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On the evening of August 5, 2012 Pacific Daylight
Time, NASA's Mars Rover, named Curiosity,
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entered Mars' atmosphere at 20,000km/h. Drag
slowed it down to around 1600km/hr, at which
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point a parachute opened. This parachute slowed
the Rover more, to about 320km/hr, or 90 m/s.
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Finally, after rockets decelerated it completely,
the rover was lowered to the surface of Mars.
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Every step of this dance was carefully choreographed
and rehearsed in many experiments here on
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Earth. But how could NASA engineers be sure
that their designs would work on a totally
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different planet? The answer is a problem-solving
method called dimensional analysis.
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This video is part of the Problem Solving
video series.
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Problem-solving skills, in combination with
an understanding of the natural and human-made
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world, are critical to the design and optimization
of systems and processes.
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Hi, my name is Ken Kamrin, and I am a professor
of mechanical engineering at MIT. Dimensional
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analysis is a powerful tool; I use it, NASA
uses it, and you will too.
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Before watching this video, you should be
familiar with unit analysis, and understand
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the difference between dependent and independent
variables.
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By the end of this video, you will be able
to use dimensional analysis to estimate the
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size of a parachute canopy that can slow the
Rover down to 90 m/s on its descent to Mars.
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Before we talk about dimensional analysis,
we need to know what dimension is. Dimensions
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and units are related, but different, concepts.
Physical quantities are measured in units.
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The dimension of the physical quantity is
independent of the particular units choosen.
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For example:
Both grams and kilograms are units, but they
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are units of mass.And mass is what we'll call
the dimension.
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There are 5 fundamental dimensions that we
commonly deal with: length, mass, time, temperature,
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and charge.
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All other dimensions are obtained by taking
products and powers of these fundamental dimensions.
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In this video, we'll be dealing
with length, which we denote by the letter
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L, mass, which we denote M, and time, T. For
example, no matter how you measure the physical
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quantity velocity, it has the dimension, which
we denote with square brackets, of length
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divided by time, or length times time to the
negative 1 power.
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Pause the video here and determine the dimension
of energy.
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Energy has dimension Mass times
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Length squared over Time squared.
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Okay, great, so what's the big deal? How is
this useful?
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Remember NASA's rover? Part of the landing
sequence calls for a parachute to slow the
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vehicle down. Suppose it is our job to design
the parachute to slow the Rover to exactly
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90m/s. The terminal velocity of the Rover
depends on the mass of the rover itself and
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its heat shield, and several different variables
related to the parachute design: the material
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of the canopy, the diameter of the hemispherical
parachute canopy, the number of suspension
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lines, etc.
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For simplicity, let's suppose that all parachute
parameters other than the diameter of the
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canopy have already been determined. Our goal
is to find the canopy diameter that is as
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small as possible, but will correspond to
the desired terminal velocity of 90 m/s.
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Clearly, we can't test our designs on Mars.
The question is: how can we get meaningful
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data here on Earth that will allow us to appropriately
size the parachute for use on Mars? How do
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we predict the behavior of a parachute on
Mars based on an Earth experiment? And what
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variables do we need to consider in designing
our experiment on Earth? This is where a problem
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solving method called dimensional analysis
can help us.
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Before we get started, we must first determine
what the dependent and independent variables
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are in our system.
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The dependent variable is terminal velocity.
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This is the quantity that we wish to constrain
by our parachute design. So what variables
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affect the terminal velocity of the parachute
and rover system?
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The diameter of the parachute canopy is one
such independent variable. Take a moment to
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pause the video and identify others.
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Ok. Here's our list: canopy diameter, mass
of the Rover (we assume the mass of the parachute
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to be negligible), acceleration due to gravity,
and the density and viscosity of the atmosphere.
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For this problem, we can assume that the dependence
of the terminal velocity on atmospheric viscosity
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is negligible, because the atmosphere on Mars
like the atmosphere on Earth is not very viscous.
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If we wanted to derive a functional relationship
that would work, for example, underwater,
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it would be important that we include viscosity
as an independent variable.
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Why didn't we include the surface area of
the parachute canopy in our list of independent
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variables?
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Pause the video and take a moment to discuss
with a classmate.
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We didn't include the surface area of the
canopy, because it is not independent from
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the diameter of the canopy.
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In fact, we can determine the area as a function
of diameter. So we don't need both!
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Question: Could we use the surface area instead
of the diameter? Absolutely. We need the variables
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to be independent, but it doesn't matter which
variables we use! The key is to have identified
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all of the correct variables to begin with.
This is where human error can come into play.
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If our list of variables isn't exhaustive,
the relationship we develop through dimensional
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analysis may not be correct!
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Once we have the full list of independent
variables, we can express the terminal velocity
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as some function of these independent variables.
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In order to find the function that describes
the relationship, we need do several experiments
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involving 4 independent variables, and fit
the data. Phew, that's a lot of work! Especially
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because we don't know what the function might
look like.
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But whenever you have an equation, all terms
in the equation must have the same dimension.
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Multiplying two terms multiplies the dimensions.
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This restricts the possible form that a function
describing the terminal velocity in terms
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of our 4 other variables can take, because
the function must combine the variables in
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some way that has the same dimension as velocity.
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And many functionsâ€”exponential, logarithmic,
trigonometric --cannot have input variables
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that have dimension. What would e to the 1kg
mean? What units could it possibly have?
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We're going to show you a problem solving
method that will allow you to find the most
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general form of such a function. This method
is called dimensional analysis.
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We begin this process by creating dimensionless
versions of the variables in our system. We
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create these dimensionless expressions out
of the variables in our system, so we don't
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introduce any new physical parameters.
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The first step is to take our list of variables,
and distill them down to their fundamental
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dimensions.
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Remember our fundamental dimensions are length,
mass, and time.
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Distilling gravity to its fundamental dimension,
we get length per time squared. Now you distill
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the remaining variables of velocity, diameter,
mass, and density into their fundamental dimensions.
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Pause the video here.
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Velocity is length per time; diameter is length;
mass is mass; and density is mass per length
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cubed.
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The second step is to express the fundamental
dimensions of mass, length, and time in terms
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of our independent variables.
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We can write M as little m.
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We can write L as mass divided by density
to the 1/3 power.
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We can write T as velocity divided by gravity.
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We had many choices as to how to write these
fundamental dimensions in terms of our variables.
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In the end, it doesn't matter which expressions
you choose.
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The third step is to use these fundamental
dimensions to turn all of the variables involved
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into dimensionless quantities. For example,
the terminal velocity v has dimension of length
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over time. So we multiply v by the dimension
of time, and divide by the dimension of length
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to get a dimension of one.
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We define a new dimensionless variable vbar
as this dimensionless version of v.
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Now you try; find dbar, mbar, gbar, and rhobar. Pause
the video here.
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You should have found that dbar is d times rho over m to the 1/3.
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Mbar is 1. Gbar is v squared over g times rho over
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m to the 1/3., and rho bar is 1.
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Now we can rewrite the equation for velocity
in terms of the new dimensionless variables.
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It is a new function, because the variables
have been modified. Notice that vbar is equal
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to gbar. This means that vbar and gbar are
not independent! So our function for vbar
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cannot depend on gbar. Also, notice that mbar
and rho bar are both equal to one, so our
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function doesn't depend on them either.
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This has simplified our relationship: vbar
is a function of only one variable, dbar.
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And remember that dbar is dimensionless, so
it is just a real number. This means that,
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phi can be any function.
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The forth and final step is to rearrange to
find a formula for the terminal velocity.
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The key here is that this equation for the
terminal velocity has the correct units.
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And this formula is so general, that any expression
with dimension of Length over Time can be
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written in terms of this formula by defining
phi in different ways. Let's see how. First,
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create an expression from the independent
variables that has the same dimension as velocity.
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One such expression is the square root of
g times d. By setting the formula for v equal
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to the square root of gd, we see that by setting
phi equal to the identity function phi(x)
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= x, the two sides of the equation can be
made equal.
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And in fact, we claim that any expression
with the correct dimension of Length over
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time created using these variables can be
written in terms of this formula by simply
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changing the definition of phi!
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It can be fun to try this. Come up with different
formulas that have the correct dimension.
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You can even add them together. Then see if
you can find a way to define phi so that our
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formula is equal to the expression you wrote.
Pause the video here.
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Now you may be concerned because this formula
is not unique. We made some choices about
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how to represent our fundamental dimensions.
What happens if we make different choices?
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Here we chose M, L, and T this way: M was
little m, L was d, and T was v over g. Running through
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the dimensional analysis process with this
choice of fundamental dimensions, we obtain
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an equation for v that looks like this.
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To see that these two formulas are equivalent,
we set the arguments under the square root
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equal, and find that we can express phi as
a function of psi.
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So any formula with the
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correct dimension can be expressed by this
general formula.
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And this general formula works for any rover
on any planet whose terminal velocity through
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the atmosphere depends on the same variables.
Because it is a general law!
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Of course, we still don't know what this function
phi is! In order to find phi, we can fit experimental
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data from any planet, for example, Earth.
On Earth, we know the gravity and atmospheric
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density. We can specify the mass of a test
rover to be 10kg.
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Then we might set up Earth bound experiments
by varying the canopy diameter of a parachute
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between 1m to 20m and measuring the terminal
velocity. For example, suppose we obtained
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the following data on Earth.
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Then we could convert this data to the dbar,
vbar axes, by scaling the variables v and
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d according to the Earth values for the mass,
gravity, and atmospheric density. We can fit
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this data to some best-fit curve. And this
best fit curve is our best approximation to
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the function phi.
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Now that we have phi, we can transform the
axes again to represent the canopy diameter
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and terminal velocity on Mars. This is done
by converting vbar and dbar to v and d by
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scaling according to known values of the gravity,
atmospheric density, and mass of the rover
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on Mars!
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To find the diameter, we find the point on
the v-axis that corresponds to a terminal
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velocity of 90m/s, and use our curve to determine
the diameter that corresponds to this terminal
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velocity! Now we have the specification we
need to design the size of our parachute to
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be used on the descent to Mars!
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In this example, we used dimensional analysis
to restrict the possible form of a function
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describing the terminal velocity of the Mars
rover as a function of parachute canopy diameter,
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gravitational acceleration, atmospheric density,
and the mass of the rover.
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This allowed us to design a parachute for
use on Mars based on Earth bound experiments.
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In general, the process of dimensional analysis
involves... 1.
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Identifying the dependent variable and independent
variables,
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2. Expressing the relevant fundamental dimensions
in terms of the variables found in step (1),
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3. Generating dimensionless expressions for all
of the variables using expressions from step
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(2).
4. Producing a functional relationship between
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the dimensionless dependent variable in terms
of the remaining independent dimensionless
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expressions.
5. Rearranging to determine a formula for the
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variable of interest. And
6. Performing experiments to determine the form
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of the general real valued function that appears
in the formula.
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We've just shown you a powerful tool which can
save you a lot of time. So the next time you
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encounter a difficult challenge, you might just
want to try... Dimensional Analysis.