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12.4 Wielding Your
Sword
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12.4.1 The
Effective Mass or Apparent Inertia
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What is the Effective Mass? |
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Finally! In this last subchapter we
are going to wield a sword. First we will just swing it around to get a feeling
of how it handles but then we will actually hit something or somebody with it.
For example you. Right on the head. That will give us an idea of what I mean
with the term "effective
mass" or "apparent
inertia" in the headline.
I'm going to use Damocles sword
to hit you with. To make it easier for me (and you) I will use an idealized
version of that, just a plain, uniform rod. Consider it dangling above your
head in various positions relative to your head, and then the thin string rips.
What will you experience? |
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Damocles sword hits you
Note that my drawings are gender mainstreamed |
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Let's start with the center of mass
(COM) of the sword exactly above your head as in the upper pictures above. In
the ideal physical world so beloved by basic physics teachers, the sword will
hit you and come to rest in the dent on your head. That happens because your
mass is much larger than that of the sword, and your head is rather soft. What
you fell is painful since the total energy of the falling sword is transferred
to your head. This energy is directly given by by the total mass of the sword.
If you remember anything from your physics teaching, you know that it is simply
mass m times the distance d between the hanging
sword and your head (times the constant acceleration of gravity = 9.81 m /
s2). This potential energy of the sword relative to your head is
converted to kinetic energy (one-half of mass times velocity squared), and
that's what you feel.
Note that the sword is rather stupid, It doesn't know if its velocity just
before impact is due to free fall in earth's gravity or because somebody is
moving it with his hands.
Now let's consider what you experience if the swords COM is not lined up with your head as in the lower pictures
above. |
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The sword will hit you somewhere
along its length and then tumble down rotating. What do you feel? An impact
less painful then the first one. The pain decreases with the distance between
the COM of the sword and the point where it hits you. The least pain is
encountered if you are hit by its very end. |
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How can that be? After all, the
energy of the sword at the moment of impact is the same in all scenarios? Yes,
indeed - but not all of that energy is transferred to your head. Some kinetic
energy remains in the sword because it keeps moving and some rotational energy
is building up. It is not too difficult to calculate what exactly is happening
for a simple uniform rod, a sword, an axe, or whatever. I'll do that in an
science module but here I'll
keep it simple |
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And I have good news: There is an ingenious
shortcut: Since any point of a falling object has the same velocity when it
hits you, you can describe the different energies transferred to you by
assigning an effective mass to the
sword that depends on position along its length.
Turner calls the effective mass
"apparent inertia" or
"apparent mass" of the
sword, which is fine too. I'll use the term "effective mass" because
that is something physicists are used to, it comes up, for example, in
semiconductor physics all the time
In the experiment with a uniform rod discussed above, we can deduce without any
calculations that the effective mass must be equal to the real mass at the
position of the center of mass, right in the center of the rod, and then it
must decrease somehow towards both ends. That is exactly what it does if you
run through the calculations: Here is what we get for a 0.3
kg sword: |
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| Effective mass distribution |
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What we also get is that the only
necessary parameter of the object in question that we need to know is its
moment of inertia relative to its center of
mass (COM). That is rather amazing and allows many insights into the
optimization of real swords, as we shall see.
I use the term "optimization" because we want to optimize as opposed to maximize properties. That involves going for
compromises since measures that might be good for one property will turn out to
be bad for another one. With the effective mass we have yet another key
parameter - sort of a mixture of the mass and the moment of inertia - that will
not only prove quite helpful but is essentialfor understanding what
follows. |
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Applications of the Effective Mass Concept |
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First let's look at what happens of
we change the moment of inertia of our rod from above without changing its
weight. We might do that by putting a sphere containing most of the weight in
the middle (while making the rod thinner to keep the mass constant) or two
half-spheres at the ends as shown below. The first measure decrease the moment
of inertia quite a bit, the second measure increases it. |
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Effective mass of three rods with the same
mass (0.3 kg) but different mass distributions (as schematically indicated) and
thus moments of inertia I
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The effective mass curve responds
simply by constricting or widening as shown for a rod weighing 0.3 kg. What you
would feel in the Damocles sword experiment is also quite clear. The impact
with the COM hitting your head is unchanged. It hurts the same for all three
rods. Getting hit by the end of the rod is different, however. Far less pain
with the low-I rod. far more with the high-I
rod. |
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Next we look at the effective mass
curve of a real sword and an axe. We get something like this: |
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| Effective mass of sword and axe |
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The curves, by the way, are not
actually calculated for the weapons shown but just give the idea qualitatively.
What you perceive for sure is that you rather have the business end of a sword
fall on you than the business end of axe. Its effective mass at this point is
much higher than that of the sword and thus also the energy transmitted to your
head. |
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Well - the sword is preferable to the
axe only as long as both objects fall and thus hit you with the same velocity.
If somebody swings these weapons, chances are that he can swing the sword much
faster than the axe, so it hits you at higher velocity. And the energy,
remember, goes with the mass and the square
of the velocity! There is a reason why you might prefer your axe whenever you
have all the time in the world to get it up to speed but go for your far more
agile sword if there is high-speed fighting. |
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What you want is an appreciable
effective mass at the tip of your sword so its hits will be felt but without
compromising on all the other points like manageable total mass, small moment
of inertia for agility, and a percussion point (plus the important vibration
nodes) at the right place.
And now you start to see the magnitude of the problem we are facing when we
want to optimize a sword. |
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© H. Föll (Iron, Steel and Swords script)
