WEBVTT

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 --> 00:00:16.329
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.