WEBVTT
00:00:02.919 --> 00:00:09.139
On the evening of August 5, 2012 Pacific Daylight
Time, NASA's Mars Rover, named Curiosity,
00:00:09.139 --> 00:00:19.439
entered Mars' atmosphere at 20,000km/h. Drag
slowed it down to around 1600km/hr, at which
00:00:19.449 --> 00:00:26.449
point a parachute opened. This parachute slowed
the Rover more, to about 320km/hr, or 90 m/s.
00:00:28.769 --> 00:00:34.699
Finally, after rockets decelerated it completely,
the rover was lowered to the surface of Mars.
00:00:34.699 --> 00:00:40.780
Every step of this dance was carefully choreographed
and rehearsed in many experiments here on
00:00:40.780 --> 00:00:47.129
Earth. But how could NASA engineers be sure
that their designs would work on a totally
00:00:47.129 --> 00:00:54.129
different planet? The answer is a problem-solving
method called dimensional analysis.
00:00:54.780 --> 00:00:59.299
This video is part of the Problem Solving
video series.
00:00:59.299 --> 00:01:03.319
Problem-solving skills, in combination with
an understanding of the natural and human-made
00:01:03.319 --> 00:01:07.009
world, are critical to the design and optimization
of systems and processes.
00:01:07.009 --> 00:01:13.899
Hi, my name is Ken Kamrin, and I am a professor
of mechanical engineering at MIT. Dimensional
00:01:13.899 --> 00:01:20.600
analysis is a powerful tool; I use it, NASA
uses it, and you will too.
00:01:20.600 --> 00:01:24.960
Before watching this video, you should be
familiar with unit analysis, and understand
00:01:24.960 --> 00:01:28.280
the difference between dependent and independent
variables.
00:01:28.280 --> 00:01:32.109
By the end of this video, you will be able
to use dimensional analysis to estimate the
00:01:32.109 --> 00:01:39.109
size of a parachute canopy that can slow the
Rover down to 90 m/s on its descent to Mars.
00:01:44.060 --> 00:01:49.630
Before we talk about dimensional analysis,
we need to know what dimension is. Dimensions
00:01:49.630 --> 00:01:56.100
and units are related, but different, concepts.
Physical quantities are measured in units.
00:01:56.100 --> 00:02:01.090
The dimension of the physical quantity is
independent of the particular units choosen.
00:02:01.090 --> 00:02:05.569
For example:
Both grams and kilograms are units, but they
00:02:05.569 --> 00:02:10.810
are units of mass.And mass is what we'll call
the dimension.
00:02:10.810 --> 00:02:19.319
There are 5 fundamental dimensions that we
commonly deal with: length, mass, time, temperature,
00:02:19.319 --> 00:02:21.260
and charge.
00:02:21.260 --> 00:02:27.510
All other dimensions are obtained by taking
products and powers of these fundamental dimensions.
00:02:27.510 --> 00:02:31.620
In this video, we'll be dealing
with length, which we denote by the letter
00:02:31.620 --> 00:02:41.140
L, mass, which we denote M, and time, T. For
example, no matter how you measure the physical
00:02:41.150 --> 00:02:47.450
quantity velocity, it has the dimension, which
we denote with square brackets, of length
00:02:47.450 --> 00:02:52.950
divided by time, or length times time to the
negative 1 power.
00:02:52.950 --> 00:02:57.530
Pause the video here and determine the dimension
of energy.
00:03:03.260 --> 00:03:06.120
Energy has dimension Mass times
00:03:06.129 --> 00:03:10.060
Length squared over Time squared.
00:03:10.060 --> 00:03:17.480
Okay, great, so what's the big deal? How is
this useful?
00:03:17.480 --> 00:03:22.720
Remember NASA's rover? Part of the landing
sequence calls for a parachute to slow the
00:03:22.720 --> 00:03:28.620
vehicle down. Suppose it is our job to design
the parachute to slow the Rover to exactly
00:03:28.620 --> 00:03:34.680
90m/s. The terminal velocity of the Rover
depends on the mass of the rover itself and
00:03:34.680 --> 00:03:40.480
its heat shield, and several different variables
related to the parachute design: the material
00:03:40.480 --> 00:03:45.939
of the canopy, the diameter of the hemispherical
parachute canopy, the number of suspension
00:03:45.939 --> 00:03:48.540
lines, etc.
00:03:48.540 --> 00:03:52.920
For simplicity, let's suppose that all parachute
parameters other than the diameter of the
00:03:52.920 --> 00:03:59.319
canopy have already been determined. Our goal
is to find the canopy diameter that is as
00:03:59.319 --> 00:04:06.319
small as possible, but will correspond to
the desired terminal velocity of 90 m/s.
00:04:06.599 --> 00:04:13.439
Clearly, we can't test our designs on Mars.
The question is: how can we get meaningful
00:04:13.439 --> 00:04:20.899
data here on Earth that will allow us to appropriately
size the parachute for use on Mars? How do
00:04:20.910 --> 00:04:25.790
we predict the behavior of a parachute on
Mars based on an Earth experiment? And what
00:04:25.790 --> 00:04:31.300
variables do we need to consider in designing
our experiment on Earth? This is where a problem
00:04:31.300 --> 00:04:36.880
solving method called dimensional analysis
can help us.
00:04:40.720 --> 00:04:45.780
Before we get started, we must first determine
what the dependent and independent variables
00:04:45.780 --> 00:04:48.380
are in our system.
00:04:48.380 --> 00:04:52.440
The dependent variable is terminal velocity.
00:04:52.440 --> 00:04:58.350
This is the quantity that we wish to constrain
by our parachute design. So what variables
00:04:58.350 --> 00:05:02.700
affect the terminal velocity of the parachute
and rover system?
00:05:02.700 --> 00:05:08.080
The diameter of the parachute canopy is one
such independent variable. Take a moment to
00:05:08.080 --> 00:05:10.680
pause the video and identify others.
00:05:16.500 --> 00:05:24.220
Ok. Here's our list: canopy diameter, mass
of the Rover (we assume the mass of the parachute
00:05:24.220 --> 00:05:32.520
to be negligible), acceleration due to gravity,
and the density and viscosity of the atmosphere.
00:05:33.370 --> 00:05:38.440
For this problem, we can assume that the dependence
of the terminal velocity on atmospheric viscosity
00:05:38.440 --> 00:05:45.440
is negligible, because the atmosphere on Mars
like the atmosphere on Earth is not very viscous.
00:05:46.120 --> 00:05:51.060
If we wanted to derive a functional relationship
that would work, for example, underwater,
00:05:51.060 --> 00:05:56.020
it would be important that we include viscosity
as an independent variable.
00:05:56.020 --> 00:06:00.090
Why didn't we include the surface area of
the parachute canopy in our list of independent
00:06:00.090 --> 00:06:01.850
variables?
00:06:01.850 --> 00:06:04.850
Pause the video and take a moment to discuss
with a classmate.
00:06:11.260 --> 00:06:15.520
We didn't include the surface area of the
canopy, because it is not independent from
00:06:15.520 --> 00:06:16.920
the diameter of the canopy.
00:06:16.920 --> 00:06:23.150
In fact, we can determine the area as a function
of diameter. So we don't need both!
00:06:23.150 --> 00:06:30.150
Question: Could we use the surface area instead
of the diameter? Absolutely. We need the variables
00:06:30.770 --> 00:06:37.090
to be independent, but it doesn't matter which
variables we use! The key is to have identified
00:06:37.090 --> 00:06:43.330
all of the correct variables to begin with.
This is where human error can come into play.
00:06:43.330 --> 00:06:47.430
If our list of variables isn't exhaustive,
the relationship we develop through dimensional
00:06:47.430 --> 00:06:51.750
analysis may not be correct!
00:06:51.750 --> 00:06:56.159
Once we have the full list of independent
variables, we can express the terminal velocity
00:06:56.159 --> 00:07:00.780
as some function of these independent variables.
00:07:00.780 --> 00:07:05.880
In order to find the function that describes
the relationship, we need do several experiments
00:07:05.880 --> 00:07:12.620
involving 4 independent variables, and fit
the data. Phew, that's a lot of work! Especially
00:07:12.620 --> 00:07:16.970
because we don't know what the function might
look like.
00:07:16.970 --> 00:07:23.640
But whenever you have an equation, all terms
in the equation must have the same dimension.
00:07:23.640 --> 00:07:28.590
Multiplying two terms multiplies the dimensions.
00:07:28.590 --> 00:07:33.240
This restricts the possible form that a function
describing the terminal velocity in terms
00:07:33.240 --> 00:07:38.860
of our 4 other variables can take, because
the function must combine the variables in
00:07:38.860 --> 00:07:44.550
some way that has the same dimension as velocity.
00:07:44.550 --> 00:07:51.270
And many functionsâ€”exponential, logarithmic,
trigonometric --cannot have input variables
00:07:51.270 --> 00:07:58.270
that have dimension. What would e to the 1kg
mean? What units could it possibly have?
00:07:59.370 --> 00:08:03.780
We're going to show you a problem solving
method that will allow you to find the most
00:08:03.780 --> 00:08:10.780
general form of such a function. This method
is called dimensional analysis.
00:08:14.210 --> 00:08:20.360
We begin this process by creating dimensionless
versions of the variables in our system. We
00:08:20.360 --> 00:08:24.390
create these dimensionless expressions out
of the variables in our system, so we don't
00:08:24.390 --> 00:08:28.510
introduce any new physical parameters.
00:08:28.510 --> 00:08:32.719
The first step is to take our list of variables,
and distill them down to their fundamental
00:08:32.719 --> 00:08:34.360
dimensions.
00:08:34.360 --> 00:08:39.000
Remember our fundamental dimensions are length,
mass, and time.
00:08:39.000 --> 00:08:46.000
Distilling gravity to its fundamental dimension,
we get length per time squared. Now you distill
00:08:46.500 --> 00:08:53.500
the remaining variables of velocity, diameter,
mass, and density into their fundamental dimensions.
00:08:54.170 --> 00:08:56.470
Pause the video here.
00:09:02.040 --> 00:09:08.880
Velocity is length per time; diameter is length;
mass is mass; and density is mass per length
00:09:08.889 --> 00:09:11.290
cubed.
00:09:11.290 --> 00:09:15.910
The second step is to express the fundamental
dimensions of mass, length, and time in terms
00:09:15.910 --> 00:09:18.040
of our independent variables.
00:09:18.040 --> 00:09:20.920
We can write M as little m.
00:09:20.920 --> 00:09:26.819
We can write L as mass divided by density
to the 1/3 power.
00:09:26.819 --> 00:09:32.439
We can write T as velocity divided by gravity.
00:09:32.439 --> 00:09:37.980
We had many choices as to how to write these
fundamental dimensions in terms of our variables.
00:09:37.980 --> 00:09:42.769
In the end, it doesn't matter which expressions
you choose.
00:09:42.769 --> 00:09:47.980
The third step is to use these fundamental
dimensions to turn all of the variables involved
00:09:47.980 --> 00:09:54.980
into dimensionless quantities. For example,
the terminal velocity v has dimension of length
00:09:55.339 --> 00:10:02.339
over time. So we multiply v by the dimension
of time, and divide by the dimension of length
00:10:09.639 --> 00:10:15.689
to get a dimension of one.
00:10:15.689 --> 00:10:22.689
We define a new dimensionless variable vbar
as this dimensionless version of v.
00:10:26.040 --> 00:10:36.120
Now you try; find dbar, mbar, gbar, and rhobar. Pause
the video here.
00:10:42.480 --> 00:10:48.840
You should have found that dbar is d times rho over m to the 1/3.
00:10:48.840 --> 00:10:56.339
Mbar is 1. Gbar is v squared over g times rho over
00:10:56.339 --> 00:11:01.240
m to the 1/3., and rho bar is 1.
00:11:01.240 --> 00:11:07.110
Now we can rewrite the equation for velocity
in terms of the new dimensionless variables.
00:11:07.110 --> 00:11:12.709
It is a new function, because the variables
have been modified. Notice that vbar is equal
00:11:12.709 --> 00:11:21.339
to gbar. This means that vbar and gbar are
not independent! So our function for vbar
00:11:21.339 --> 00:11:28.999
cannot depend on gbar. Also, notice that mbar
and rho bar are both equal to one, so our
00:11:29.009 --> 00:11:32.939
function doesn't depend on them either.
00:11:32.939 --> 00:11:39.939
This has simplified our relationship: vbar
is a function of only one variable, dbar.
00:11:40.790 --> 00:11:46.819
And remember that dbar is dimensionless, so
it is just a real number. This means that,
00:11:46.819 --> 00:11:50.949
phi can be any function.
00:11:50.949 --> 00:11:56.709
The forth and final step is to rearrange to
find a formula for the terminal velocity.
00:11:56.709 --> 00:12:03.709
The key here is that this equation for the
terminal velocity has the correct units.
00:12:04.029 --> 00:12:09.879
And this formula is so general, that any expression
with dimension of Length over Time can be
00:12:09.879 --> 00:12:18.079
written in terms of this formula by defining
phi in different ways. Let's see how. First,
00:12:18.079 --> 00:12:23.559
create an expression from the independent
variables that has the same dimension as velocity.
00:12:23.559 --> 00:12:30.550
One such expression is the square root of
g times d. By setting the formula for v equal
00:12:30.550 --> 00:12:38.579
to the square root of gd, we see that by setting
phi equal to the identity function phi(x)
00:12:38.580 --> 00:12:44.040
= x, the two sides of the equation can be
made equal.
00:12:44.050 --> 00:12:49.220
And in fact, we claim that any expression
with the correct dimension of Length over
00:12:49.220 --> 00:12:54.800
time created using these variables can be
written in terms of this formula by simply
00:12:54.800 --> 00:12:57.949
changing the definition of phi!
00:12:57.949 --> 00:13:03.339
It can be fun to try this. Come up with different
formulas that have the correct dimension.
00:13:03.339 --> 00:13:08.069
You can even add them together. Then see if
you can find a way to define phi so that our
00:13:08.069 --> 00:13:13.209
formula is equal to the expression you wrote.
Pause the video here.
00:13:18.699 --> 00:13:23.850
Now you may be concerned because this formula
is not unique. We made some choices about
00:13:23.850 --> 00:13:30.850
how to represent our fundamental dimensions.
What happens if we make different choices?
00:13:31.490 --> 00:13:44.069
Here we chose M, L, and T this way: M was
little m, L was d, and T was v over g. Running through
00:13:44.069 --> 00:13:48.779
the dimensional analysis process with this
choice of fundamental dimensions, we obtain
00:13:48.779 --> 00:13:51.980
an equation for v that looks like this.
00:13:51.980 --> 00:13:57.499
To see that these two formulas are equivalent,
we set the arguments under the square root
00:13:57.500 --> 00:14:03.040
equal, and find that we can express phi as
a function of psi.
00:14:15.840 --> 00:14:18.040
So any formula with the
00:14:18.040 --> 00:14:24.170
correct dimension can be expressed by this
general formula.
00:14:24.170 --> 00:14:30.559
And this general formula works for any rover
on any planet whose terminal velocity through
00:14:30.559 --> 00:14:35.899
the atmosphere depends on the same variables.
Because it is a general law!
00:14:41.410 --> 00:14:47.660
Of course, we still don't know what this function
phi is! In order to find phi, we can fit experimental
00:14:47.660 --> 00:14:55.600
data from any planet, for example, Earth.
On Earth, we know the gravity and atmospheric
00:14:55.610 --> 00:15:02.129
density. We can specify the mass of a test
rover to be 10kg.
00:15:02.129 --> 00:15:06.559
Then we might set up Earth bound experiments
by varying the canopy diameter of a parachute
00:15:06.559 --> 00:15:13.719
between 1m to 20m and measuring the terminal
velocity. For example, suppose we obtained
00:15:13.720 --> 00:15:18.300
the following data on Earth.
00:15:18.300 --> 00:15:24.220
Then we could convert this data to the dbar,
vbar axes, by scaling the variables v and
00:15:24.220 --> 00:15:32.680
d according to the Earth values for the mass,
gravity, and atmospheric density. We can fit
00:15:32.689 --> 00:15:37.899
this data to some best-fit curve. And this
best fit curve is our best approximation to
00:15:37.899 --> 00:15:40.999
the function phi.
00:15:40.999 --> 00:15:47.170
Now that we have phi, we can transform the
axes again to represent the canopy diameter
00:15:47.170 --> 00:15:55.860
and terminal velocity on Mars. This is done
by converting vbar and dbar to v and d by
00:15:55.860 --> 00:16:01.759
scaling according to known values of the gravity,
atmospheric density, and mass of the rover
00:16:01.759 --> 00:16:04.579
on Mars!
00:16:04.579 --> 00:16:10.209
To find the diameter, we find the point on
the v-axis that corresponds to a terminal
00:16:10.209 --> 00:16:17.939
velocity of 90m/s, and use our curve to determine
the diameter that corresponds to this terminal
00:16:17.939 --> 00:16:24.249
velocity! Now we have the specification we
need to design the size of our parachute to
00:16:24.249 --> 00:16:27.569
be used on the descent to Mars!
00:16:31.490 --> 00:16:36.040
In this example, we used dimensional analysis
to restrict the possible form of a function
00:16:36.040 --> 00:16:42.550
describing the terminal velocity of the Mars
rover as a function of parachute canopy diameter,
00:16:42.550 --> 00:16:48.220
gravitational acceleration, atmospheric density,
and the mass of the rover.
00:16:48.220 --> 00:16:54.600
This allowed us to design a parachute for
use on Mars based on Earth bound experiments.
00:16:55.620 --> 00:17:00.360
In general, the process of dimensional analysis
involves... 1.
00:17:00.360 --> 00:17:05.079
Identifying the dependent variable and independent
variables,
00:17:05.079 --> 00:17:11.880
2. Expressing the relevant fundamental dimensions
in terms of the variables found in step (1),
00:17:11.880 --> 00:17:17.400
3. Generating dimensionless expressions for all
of the variables using expressions from step
00:17:17.400 --> 00:17:21.329
(2).
4. Producing a functional relationship between
00:17:21.329 --> 00:17:26.069
the dimensionless dependent variable in terms
of the remaining independent dimensionless
00:17:26.069 --> 00:17:31.970
expressions.
5. Rearranging to determine a formula for the
00:17:31.970 --> 00:17:37.880
variable of interest. And
6. Performing experiments to determine the form
00:17:37.880 --> 00:17:42.880
of the general real valued function that appears
in the formula.
00:17:42.880 --> 00:17:48.380
We've just shown you a powerful tool which can
save you a lot of time. So the next time you
00:17:48.390 --> 00:17:54.190
encounter a difficult challenge, you might just
want to try... Dimensional Analysis.