:35.480
So the mass is now sitting
at point A.

00:00:35.480 --> 00:00:36.710
So what's going to
happen to this?

00:00:36.710 --> 00:00:40.330
Well, as we know, the force, the
restorative force of the

00:00:40.330 --> 00:00:45.230
spring, is equal to minus some

00:00:45.230 --> 00:00:47.360
constant, times the x position.

00:00:47.360 --> 00:00:48.860
The x position starting at A.

00:00:48.860 --> 00:00:50.910
So initially the spring
is going to pull

00:00:50.910 --> 00:00:52.870
back this way, right?

00:00:52.870 --> 00:00:54.750
The spring is going to
pull back this way.

00:00:54.750 --> 00:00:57.280
It's going to get faster and
faster and faster and faster.

00:00:57.280 --> 00:00:58.710
And we learned that at this
point, it has a lot of

00:00:58.710 --> 00:00:59.870
potential energy.

00:00:59.870 --> 00:01:02.110
At this point, when it kind of
gets back to its resting

00:01:02.110 --> 00:01:06.940
state, it'll have a lot of
velocity and a lot of kinetic

00:01:06.940 --> 00:01:08.680
energy, but very little
potential energy.

00:01:08.680 --> 00:01:10.780
But then its momentum is going
to keep it going, and it's

00:01:10.780 --> 00:01:14.950
going to compress the spring all
the way, until all of that

00:01:14.950 --> 00:01:16.930
kinetic energy is turned back
into potential energy.

00:01:16.930 --> 00:01:19.370
Then the process will
start over again.

00:01:19.370 --> 00:01:22.550
So let's see if we can just get
an intuition for what x

00:01:22.550 --> 00:01:24.160
will look like as a
function of time.

00:01:24.160 --> 00:01:29.980
So our goal is to figure out x
of t, x as a function of time.

00:01:29.980 --> 00:01:31.900
That's going to be our goal
on this video and

00:01:31.900 --> 00:01:33.610
probably the next few.

00:01:33.610 --> 00:01:37.770
So let's just get an intuition
for what's happening here.

00:01:37.770 --> 00:01:41.340
So let me try to graph x
as a function of time.

00:01:41.340 --> 00:01:46.030
So time is the independent
variable.

00:01:46.030 --> 00:01:49.440
And I'll start at time
is equal to 0.

00:01:49.440 --> 00:01:52.060
So this is the time axis.

00:01:52.060 --> 00:01:53.100
Let me draw the x-axis.

00:01:53.100 --> 00:01:55.420
This might be a little unusual
for you, for me to draw the

00:01:55.420 --> 00:01:57.860
x-axis in the vertical, but
that's because x is the

00:01:57.860 --> 00:02:01.530
dependent variable in
this situation.

00:02:01.530 --> 00:02:05.880
So that's the x-axis,
very unusually.

00:02:05.880 --> 00:02:08.940
Or we could say x of t, just so
you know x is a function of

00:02:08.940 --> 00:02:12.100
time, x of t.

00:02:12.100 --> 00:02:15.500
And this state, that I've drawn
here, this is at time

00:02:15.500 --> 00:02:16.490
equals 0, right?

00:02:16.490 --> 00:02:17.400
So this is at 0.

00:02:17.400 --> 00:02:19.340
Let me switch colors.

00:02:19.340 --> 00:02:24.020
So at time equals 0, what is
the x position of the mass?

00:02:24.020 --> 00:02:26.300
Well the x position
is A, right?

00:02:26.300 --> 00:02:30.890
So if I draw this, this is A.

00:02:30.890 --> 00:02:32.280
Actually, let me draw
a line there.

00:02:32.280 --> 00:02:34.700
That might come in useful.

00:02:34.700 --> 00:02:37.560
This is A.

00:02:37.560 --> 00:02:40.030
And then this is going to be--
let me try to make it

00:02:40.030 --> 00:02:44.420
relatively-- that
is negative A.

00:02:44.420 --> 00:02:45.670
That's minus A.

00:02:49.070 --> 00:02:52.200
So at time t equals
0, where is it?

00:02:52.200 --> 00:02:52.920
Well it's at A.

00:02:52.920 --> 00:02:57.530
So this is where the
graph is, right?

00:02:57.530 --> 00:02:59.880
Actually, let's do something
interesting.

00:02:59.880 --> 00:03:01.740
Let's define the period.

00:03:01.740 --> 00:03:03.880
So the period I'll do
with a capital T.

00:03:03.880 --> 00:03:07.950
Let's say the period is how long
it takes for this mass to

00:03:07.950 --> 00:03:09.140
go from this position.

00:03:09.140 --> 00:03:11.020
It's going to accelerate,
accelerate, accelerate,

00:03:11.020 --> 00:03:12.180
accelerate.

00:03:12.180 --> 00:03:14.800
Be going really fast at this
point, all kinetic energy.

00:03:14.800 --> 00:03:17.080
And then start slowing down,
slowing down, slowing down,

00:03:17.080 --> 00:03:17.800
slowing down.

00:03:17.800 --> 00:03:20.390
And then do that whole process
all the way back.

00:03:20.390 --> 00:03:22.780
Let's say T is the amount of
time it takes to do that whole

00:03:22.780 --> 00:03:24.560
process, right?

00:03:24.560 --> 00:03:31.850
So at time 0 today, and then we
also know that at time T--

00:03:31.850 --> 00:03:38.500
this is time T-- it'll
also be at A, right?

00:03:38.500 --> 00:03:40.660
I'm just trying to graph some
points that I know of this

00:03:40.660 --> 00:03:43.050
function and just see if I can
get some intuition of what

00:03:43.050 --> 00:03:46.590
this function might
be analytically.

00:03:46.590 --> 00:03:52.060
So if it takes T seconds to go
there and back, it takes T

00:03:52.060 --> 00:03:54.090
over 2 seconds to
get here, right?

00:03:54.090 --> 00:03:56.470
The same amount of time it takes
to get here was also the

00:03:56.470 --> 00:03:58.770
same amount of time it
takes to get back.

00:03:58.770 --> 00:04:05.880
So at T over 2 what's going
to be the x position?

00:04:05.880 --> 00:04:08.780
Well at T over 2, the block
is going to be here.

00:04:08.780 --> 00:04:10.690
It will have compressed
all the way over here.

00:04:10.690 --> 00:04:12.550
So at T over 2, it'll
have been here.

00:04:15.440 --> 00:04:18.680
And then at the points in
between, it will be at x

00:04:18.680 --> 00:04:20.880
equals 0, right?

00:04:20.880 --> 00:04:23.310
It'll be there and there.

00:04:23.310 --> 00:04:24.600
Hopefully that makes sense.

00:04:24.600 --> 00:04:26.520
So now we know these points.

00:04:26.520 --> 00:04:28.900
But let's think about what the
actual function looks like.

00:04:28.900 --> 00:04:30.880
Will it just be a straight line
down, then a straight

00:04:30.880 --> 00:04:33.230
line up, and then the straight
line down, and then a

00:04:33.230 --> 00:04:34.690
straight line up.

00:04:34.690 --> 00:04:37.220
That would imply-- think about
it-- if you have a straight

00:04:37.220 --> 00:04:39.920
line down that whole time, that
means that you would have

00:04:39.920 --> 00:04:43.910
a constant rate of change
of your x value.

00:04:43.910 --> 00:04:45.550
Or another way of thinking about
that is that you would

00:04:45.550 --> 00:04:48.270
have a constant velocity,
right?

00:04:48.270 --> 00:04:51.200
Well do we have a constant
velocity this entire time?

00:04:51.200 --> 00:04:51.880
Well, no.

00:04:51.880 --> 00:04:55.200
We know that at this point right
here you have a very

00:04:55.200 --> 00:04:57.650
high velocity, right?

00:04:57.650 --> 00:04:58.850
You have a very high velocity.

00:04:58.850 --> 00:05:00.700
We know at this point you have
a very low velocity.

00:05:00.700 --> 00:05:03.410
So you're accelerating
this entire time.

00:05:03.410 --> 00:05:05.150
And you actually, the more you
think about it, you're

00:05:05.150 --> 00:05:09.480
actually accelerating at
a decreasing rate.

00:05:09.480 --> 00:05:11.610
But you're accelerating
the entire time.

00:05:11.610 --> 00:05:15.120
And then you're accelerating and
then you're decelerating

00:05:15.120 --> 00:05:16.180
this entire time.

00:05:16.180 --> 00:05:19.420
So your actual rate of change
of x is not constant, so you

00:05:19.420 --> 00:05:21.740
wouldn't have a zigzag
pattern, right?

00:05:21.740 --> 00:05:24.850
And it'll keep going here and
then you'll have a point here.

00:05:24.850 --> 00:05:25.840
So what's happening?

00:05:25.840 --> 00:05:27.980
When you start off, you're
going very slow.

00:05:27.980 --> 00:05:29.860
Your change of x is very slow.

00:05:29.860 --> 00:05:32.380
And then you start
accelerating.

00:05:32.380 --> 00:05:36.290
And then, once you get to this
point, right here, you start

00:05:36.290 --> 00:05:37.540
decelerating.

00:05:39.410 --> 00:05:44.050
Until at this point, your
velocity is exactly 0.

00:05:44.050 --> 00:05:46.820
So your rate of change, or your
slope, is going to be 0.

00:05:46.820 --> 00:05:49.890
And then you're going to start
accelerating back.

00:05:49.890 --> 00:05:51.610
Your velocity is going to get
faster, faster, faster.

00:05:51.610 --> 00:05:53.870
It's going to be really
fast at this point.

00:05:53.870 --> 00:05:57.850
And then you'll start
decelerating at that point.

00:05:57.850 --> 00:05:59.750
So at this point, what does
this point correspond to?

00:05:59.750 --> 00:06:00.820
You're back at A.

00:06:00.820 --> 00:06:04.350
So at this point your velocity
is now 0 again.

00:06:04.350 --> 00:06:06.350
So the rate of change
of x is 0.

00:06:06.350 --> 00:06:08.650
And now you're going to
start accelerating.

00:06:08.650 --> 00:06:11.290
Your slope increases, increases,
increases.

00:06:11.290 --> 00:06:14.500
This is the point of highest
kinetic energy right here.

00:06:14.500 --> 00:06:17.220
Then your velocity starts
slowing down.

00:06:17.220 --> 00:06:20.420
And notice here, your slope
at these points is 0.

00:06:20.420 --> 00:06:21.640
So that means you
have no kinetic

00:06:21.640 --> 00:06:22.710
energy at those points.

00:06:22.710 --> 00:06:25.330
And it just keeps on going.

00:06:25.330 --> 00:06:27.590
On and on and on
and on and on.

00:06:27.590 --> 00:06:28.970
So what does this look like?

00:06:28.970 --> 00:06:31.050
Well, I haven't proven it to
you, but out of all the

00:06:31.050 --> 00:06:34.610
functions that I have in my
repertoire, this looks an

00:06:34.610 --> 00:06:36.700
awful lot like a trigonometric
function.

00:06:36.700 --> 00:06:38.860
And if I had to pick one,
I would pick cosine.

00:06:38.860 --> 00:06:40.100
Well why?

00:06:40.100 --> 00:06:44.210
Because when cosine is 0--
I'll write it down here--

00:06:44.210 --> 00:06:47.360
cosine of 0 is equal
to 1, right?

00:06:47.360 --> 00:06:50.610
So when t equals 0, this
function is equal to A.

00:06:50.610 --> 00:06:59.880
So this function probably looks
something like A cosine

00:06:59.880 --> 00:07:05.730
of-- and I'll just use the
variable omega t-- it probably

00:07:05.730 --> 00:07:08.650
looks something like that,
this function.

00:07:08.650 --> 00:07:10.570
And we'll learn in a second
that it looks

00:07:10.570 --> 00:07:11.110
exactly like that.

00:07:11.110 --> 00:07:12.460
But I want to prove it
to you, so don't just

00:07:12.460 --> 00:07:13.690
take my word for it.

00:07:13.690 --> 00:07:17.490
So let's just figure out how we
can figure out what w is.

00:07:17.490 --> 00:07:21.100
And it's probably a function of
the mass of this object and

00:07:21.100 --> 00:07:22.910
also probably a function
of the spring

00:07:22.910 --> 00:07:24.400
constant, but I'm not sure.

00:07:24.400 --> 00:07:26.620
So let's see what we
can figure out.

00:07:26.620 --> 00:07:31.000
Well now I'm about to embark
into a little bit of calculus.

00:07:31.000 --> 00:07:32.380
Actually, a decent
bit of calculus.

00:07:32.380 --> 00:07:34.470
And we'll actually even touch
on differential equations.

00:07:34.470 --> 00:07:36.790
This might be the first
differential equation you see

00:07:36.790 --> 00:07:39.620
in your life, so it's a
momentous occasion.

00:07:39.620 --> 00:07:41.120
But let's just move forward.

00:07:41.120 --> 00:07:42.670
Close your eyes if you don't
want to be confused, or go

00:07:42.670 --> 00:07:46.290
watch the calculus videos at
least so you know what a

00:07:46.290 --> 00:07:47.510
derivative is.

00:07:47.510 --> 00:07:52.190
So let's write this seemingly
simple equation, or let's

00:07:52.190 --> 00:07:54.570
rewrite it in ways
that we know.

00:07:54.570 --> 00:07:57.620
So what's the definition
of force?

00:07:57.620 --> 00:08:00.030
Force is mass times
acceleration, right?

00:08:00.030 --> 00:08:05.820
So we can rewrite Hooke's law
as-- let me switch colors--

00:08:05.820 --> 00:08:11.020
mass times acceleration is
equal to minus the spring

00:08:11.020 --> 00:08:15.600
constant, times the
position, right?

00:08:15.600 --> 00:08:17.840
And I'll actually write the
position as a function of t,

00:08:17.840 --> 00:08:19.010
just so you remember.

00:08:19.010 --> 00:08:22.230
We're so used to x being the
independent variable, that if

00:08:22.230 --> 00:08:24.470
I didn't write that function of
t, it might get confusing.

00:08:24.470 --> 00:08:26.770
You're like, oh I thought x is
the independent variable.

00:08:26.770 --> 00:08:28.350
No.

00:08:28.350 --> 00:08:31.150
Because in this function that we
want to figure out, we want

00:08:31.150 --> 00:08:33.270
to know what happens as
a function of time?

00:08:33.270 --> 00:08:35.280
So actually this is also
maybe a good review

00:08:35.280 --> 00:08:38.080
of parametric equations.

00:08:38.080 --> 00:08:39.630
This is where we get
into calculus.

00:08:39.630 --> 00:08:40.880
What is acceleration?

00:08:44.670 --> 00:08:51.650
If I call my position x, my
position is equal to x as a

00:08:51.650 --> 00:08:52.980
function of t, right?

00:08:52.980 --> 00:08:56.500
I put in some time, and it tells
me what my x value is.

00:08:56.500 --> 00:08:57.510
That's my position.

00:08:57.510 --> 00:08:58.890
What's my velocity?

00:08:58.890 --> 00:09:02.300
Well my velocity is the
derivative of this, right?

00:09:02.300 --> 00:09:06.320
My velocity, at any given point,
is going to be the

00:09:06.320 --> 00:09:07.890
derivative of this function.

00:09:07.890 --> 00:09:11.130
The rate of change of this
function with respect to t.

00:09:11.130 --> 00:09:13.340
So I would take the rate
of change with

00:09:13.340 --> 00:09:17.080
respect to t, x of t.

00:09:17.080 --> 00:09:23.030
And I could write
that as dx, dt.

00:09:23.030 --> 00:09:24.490
And then what's acceleration?

00:09:24.490 --> 00:09:26.410
Well acceleration is just
the rate of change

00:09:26.410 --> 00:09:28.340
of velocity, right?

00:09:28.340 --> 00:09:30.560
So it would be taking the
derivative of this.

00:09:30.560 --> 00:09:32.940
Or another way of doing it, it's
like taking the second

00:09:32.940 --> 00:09:36.460
derivative of the position
function, right?

00:09:36.460 --> 00:09:41.850
So in this situation,
acceleration is equal to, we

00:09:41.850 --> 00:09:44.930
could write it as-- I'm just
showing you all different

00:09:44.930 --> 00:09:49.650
notations-- x prime prime of t,
second derivative of x with

00:09:49.650 --> 00:09:50.480
respect to t.

00:09:50.480 --> 00:09:53.930
Or-- these are just notational--
d squared x over

00:09:53.930 --> 00:09:55.910
dt squared.

00:09:55.910 --> 00:09:57.000
So that's the second
derivative.

00:09:57.000 --> 00:09:58.280
Oh it looks like I'm running
out of time.

00:09:58.280 --> 00:09:59.480
So I'll see you in
the next video.

00:09:59.480 --> 00:10:01.510
Remember what I just
wrote. just wrote


Spring potential energy example (mistake in math)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.460
Welcome back.

00:00:01.460 --> 00:00:04.490
So let's do a potential
energy problem with

00:00:04.490 --> 00:00:05.510
a compressed spring.

00:00:05.510 --> 00:00:08.060
So let's make this an
interesting problem.

00:00:08.060 --> 00:00:10.220
Let's say I have
a loop-d-loop.

00:00:10.220 --> 00:00:12.120
A loop-d-loop made out of ice.

00:00:12.120 --> 00:00:15.080
And I made it out of ice so that
we don't have friction.

00:00:15.080 --> 00:00:16.330
Let me draw my loop-d-loop.

00:00:19.900 --> 00:00:22.850
There's the loop, there's
the d-loop.

00:00:22.850 --> 00:00:24.020
All right.

00:00:24.020 --> 00:00:29.475
And let's say this loop-d-loop
has a radius of 1 meter.

00:00:29.475 --> 00:00:34.220
Let's say this is-- this right
here-- is 1 meter.

00:00:34.220 --> 00:00:36.700
So of course the loop-d-loop
is 2 meters high.

00:00:40.050 --> 00:00:42.380
And let's say I have a
spring here-- it's

00:00:42.380 --> 00:00:43.760
a compressed spring.

00:00:43.760 --> 00:00:45.200
Let's say this is the wall.

00:00:45.200 --> 00:00:46.942
This is my spring, it's
compressed, so it's

00:00:46.942 --> 00:00:49.000
all tight like that.

00:00:49.000 --> 00:00:53.050
And let's say its spring
constant, k, is,

00:00:53.050 --> 00:00:56.150
I don't know, 10.

00:00:56.150 --> 00:00:59.650
Attached to that compressed
spring-- so I have a block of

00:00:59.650 --> 00:01:03.860
ice, because I need ice on ice,
so I have no friction.

00:01:03.860 --> 00:01:07.740
This is my block of
ice, shining.

00:01:07.740 --> 00:01:16.930
And let's say the block of ice
is, I don't know, 4 kilograms.

00:01:16.930 --> 00:01:19.860
And we also know that we are
on Earth, and that's

00:01:19.860 --> 00:01:21.330
important, because this problem
might have been

00:01:21.330 --> 00:01:23.660
different if we were
on another planet.

00:01:23.660 --> 00:01:28.520
And my question to you is how
much do we have to compress

00:01:28.520 --> 00:01:31.380
the spring-- so, let's say
that the spring's natural

00:01:31.380 --> 00:01:36.000
state was here, right, if
we didn't push on it.

00:01:36.000 --> 00:01:37.230
And now it's here.

00:01:37.230 --> 00:01:38.740
So what is this distance?

00:01:38.740 --> 00:01:42.800
How much do I have to compress
this spring, in order for when

00:01:42.800 --> 00:01:47.615
I let go of the spring, the
block goes with enough speed

00:01:47.615 --> 00:01:50.920
and enough energy, that it's
able to complete the

00:01:50.920 --> 00:01:56.320
loop-d-loop, and reach safely
to the other end?

00:01:56.320 --> 00:01:58.640
So, how do we do this problem?

00:01:58.640 --> 00:02:02.440
Well, in order-- any loop-d-loop
problem, the hard

00:02:02.440 --> 00:02:04.870
part is completing the
high point of the

00:02:04.870 --> 00:02:07.240
loop-d-loop, right?

00:02:07.240 --> 00:02:09.490
The hard part is making sure
you have enough velocity at

00:02:09.490 --> 00:02:12.050
this point, so that you
don't fall down.

00:02:12.050 --> 00:02:15.430
Your velocity has to offset the
downward acceleraton, in

00:02:15.430 --> 00:02:17.530
which case-- and here, is going
to be the centripetal

00:02:17.530 --> 00:02:19.320
acceleration, right?

00:02:19.320 --> 00:02:20.740
So that's one thing
to think about.

00:02:20.740 --> 00:02:23.180
And you might say, wow this is
complicated, I have a spring

00:02:23.180 --> 00:02:25.150
here, it's going to accelerate
the block.

00:02:25.150 --> 00:02:26.720
And then the block's going to
get here, and then it's going

00:02:26.720 --> 00:02:28.720
to decelerate, decelerate.

00:02:28.720 --> 00:02:30.720
This is probably where it's
going to be at its slowest,

00:02:30.720 --> 00:02:32.610
then it's going to accelerate
back here.

00:02:32.610 --> 00:02:34.430
It's a super complicated
problem.

00:02:34.430 --> 00:02:36.400
And in physics, whenever you
have a super complicated

00:02:36.400 --> 00:02:38.980
problem, it's probably because
you are approaching it in a

00:02:38.980 --> 00:02:40.810
super complicated way,
but there might be a

00:02:40.810 --> 00:02:41.610
simple way to do it.

00:02:41.610 --> 00:02:44.980
And that's using energy--
potential and kinetic energy.

00:02:44.980 --> 00:02:47.280
And what we learned when we
learned about potential and

00:02:47.280 --> 00:02:50.190
kinetic energy, is that the
total energy in the system

00:02:50.190 --> 00:02:51.520
doesn't change.

00:02:51.520 --> 00:02:53.370
It just gets converted from
one form to another.

00:02:53.370 --> 00:02:55.820
So it goes from potential
energy to kinetic

00:02:55.820 --> 00:02:58.680
energy, or to heat.

00:02:58.680 --> 00:02:59.890
And we assume that
there's no heat,

00:02:59.890 --> 00:03:00.780
because there's no friction.

00:03:00.780 --> 00:03:02.940
So let's do this problem.

00:03:02.940 --> 00:03:05.970
So what we want to know is, how
much do I have to compress

00:03:05.970 --> 00:03:06.760
this spring?

00:03:06.760 --> 00:03:09.580
So what I'm essentially saying
is, how much potential energy

00:03:09.580 --> 00:03:13.680
do I have to start off with--
with this compressed spring--

00:03:13.680 --> 00:03:15.900
in order to make it up here?

00:03:15.900 --> 00:03:17.310
So what's the potential
energy?

00:03:17.310 --> 00:03:19.675
Let's say I compress the
spring x meters.

00:03:22.340 --> 00:03:24.880
And in the last video, how
much potential energy

00:03:24.880 --> 00:03:26.410
would I then have?

00:03:26.410 --> 00:03:28.720
Well, we learned that the
potential energy of a

00:03:28.720 --> 00:03:32.040
compressed spring-- and I'll
call this the initial

00:03:32.040 --> 00:03:37.110
potential energy-- the initial
potential energy, with an i--

00:03:37.110 --> 00:03:42.720
is equal to 1/2 kx squared.

00:03:42.720 --> 00:03:44.180
And we know what k is.

00:03:44.180 --> 00:03:47.140
I told you that the spring
constant for the spring is 10.

00:03:47.140 --> 00:03:52.990
So my initial potential energy
is going to be 1/2 times 10,

00:03:52.990 --> 00:03:54.240
times x squared.

00:03:58.010 --> 00:04:00.340
So what are all of the energy
components here?

00:04:00.340 --> 00:04:02.520
Well, obviously, at this point,
the block's going to

00:04:02.520 --> 00:04:05.160
have to be moving, in order
to not fall down.

00:04:05.160 --> 00:04:07.990
So it's going to have
some velocity, v.

00:04:07.990 --> 00:04:10.770
It's going tangential
to the loop-d-loop.

00:04:10.770 --> 00:04:14.020
And it also is going to have
some potential energy still.

00:04:14.020 --> 00:04:15.850
And where is that potential
energy coming from?

00:04:15.850 --> 00:04:18.790
Well, it's going to come because
it's up in the air.

00:04:18.790 --> 00:04:22.089
It's above the surface
of the loop-d-loop.

00:04:22.089 --> 00:04:24.780
So it's going to have some
gravitational potential

00:04:24.780 --> 00:04:26.450
energy, right?

00:04:26.450 --> 00:04:31.370
So at this point, we're going
to have some kinetic energy.

00:04:31.370 --> 00:04:34.460
We'll call that-- well, I'll
just call that kinetic energy

00:04:34.460 --> 00:04:36.690
final-- because this is while
we care about alpha, maybe

00:04:36.690 --> 00:04:38.410
here it might be the kinetic
energy final, but I'll just

00:04:38.410 --> 00:04:40.240
define this as kinetic
energy final.

00:04:40.240 --> 00:04:45.580
And then plus the potential
energy final.

00:04:45.580 --> 00:04:48.480
And that of course, has to
add up to 10x squared.

00:04:48.480 --> 00:04:51.510
And this, of course, now, this
was kind of called the spring

00:04:51.510 --> 00:04:52.850
potential energy,
and now this is

00:04:52.850 --> 00:04:55.080
gravitational potential energy.

00:04:55.080 --> 00:04:57.780
So what's the energy
at this point?

00:04:57.780 --> 00:04:59.660
Well, what's kinetic energy?

00:04:59.660 --> 00:05:06.590
Kinetic energy final is going
to have to be 1/2 times the

00:05:06.590 --> 00:05:11.200
mass times the velocity
squared, right?

00:05:11.200 --> 00:05:13.690
And then what's the potential
energy at this point?

00:05:13.690 --> 00:05:16.660
It's gravitational potential
energy, so it's the mass times

00:05:16.660 --> 00:05:19.380
gravity times this height.

00:05:19.380 --> 00:05:21.150
Right?

00:05:21.150 --> 00:05:22.070
So I'll write that here.

00:05:22.070 --> 00:05:27.250
Potential energy final is going
to be mass times gravity

00:05:27.250 --> 00:05:29.940
times the height, which also
stands for Mass General

00:05:29.940 --> 00:05:33.020
Hospital, anyway.

00:05:33.020 --> 00:05:35.750
You can tell my wife's
a doctor, so my

00:05:35.750 --> 00:05:38.130
brain just-- anyway.

00:05:38.130 --> 00:05:41.360
So let's figure out the kinetic
energy at this point.

00:05:41.360 --> 00:05:44.320
So what does the velocity
have to be?

00:05:44.320 --> 00:05:46.430
Well, we have to figure out
what the centripetal

00:05:46.430 --> 00:05:50.580
acceleration is, and then, given
that, we can figure out

00:05:50.580 --> 00:05:51.120
the velocity.

00:05:51.120 --> 00:05:52.915
Because we know that the
centripetal acceleration-- and

00:05:52.915 --> 00:05:55.730
I'll change colors for
variety-- centripetal

00:05:55.730 --> 00:06:00.830
acceleration has to be the
velocity squared, over the

00:06:00.830 --> 00:06:03.900
radius, right?

00:06:03.900 --> 00:06:06.780
Or we could say-- and what is
the centripetal acceleration

00:06:06.780 --> 00:06:07.490
at this point?

00:06:07.490 --> 00:06:09.450
Well it's just the acceleration
of gravity, 9.8

00:06:09.450 --> 00:06:11.410
meters per second squared.

00:06:11.410 --> 00:06:14.750
So 9.8 meters per second
squared is equal to v

00:06:14.750 --> 00:06:16.470
squared over r.

00:06:16.470 --> 00:06:18.900
And what's the radius
of this loop-d-loop?

00:06:18.900 --> 00:06:20.420
Well it's 1.

00:06:20.420 --> 00:06:21.940
So v squared over r
is just going to

00:06:21.940 --> 00:06:23.420
be equal to v squared.

00:06:23.420 --> 00:06:26.110
So v squared equals 9.8-- we
could take the square root, or

00:06:26.110 --> 00:06:27.740
we could just substitute the
9.8 straight into this

00:06:27.740 --> 00:06:29.420
equation, right?

00:06:29.420 --> 00:06:36.930
So the kinetic energy final is
going to be equal to 1/2 times

00:06:36.930 --> 00:06:45.050
the mass times 4 times
v squared times 9.8.

00:06:45.050 --> 00:06:50.770
And that equals-- let's just use
g for 9.8, because I think

00:06:50.770 --> 00:06:53.110
that might keep it
interesting.

00:06:53.110 --> 00:06:54.490
So this is just g, right?

00:06:54.490 --> 00:06:56.340
So it's 2 times g.

00:06:56.340 --> 00:07:03.610
So the kinetic energy final
is equal to 2g-- and g is

00:07:03.610 --> 00:07:06.680
normally kilogram meters per
second squared, but now it's

00:07:06.680 --> 00:07:07.600
energy, right?

00:07:07.600 --> 00:07:09.360
So it's going to be in joules.

00:07:09.360 --> 00:07:11.640
But it's 2g, right?

00:07:11.640 --> 00:07:13.260
And what is the potential
energy at this point?

00:07:13.260 --> 00:07:18.470
Well, it's the mass, which is
4, times g times the height,

00:07:18.470 --> 00:07:19.490
which is 2.

00:07:19.490 --> 00:07:22.290
So it's equal to 8g.

00:07:22.290 --> 00:07:22.800
Right.

00:07:22.800 --> 00:07:24.770
So what's the total energy
at this point?

00:07:24.770 --> 00:07:29.080
The kinetic energy is 2g, the
potential energy is 8g, so the

00:07:29.080 --> 00:07:32.730
total energy at this
point is 10g.

00:07:32.730 --> 00:07:36.580
10g total energy.

00:07:36.580 --> 00:07:38.950
So if the total energy at this
point is 10g, and we didn't

00:07:38.950 --> 00:07:42.000
lose any energy to friction
and heat, and all of that.

00:07:42.000 --> 00:07:44.800
So then the total energy
at this point has also

00:07:44.800 --> 00:07:46.240
got to equal 10g.

00:07:46.240 --> 00:07:49.530
And at this point we have no
kinetic energy, because this

00:07:49.530 --> 00:07:51.400
block hasn't started
moving yet.

00:07:51.400 --> 00:07:53.210
So all the energy is
a potential energy.

00:07:53.210 --> 00:07:56.280
So this also has to equal 10g.

00:07:56.280 --> 00:07:58.620
And this g, I keep saying,
is just 9.8.

00:07:58.620 --> 00:08:00.750
I just wanted to do that just
so you see that it's a

00:08:00.750 --> 00:08:04.140
multiple of 9.8, just for
you to think about.

00:08:04.140 --> 00:08:04.900
So what do we have here?

00:08:04.900 --> 00:08:05.110
[? I'll do ?]

00:08:05.110 --> 00:08:07.060
these numbers worked out well.

00:08:07.060 --> 00:08:09.410
So let's divide both
sides by 10.

00:08:09.410 --> 00:08:13.730
You get x squared is equal
to g, which is 9.8.

00:08:13.730 --> 00:08:16.560
So the x is going to be equal to
the square root of g, which

00:08:16.560 --> 00:08:19.420
is going to be equal to what?

00:08:19.420 --> 00:08:24.350
Let's see-- if I take 9.8, take
the square root of it,

00:08:24.350 --> 00:08:26.360
it's like 3.13.

00:08:26.360 --> 00:08:30.170
So x is 3.13.

00:08:30.170 --> 00:08:34.049
So we just did a fairly-- what
seemed to be a difficult

00:08:34.049 --> 00:08:35.179
problem, but it wasn't so bad.

00:08:35.179 --> 00:08:37.500
We just said that, well the
energy in the beginning has to

00:08:37.500 --> 00:08:40.340
be the energy at any point in
this, assuming that none of

00:08:40.340 --> 00:08:42.340
the energy is lost to heat.

00:08:42.340 --> 00:08:45.990
And so we just figured out
that if we compress this

00:08:45.990 --> 00:08:48.540
spring, with the spring
constant of 10.

00:08:48.540 --> 00:08:54.920
If we compress it 3.3 meters--
3.13 meters-- we will have

00:08:54.920 --> 00:08:57.670
created enough potential
energy-- and in this case, the

00:08:57.670 --> 00:09:01.730
potential energy is 10 times
9.8, so roughly 98 joules.

00:09:01.730 --> 00:09:06.400
98 joules of potential energy
to carry this object all the

00:09:06.400 --> 00:09:09.150
way with enough velocity at the
top of the loop-d-loop to

00:09:09.150 --> 00:09:11.330
complete it, and then come
back down safely.

00:09:11.330 --> 00:09:13.280
And so if we wanted to think
about it, what's the kinetic

00:09:13.280 --> 00:09:14.140
energy at this point?

00:09:14.140 --> 00:09:16.730
Well we figured out it
was 2 times g, so

00:09:16.730 --> 00:09:23.120
it's like 19.6 joules.

00:09:23.120 --> 00:09:24.020
Right.

00:09:24.020 --> 00:09:30.590
And then at this point,
it is 98 joules.

00:09:30.590 --> 00:09:30.950
Right?

00:09:30.950 --> 00:09:32.000
Did I do that right?

00:09:32.000 --> 00:09:35.160
Well, anyway I'm running out
of time, so I hope I did do

00:09:35.160 --> 00:09:36.150
that last part right.

00:09:36.150 --> 00:09:38.130
But I'll see you in
the next video.


Potential energy stored in a spring

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.770
Welcome back.

00:00:01.770 --> 00:00:03.750
So we have this green spring
here, and let's see,

00:00:03.750 --> 00:00:04.910
there's a wall here.

00:00:04.910 --> 00:00:06.080
This connected to the wall.

00:00:06.080 --> 00:00:09.690
And let's say that this is where
the spring is naturally.

00:00:09.690 --> 00:00:12.560
So if I were not to push on the
spring, it would stretch

00:00:12.560 --> 00:00:14.040
all the way out here.

00:00:14.040 --> 00:00:16.410
But in this situation, I pushed
on the spring, so it

00:00:16.410 --> 00:00:19.560
has a displacement
of x to the left.

00:00:19.560 --> 00:00:21.450
And we'll just worry about
magnitude, so we won't worry

00:00:21.450 --> 00:00:23.770
too much about direction.

00:00:23.770 --> 00:00:25.680
So what I want to do is think
a little bit-- well, first I

00:00:25.680 --> 00:00:29.710
want to graph how much force
I've applied at different

00:00:29.710 --> 00:00:31.920
points as I compress
this spring.

00:00:31.920 --> 00:00:35.140
And then I want to use that
graph to maybe figure out how

00:00:35.140 --> 00:00:38.270
much work we did in compressing
the spring.

00:00:38.270 --> 00:00:45.300
So let's look at-- I know I'm
compressing to the left.

00:00:45.300 --> 00:00:48.990
Maybe I should compress to the
right, so that you can-- well,

00:00:48.990 --> 00:00:51.050
we're just worrying about the
magnitude of the x-axis.

00:00:51.050 --> 00:00:53.960
Let's draw a little
graph here.

00:00:53.960 --> 00:00:59.050
That's my y-axis, x-axis.

00:01:03.960 --> 00:01:09.350
So this axis is how much I've
compressed it, x, and then

00:01:09.350 --> 00:01:14.170
this axis, the y-axis, is how
much force I have to apply.

00:01:14.170 --> 00:01:18.850
So when the spring was initially
all the way out

00:01:18.850 --> 00:01:21.180
here, to compress it a little
bit, how much force

00:01:21.180 --> 00:01:22.670
do I have to apply?

00:01:22.670 --> 00:01:25.950
Well, this was its natural
state, right?

00:01:25.950 --> 00:01:29.190
And we know from-- well, Hooke's
Law told us that the

00:01:29.190 --> 00:01:34.570
restorative force-- I'll write
a little r down here-- is

00:01:34.570 --> 00:01:39.400
equal to negative K, where K is
the spring constant, times

00:01:39.400 --> 00:01:42.010
the displacement, right?

00:01:42.010 --> 00:01:44.580
That's the restorative force,
so that's the force that the

00:01:44.580 --> 00:01:47.470
spring applies to whoever's
pushing on it.

00:01:47.470 --> 00:01:50.260
The force to compress it is just
the same thing, but it's

00:01:50.260 --> 00:01:51.980
going in the same direction
as the x.

00:01:51.980 --> 00:01:55.310
If I'm moving the spring, if I'm
compressing the spring to

00:01:55.310 --> 00:01:58.310
the left, then the force I'm
applying is also to the left.

00:01:58.310 --> 00:02:01.030
So I'll call that the force
of compression.

00:02:01.030 --> 00:02:03.290
The force of compression
is going to be

00:02:03.290 --> 00:02:05.270
equal to K times x.

00:02:05.270 --> 00:02:07.440
And when the spring is
compressed and not

00:02:07.440 --> 00:02:09.680
accelerating in either
direction, the force of

00:02:09.680 --> 00:02:11.320
compression is going
to be equal to

00:02:11.320 --> 00:02:12.730
the restorative force.

00:02:12.730 --> 00:02:16.120
So what I want to do here is
plot the force of compression

00:02:16.120 --> 00:02:17.450
with respect to x.

00:02:17.450 --> 00:02:20.320
And I should have drawn it the
other way, but I think you

00:02:20.320 --> 00:02:22.810
understand that x is increasing
to the left in my

00:02:22.810 --> 00:02:23.760
example, right?

00:02:23.760 --> 00:02:30.270
This is where x is equal
to 0 right here.

00:02:30.270 --> 00:02:33.220
And say, this might be x is
equal to 10 because we've

00:02:33.220 --> 00:02:36.030
compressed it by 10 meters.

00:02:36.030 --> 00:02:38.530
So let's see how much
force we've applied.

00:02:38.530 --> 00:02:43.120
So when x is 0, which is right
here, how much force do we

00:02:43.120 --> 00:02:45.300
need to apply to compress
the spring?

00:02:45.300 --> 00:02:48.600
Well, if we give zero force, the
spring won't move, but if

00:02:48.600 --> 00:02:52.640
we just give a little, little
bit of force, if we just give

00:02:52.640 --> 00:02:55.370
infinitesimal, super-small
amount of force, we'll

00:02:55.370 --> 00:02:58.980
compress the spring just
a little bit, right?

00:02:58.980 --> 00:03:01.530
Because at that point, the force
of compression is going

00:03:01.530 --> 00:03:03.340
to be pretty much zero.

00:03:03.340 --> 00:03:06.830
So when the spring is barely
compressed, we're going to

00:03:06.830 --> 00:03:12.380
apply a little, little bit of
force, so almost at zero.

00:03:12.380 --> 00:03:15.330
To displace the spring zero,
we apply zero force.

00:03:15.330 --> 00:03:17.630
To displace the spring a little
bit, we have to apply a

00:03:17.630 --> 00:03:19.290
little bit more force.

00:03:19.290 --> 00:03:22.830
To displace soon. the spring 1
meter, so if this is say, 1

00:03:22.830 --> 00:03:28.870
meter, how much force
will we have to

00:03:28.870 --> 00:03:31.380
apply to keep it there?

00:03:31.380 --> 00:03:36.690
So let's say if this is
1 meter, the force of

00:03:36.690 --> 00:03:38.630
compression is going to
be K times 1, so it's

00:03:38.630 --> 00:03:39.880
just going to be K.

00:03:42.770 --> 00:03:46.790
And realize, you didn't apply
zero and then apply K force.

00:03:46.790 --> 00:03:49.650
You keep applying a little
bit more force.

00:03:49.650 --> 00:03:52.410
Every time you compress the
spring a little bit, it takes

00:03:52.410 --> 00:03:55.670
a little bit more force to
compress it a little bit more.

00:03:55.670 --> 00:03:58.950
So to compress it 1 meters,
you need to apply K.

00:03:58.950 --> 00:04:01.770
And to get it there, you have to
keep increasing the amount

00:04:01.770 --> 00:04:02.735
of force you apply.

00:04:02.735 --> 00:04:10.330
At 2 meters, you would've been
up to 2K, et cetera.

00:04:10.330 --> 00:04:11.900
I think you see a
line is forming.

00:04:11.900 --> 00:04:14.650
Let me draw that line.

00:04:14.650 --> 00:04:17.480
The line looks something
like that.

00:04:17.480 --> 00:04:20.920
And so this is how much force
you need to apply as a

00:04:20.920 --> 00:04:24.210
function of the displacement of
the spring from its natural

00:04:24.210 --> 00:04:26.190
rest state, right?

00:04:26.190 --> 00:04:28.530
And here I have positive x going
to the right, but in

00:04:28.530 --> 00:04:30.790
this case, positive
x is to the left.

00:04:30.790 --> 00:04:32.470
I'm just measuring its
actual displacement.

00:04:32.470 --> 00:04:36.460
I'm not worried too much about
direction right now.

00:04:36.460 --> 00:04:38.320
So I just want you to think
a little bit about what's

00:04:38.320 --> 00:04:39.350
happening here.

00:04:39.350 --> 00:04:42.330
You just have to slowly keep
on-- you could apply a very

00:04:42.330 --> 00:04:43.920
large force initially.

00:04:43.920 --> 00:04:46.820
If you apply a very large force
initially, the spring

00:04:46.820 --> 00:04:48.620
will actually accelerate much
faster, because you're

00:04:48.620 --> 00:04:52.150
applying a much larger force
than its restorative force,

00:04:52.150 --> 00:04:54.210
and so it might accelerate and
then it'll spring back, and

00:04:54.210 --> 00:04:55.910
actually, we'll do a little
example of that.

00:04:55.910 --> 00:04:59.260
But really, just to displace the
spring a certain distance,

00:04:59.260 --> 00:05:01.730
you have to just gradually
increase the force, just so

00:05:01.730 --> 00:05:04.530
that you offset the
restorative force.

00:05:04.530 --> 00:05:06.780
Hopefully, that makes sense,
and you understand that the

00:05:06.780 --> 00:05:09.280
force just increases
proportionally as a function

00:05:09.280 --> 00:05:11.250
of the distance, and
that's just because

00:05:11.250 --> 00:05:12.700
this is a linear equation.

00:05:12.700 --> 00:05:14.430
And what's the slope of this?

00:05:14.430 --> 00:05:17.250
Well, slope is rise
over run, right?

00:05:17.250 --> 00:05:23.350
So if I run 1, this is
1, what's my rise?

00:05:23.350 --> 00:05:24.340
It's K.

00:05:24.340 --> 00:05:28.590
So the slope of this
graph is K.

00:05:28.590 --> 00:05:31.400
So using this graph, let's
figure out how much work we

00:05:31.400 --> 00:05:35.400
need to do to compress
this spring.

00:05:35.400 --> 00:05:39.650
I don't know, let's
say this is x0.

00:05:39.650 --> 00:05:41.115
So x is where it's the
general variable.

00:05:41.115 --> 00:05:42.840
X0 is a particular
value for x.

00:05:42.840 --> 00:05:44.390
That could be 10 or whatever.

00:05:44.390 --> 00:05:46.360
Let's see how much
work we need.

00:05:46.360 --> 00:05:48.380
So what's the definition
of work?

00:05:48.380 --> 00:05:52.750
Work is equal to the force
in the direction of your

00:05:52.750 --> 00:05:57.770
displacement times the
displacement, right?

00:05:57.770 --> 00:05:59.550
So let's see how much
we've displaced.

00:05:59.550 --> 00:06:05.370
So when we go from zero
to here, we've

00:06:05.370 --> 00:06:06.810
displaced this much.

00:06:06.810 --> 00:06:08.950
And what was the force
of the displacement?

00:06:08.950 --> 00:06:11.890
Well, the force was gradually
increasing the entire time, so

00:06:11.890 --> 00:06:16.330
the force is going to be be
roughly about that big.

00:06:16.330 --> 00:06:17.920
I'm approximating.

00:06:17.920 --> 00:06:19.960
And I'll show you that you
actually have to approximate.

00:06:19.960 --> 00:06:22.480
So the force is kind of that
square right there.

00:06:25.980 --> 00:06:31.560
And then to displace the next
little distance-- that's not

00:06:31.560 --> 00:06:34.830
bright enough-- my force is
going to increase a little

00:06:34.830 --> 00:06:35.770
bit, right?

00:06:35.770 --> 00:06:38.120
So this is the force, this
is the distance.

00:06:38.120 --> 00:06:41.000
So if you you see, the work I'm
doing is actually going to

00:06:41.000 --> 00:06:42.880
be the area under the
curve, each of

00:06:42.880 --> 00:06:43.890
these rectangles, right?

00:06:43.890 --> 00:06:46.080
Because the height of the
rectangle is the force I'm

00:06:46.080 --> 00:06:50.340
applying and the width is
the distance, right?

00:06:50.340 --> 00:06:52.960
So the work is just going to
be the sum of all of these

00:06:52.960 --> 00:06:53.740
rectangles.

00:06:53.740 --> 00:06:56.450
And the rectangles I drew are
just kind of approximations,

00:06:56.450 --> 00:06:57.590
because they don't get
right under the line.

00:06:57.590 --> 00:06:59.590
You have to keep making the
rectangle smaller, smaller,

00:06:59.590 --> 00:07:02.660
smaller, and smaller, and just
sum up more and more and more

00:07:02.660 --> 00:07:04.360
rectangles, right?

00:07:04.360 --> 00:07:07.880
And actually I'm touching on
integral calculus right now.

00:07:07.880 --> 00:07:09.130
But if you don't know
integral calculus,

00:07:09.130 --> 00:07:09.950
don't worry about it.

00:07:09.950 --> 00:07:12.310
But the bottom line is the work
we're doing-- hopefully I

00:07:12.310 --> 00:07:16.290
showed you-- is just going to
be the area under this line.

00:07:16.290 --> 00:07:20.910
So the work I'm doing to
displace the spring x meters

00:07:20.910 --> 00:07:26.020
is the area from here to here.

00:07:26.020 --> 00:07:27.430
And what's that area?

00:07:27.430 --> 00:07:30.590
Well, this is a triangle, so we
just need to know the base,

00:07:30.590 --> 00:07:32.810
the height, and multiply
it times 1/2, right?

00:07:32.810 --> 00:07:34.530
That's just the area
of a triangle.

00:07:34.530 --> 00:07:36.050
So what's the base?

00:07:36.050 --> 00:07:39.860
So this is just x0.

00:07:39.860 --> 00:07:41.520
What's the height?

00:07:41.520 --> 00:07:44.360
Well, we know the slope is K, so
this height is going to be

00:07:44.360 --> 00:07:47.750
x0 times K.

00:07:47.750 --> 00:07:51.370
So this point right here
is the point x0, and

00:07:51.370 --> 00:07:54.610
then x0 times K.

00:07:54.610 --> 00:07:57.840
And so what's the area under the
curve, which is the total

00:07:57.840 --> 00:08:02.090
work I did to compress
the spring x0 meters?

00:08:02.090 --> 00:08:09.500
Well, it's the base, x0, times
the height, x0, times K.

00:08:09.500 --> 00:08:12.280
And then, of course, multiply by
1/2, because we're dealing

00:08:12.280 --> 00:08:14.420
with a triangle, right?

00:08:14.420 --> 00:08:19.230
So that equals 1/2K
x0 squared.

00:08:19.230 --> 00:08:22.050
And for those of you who know
calculus, that, of course, is

00:08:22.050 --> 00:08:25.910
the same thing as the
integral of Kx dx.

00:08:25.910 --> 00:08:27.080
And that should make sense.

00:08:27.080 --> 00:08:28.180
Each of these are little dx's.

00:08:28.180 --> 00:08:29.490
But I don't want to go too
much into calculus now.

00:08:29.490 --> 00:08:31.340
It'll confuse people.

00:08:31.340 --> 00:08:33.950
So that's the total work
necessary to compress the

00:08:33.950 --> 00:08:36.950
spring by distance of x0.

00:08:36.950 --> 00:08:38.659
Or if we set a distance
of x, you can just get

00:08:38.659 --> 00:08:40.340
rid of this 0 here.

00:08:40.340 --> 00:08:41.770
And why is that useful?

00:08:41.770 --> 00:08:44.700
Because the work necessary to
compress the spring that much

00:08:44.700 --> 00:08:47.800
is also how much potential
energy there is

00:08:47.800 --> 00:08:49.340
stored in the spring.

00:08:49.340 --> 00:08:57.750
So if I told you that I had a
spring and its spring constant

00:08:57.750 --> 00:09:03.890
is 10, and I compressed it 5
meters, so x is equal to 5

00:09:03.890 --> 00:09:08.800
meters, at the time that it's
compressed, how much potential

00:09:08.800 --> 00:09:10.780
energy is in that spring?

00:09:10.780 --> 00:09:15.940
We can just say the potential
energy is equal to 1/2K times

00:09:15.940 --> 00:09:18.790
x squared equals 1/2.

00:09:18.790 --> 00:09:25.050
K is 10 times 25, and
that equals 125.

00:09:25.050 --> 00:09:26.920
And, of course, work and
potential energy

00:09:26.920 --> 00:09:29.980
are measured in joules.

00:09:29.980 --> 00:09:33.110
So this is really what you
just have to memorize.

00:09:33.110 --> 00:09:34.240
Or hopefully you don't
memorize it.

00:09:34.240 --> 00:09:36.700
Hopefully, you understand where
I got it, and that's why

00:09:36.700 --> 00:09:38.030
I spent 10 minutes doing it.

00:09:38.030 --> 00:09:39.940
But this is how much work is
necessary to compress the

00:09:39.940 --> 00:09:43.030
spring to that point and how
much potential energy is

00:09:43.030 --> 00:09:45.790
stored once it is compressed
to that point, or actually

00:09:45.790 --> 00:09:47.490
stretched that much.

00:09:47.490 --> 00:09:48.520
We've been compressing,
but you can

00:09:48.520 --> 00:09:50.690
also stretch the spring.

00:09:50.690 --> 00:09:53.730
If you know that, then we can
start doing some problems with

00:09:53.730 --> 00:09:56.060
potential energy in springs,
which I will

00:09:56.060 --> 00:09:57.110
do in the next video.

00:09:57.110 --> 00:09:58.360
See


Intro to springs and Hooke's law

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.876
Let's learn a little
bit about springs.

00:00:02.876 --> 00:00:05.100
So let's say I have a spring.

00:00:05.100 --> 00:00:08.870
Let me draw the ground so that
we know what's going on with

00:00:08.870 --> 00:00:09.490
the spring.

00:00:09.490 --> 00:00:13.280
So let me see, this
is the floor.

00:00:13.280 --> 00:00:16.830
That's the floor, and
I have a spring.

00:00:16.830 --> 00:00:18.680
It's along the floor.

00:00:18.680 --> 00:00:21.000
I'll use a thicker one, just
to show it's a spring.

00:00:21.000 --> 00:00:24.060
Let's say the spring looks
something like this.

00:00:24.060 --> 00:00:26.170
Whoops, I'm still using
the line tool.

00:00:26.170 --> 00:00:27.430
So the spring looks like this.

00:00:27.430 --> 00:00:32.299
This is my spring, my amazingly
drawn spring.

00:00:32.299 --> 00:00:36.170
Let's say at this end it's
attached to a wall.

00:00:36.170 --> 00:00:38.980
That's a wall.

00:00:38.980 --> 00:00:41.470
And so this is a spring when I
don't have any force acting on

00:00:41.470 --> 00:00:44.880
it, this is just the natural
state of the spring.

00:00:44.880 --> 00:00:50.050
And we could call this, where it
just naturally rests, this

00:00:50.050 --> 00:00:51.340
tip of the spring.

00:00:51.340 --> 00:00:57.640
And let's say that when I were
to apply a force of 5 Newtons

00:00:57.640 --> 00:01:00.410
into the spring, it looks
something like this.

00:01:00.410 --> 00:01:03.210
Redraw everything.

00:01:03.210 --> 00:01:11.355
So when I apply a force of 5
Newtons-- I'll draw the wall

00:01:11.355 --> 00:01:12.700
in magenta now.

00:01:16.180 --> 00:01:19.300
When I apply a force
of 5 Newtons, the

00:01:19.300 --> 00:01:20.550
spring looks like this.

00:01:25.510 --> 00:01:26.880
It compresses, right?

00:01:26.880 --> 00:01:28.060
We're all familiar with this.

00:01:28.060 --> 00:01:30.320
We sit on a bed every
day or a sofa.

00:01:30.320 --> 00:01:31.620
So let's say it compresses
to here.

00:01:37.220 --> 00:01:40.960
If this was the normal resting--
so this is where the

00:01:40.960 --> 00:01:44.000
spring was when I applied no
force, but when I applied 5

00:01:44.000 --> 00:01:53.420
Newtons in that direction, let's
say that this distance

00:01:53.420 --> 00:02:01.290
right here is 10 meters.

00:02:01.290 --> 00:02:03.940
And so a typical question that
you'll see, and we'll explain

00:02:03.940 --> 00:02:08.759
how to do it, is a spring
compresses or elongates when

00:02:08.759 --> 00:02:11.950
you apply a certain force
by some distance.

00:02:11.950 --> 00:02:14.010
How much will it compress when
you apply a different force?

00:02:14.010 --> 00:02:16.615
So my question is how much will
it compress when I apply

00:02:16.615 --> 00:02:19.260
a 10-Newton force?

00:02:19.260 --> 00:02:22.480
So your intuition that it'll
compress more is correct, but

00:02:22.480 --> 00:02:27.220
is it linear to how much
I compress it?

00:02:27.220 --> 00:02:30.150
Is it a square of how
much I compress it?

00:02:30.150 --> 00:02:32.175
How does it relate?

00:02:32.175 --> 00:02:34.480
I think you probably
could guess.

00:02:34.480 --> 00:02:36.880
It's actually worth
an experiment.

00:02:36.880 --> 00:02:39.280
Or you could just keep
watching the video.

00:02:39.280 --> 00:02:42.020
So let's say I apply
a 10-Newton force.

00:02:42.020 --> 00:02:43.180
What will the spring
look like?

00:02:43.180 --> 00:02:44.740
Well, it'll be more
compressed.

00:02:49.900 --> 00:02:55.780
Drop my force to 10 Newtons.

00:02:55.780 --> 00:02:58.060
And if this was the natural
place where the spring would

00:02:58.060 --> 00:03:00.580
rest, what is this distance?

00:03:00.580 --> 00:03:02.280
Well, it turns out that
it is linear.

00:03:02.280 --> 00:03:03.590
What do I mean by linear?

00:03:03.590 --> 00:03:07.860
Well, it means that the more
the force-- it's equally

00:03:07.860 --> 00:03:10.750
proportional to how much the
spring will compress.

00:03:10.750 --> 00:03:12.060
And it actually works
the other way.

00:03:12.060 --> 00:03:15.040
If you applied 5 Newtons in this
direction, to the right,

00:03:15.040 --> 00:03:18.670
you would have gone 10 meters
in this direction.

00:03:18.670 --> 00:03:20.940
So it goes whether you're
elongating the spring or

00:03:20.940 --> 00:03:24.450
compressing the spring within
some reasonable tolerance.

00:03:24.450 --> 00:03:26.620
We've all had this experience.

00:03:26.620 --> 00:03:29.300
If you compress something too
much or you stretch it too

00:03:29.300 --> 00:03:32.070
much, it doesn't really go back
to where it was before.

00:03:32.070 --> 00:03:34.430
But within some reasonable
tolerance, it's proportional.

00:03:34.430 --> 00:03:35.830
So what does that mean?

00:03:35.830 --> 00:03:43.990
That means that the restoring
force of the spring is minus

00:03:43.990 --> 00:03:47.840
some number, times the
displacement of the spring.

00:03:47.840 --> 00:03:49.040
So what does this mean?

00:03:49.040 --> 00:03:52.110
So in this example right here,
what was the displacement of

00:03:52.110 --> 00:03:53.140
the spring?

00:03:53.140 --> 00:03:56.528
Well, if we take positive x to
the right and negative x to

00:03:56.528 --> 00:04:00.980
the left, the displacement
of the spring was what?

00:04:00.980 --> 00:04:05.010
The displacement, in this
example right here, x is equal

00:04:05.010 --> 00:04:06.090
to minus 10, right?

00:04:06.090 --> 00:04:08.180
Because I went 10 to the left.

00:04:08.180 --> 00:04:13.520
And so it says that the
restorative force is going to

00:04:13.520 --> 00:04:21.310
be equal to minus K times
how much it's

00:04:21.310 --> 00:04:23.260
distorted times minus 10.

00:04:23.260 --> 00:04:27.470
So the minuses cancel out,
so it equals 10K.

00:04:27.470 --> 00:04:30.420
What's the restorative force
in this example?

00:04:30.420 --> 00:04:32.210
Well, you might say, it's 5
Newtons, just because that's

00:04:32.210 --> 00:04:37.550
the only force I've drawn here,
and you would be to some

00:04:37.550 --> 00:04:38.160
degree correct.

00:04:38.160 --> 00:04:40.260
And actually, since we're doing
positive and negative,

00:04:40.260 --> 00:04:43.570
and this 5 Newton is to the
left, so to the negative

00:04:43.570 --> 00:04:45.770
x-direction, actually, I should
call this minus 5

00:04:45.770 --> 00:04:48.090
Newtons and I should call this
minus 10 Newtons, because

00:04:48.090 --> 00:04:50.340
obviously, these are vectors and
we're going to the left.

00:04:50.340 --> 00:04:53.630
I picked the convention that
to the left means negative.

00:04:53.630 --> 00:04:54.980
So what's the restorative
force?

00:04:54.980 --> 00:04:57.820
Well, in this example-- and we
assume that K is a positive

00:04:57.820 --> 00:05:00.500
number for our purposes.

00:05:00.500 --> 00:05:02.370
In this example, the restorative
force is a

00:05:02.370 --> 00:05:03.660
positive number.

00:05:03.660 --> 00:05:05.310
So what is the restorative
force?

00:05:05.310 --> 00:05:08.970
So that's actually the force,
the counteracting force, of

00:05:08.970 --> 00:05:10.030
the spring.

00:05:10.030 --> 00:05:12.680
That's what this formula
gives us.

00:05:12.680 --> 00:05:15.220
So if this spring is stationary
when I apply this

00:05:15.220 --> 00:05:18.030
5-Newton force, that means that
there must be another

00:05:18.030 --> 00:05:19.830
equal and opposite force that's

00:05:19.830 --> 00:05:22.160
positive 5 Newtons, right?

00:05:22.160 --> 00:05:24.460
If there weren't, the spring
would keep compressing.

00:05:24.460 --> 00:05:27.240
And if the force was more than 5
Newtons, the spring would go

00:05:27.240 --> 00:05:28.620
back this way.

00:05:28.620 --> 00:05:32.010
So the fact that I know that
when I apply a 5-Newton force

00:05:32.010 --> 00:05:34.920
to the left, or a negative
5-Newton force, the spring is

00:05:34.920 --> 00:05:37.130
no longer moving, it means that
there must be-- or no

00:05:37.130 --> 00:05:39.830
longer accelerating, actually,
it means that there must be an

00:05:39.830 --> 00:05:42.330
equal and opposite force to
the right, and that's the

00:05:42.330 --> 00:05:43.650
restorative force.

00:05:43.650 --> 00:05:47.370
Another way to think about it is
if I were to let-- well, I

00:05:47.370 --> 00:05:48.510
won't go in there now.

00:05:48.510 --> 00:05:50.990
So in this case, the restorative
force is 5

00:05:50.990 --> 00:05:52.950
Newtons, so we can
solve for K.

00:05:52.950 --> 00:05:56.940
We could say 5 is
equal to 10K.

00:05:56.940 --> 00:05:58.110
Divide both sides by 10.

00:05:58.110 --> 00:05:59.380
You get K is equal to 1/2.

00:06:03.850 --> 00:06:06.610
So now we can use that
information to figure out what

00:06:06.610 --> 00:06:10.750
is the displacement
when I apply a

00:06:10.750 --> 00:06:12.050
negative 10-Newton force?

00:06:12.050 --> 00:06:15.410
When I push the spring
in with 10 Newtons in

00:06:15.410 --> 00:06:16.790
the leftward direction?

00:06:16.790 --> 00:06:19.050
So first of all, what's the
restorative force here?

00:06:19.050 --> 00:06:22.790
Well, if the spring is no longer
accelerating in either

00:06:22.790 --> 00:06:25.710
direction, or the tip of
the spring is no longer

00:06:25.710 --> 00:06:28.120
accelerating in either
direction, we know that the

00:06:28.120 --> 00:06:30.540
restorative force must be
counterbalancing this force

00:06:30.540 --> 00:06:32.290
that I'm compressing
with, right?

00:06:32.290 --> 00:06:35.810
The force that the spring wants
to expand back with is

00:06:35.810 --> 00:06:39.030
10 Newtons, positive
10 Newtons, right?

00:06:39.030 --> 00:06:42.550
And we know the spring constant,
this K for this

00:06:42.550 --> 00:06:46.780
spring, for this material,
whatever it might be, is 1/2.

00:06:46.780 --> 00:06:53.320
So we know the restorative force
is equal to 1/2 times

00:06:53.320 --> 00:06:55.280
the distance, right?

00:06:55.280 --> 00:06:58.940
And the formula is
minus K, right?

00:06:58.940 --> 00:07:00.670
And then, what is
the restorative

00:07:00.670 --> 00:07:02.340
force in this example?

00:07:02.340 --> 00:07:06.190
Well I said it's 10 Newtons, so
we know that 10 Newtons is

00:07:06.190 --> 00:07:07.440
equal to minus 1/2x.

00:07:09.950 --> 00:07:11.750
And so what is x?

00:07:11.750 --> 00:07:14.030
Well, multiply both sides
by minus 1/2, and

00:07:14.030 --> 00:07:15.920
you get minus 20.

00:07:15.920 --> 00:07:18.455
I'm sorry, multiply both sides
by minus 2, you get minus 20

00:07:18.455 --> 00:07:19.705
is equal to x.

00:07:22.900 --> 00:07:25.700
So x goes to the
left 20 units.

00:07:25.700 --> 00:07:28.390
So that's all that
it's telling us.

00:07:28.390 --> 00:07:32.710
And this law is called Hooke's
Law, and it's named after--

00:07:32.710 --> 00:07:35.810
I'll read it-- a physicist in
the 17th century, a British

00:07:35.810 --> 00:07:39.170
physicist. And he figured out
that the amount of force

00:07:39.170 --> 00:07:44.460
necessary to keep a spring
compressed is proportional to

00:07:44.460 --> 00:07:46.420
how much you've compressed it.

00:07:46.420 --> 00:07:48.570
And that's all that
this formula says.

00:07:48.570 --> 00:07:51.150
And that negative number,
remember, this formula gives

00:07:51.150 --> 00:07:52.710
us the restorative force.

00:07:52.710 --> 00:07:55.910
So it says that the force is
always in the opposite

00:07:55.910 --> 00:07:57.860
direction of how much
you displace it.

00:07:57.860 --> 00:08:01.470
So, for example, if you were
to displace this spring in

00:08:01.470 --> 00:08:04.590
this direction, if you were to
apply a force and x were a

00:08:04.590 --> 00:08:08.620
positive and you were to go in
that direction, the force-- no

00:08:08.620 --> 00:08:09.150
wait, sorry.

00:08:09.150 --> 00:08:11.320
This is where the
spring rests.

00:08:11.320 --> 00:08:14.260
If you were to apply some force
and take the spring out

00:08:14.260 --> 00:08:18.610
to here, this negative number
tells us that the spring will

00:08:18.610 --> 00:08:21.610
essentially try to pull back
with the restorative force in

00:08:21.610 --> 00:08:24.190
the other direction.

00:08:24.190 --> 00:08:26.750
Let's do one more problem
and I think this

00:08:26.750 --> 00:08:29.570
will be clear to you.

00:08:29.570 --> 00:08:33.460
So let's say I have a spring,
and all of these problems kind

00:08:33.460 --> 00:08:35.090
of go along.

00:08:35.090 --> 00:08:40.280
So let's say when I apply a
force of 2 Newtons, so this is

00:08:40.280 --> 00:08:44.880
what I apply when I apply
a force of 2 Newtons.

00:08:44.880 --> 00:08:46.180
Well, let's say it this way.

00:08:46.180 --> 00:08:49.740
Let's say when I stretch
the spring.

00:08:49.740 --> 00:08:55.650
Let's say this is the spring,
and when I apply a force of 2

00:08:55.650 --> 00:09:05.940
Newtons to the right, the spring
gets stretched 1 meter.

00:09:05.940 --> 00:09:09.280
So first of all, let's
figure out what K is.

00:09:09.280 --> 00:09:14.030
So if the spring is stretched
by 1 meter, out here, its

00:09:14.030 --> 00:09:18.650
restorative force will be 2
Newtons back this way, right?

00:09:18.650 --> 00:09:22.620
So its restorative force, this
2 Newtons, will equal minus K

00:09:22.620 --> 00:09:24.530
times how much I displaced it.

00:09:24.530 --> 00:09:28.050
Well I, displaced it by 1 meter,
so then we multiply

00:09:28.050 --> 00:09:32.380
both sides by negative 1, and we
get K is equal to minus 2.

00:09:32.380 --> 00:09:37.350
So then we can use Hooke's Law
to note the equation for

00:09:37.350 --> 00:09:40.450
this-- to figure out the
restorative force for this

00:09:40.450 --> 00:09:44.110
particular spring, and
it would be minus 2x.

00:09:44.110 --> 00:09:46.530
And then I said, well, how much
force would I have to

00:09:46.530 --> 00:09:49.730
apply to distort the
spring by 2 meters?

00:09:49.730 --> 00:09:52.420
Well, it's 2 times
2, it would be 4.

00:09:52.420 --> 00:09:57.330
4 Newtons to displace it by 2
meters, and, of course, the

00:09:57.330 --> 00:09:59.580
restorative force will then be
in the opposite direction, and

00:09:59.580 --> 00:10:01.620
that's where we get the
negative number.

00:10:01.620 --> 00:10:02.800
Anyway, I've run out of time.

00:10:02.800 --> 00:10:04.860
I'll see you in the
next video.