PHISICAL SCIENCE

THE HUMAN BODY

(SECOND EDITION)

To my mother, with love, wherever you are.

INDEX

1 THE OBLIGATION OF TALMUD page 12 RESONANCE EFFECTS TO SWIM pages 2-53 RESONANCE EFFECTS ON LEGS pages 6-154 THE NECK AND HEAD pages 16-205 THE TONGUE pages 21-286 ARMS pages 29-327 ARMS AS A SPRING pages 33-388 WHOLE BODY AS A PENDULUM pages 39-419 FINGERS OF HAND pages 42-45

Prologue:

I have a maxim, "What can not be measured is not much”.What I will write in this book is not a scientific discovery, it is simply an application of classical mechanics to the movements of arms, legs, etc ... Classical mechanics is Newtonian mechanics and the mechanics of large bodies we will not go into relativistic quantum mechanics or any considerations.Basically I'll give the halls needed for the times you need to move different parts of our body to resonate with the natural frequency of different parts of our body tools.For a person who does not understand the phenomenon of resonance hasto believe what he will tell, but I will try to give some examples to understand this phenomenon.Technically it resonates with a physical oscillator, as can be a swing or our own arms, when there is another driving force whose frequency is substantially equal to the natural frequency of the physical oscillator, and shows that the width goes to infinity as both frequencies are virtually identical.To give some examples, when a person pushes a child on the swing (ifwe ignore the friction of the swing chains) and that force has the same frequency as the natural frequency of the swing, which can be considered as a pendulum come a point where the driving force is imposed and the child would end fired, the range of motion of the swing is greatly enlarged (expect anyone to do the test).These precautions are also used in the military and when a battalionpassed through a bridge is made out of step because if the force that makes unison troop comes into resonance with the bridge structure it could come down or classic example of soprano that emits a continuous note with the same frequency of vibration of a vessel that causes it to break.We do not need a very large force, such as the soprano, but once it comes into resonance with the natural frequency of another oscillator the resulting amplitude tends to infinity.

We see that the key word is "resonance" and I'll give you the tools to halles time where you have to move your legs, arms and other parts of your body to come into resonance with the natural frequencyof these parts of your body.Finally say that always struck me a picture I once had an examination and two spherical charges hanging by a thread when released repel and if they are of the same charge produced the figure of a pendulum, obviously do not move by the repulsive force but greatly resembles the shape of a pendulum.

I do not know if the unification of electromagnetic theory with gravitational continue patterns like a pendulum, but is now fashionable string theory, that somehow have analogies with the pendulum movement, the study of the motion of a string is studied inthis branch of physics, and vibrations and waves greatly simplified if the rope is anchored at one point, so these movements so curious as to throw streamers are anchored to the fixed point of the hand that threw occur.

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CHAPTER 1

THE OBLIGATION OF TALMUD

This is a scientific book, can not be considered a novel. The book discusses aspects that can be measured, in order to reach conclusions.Although a scientific book I have tried to try to reach all readers is.That said get down to business with a curiosity that I read in a book, particularly in the French book, vibrations and waves.The Talmud in a Hebrew book about religious issues and laws is.It's funny because it was written 2,000 years ago and a legislative provision of this book said that if a hen cackling broke with her a jar poking his head inside, the owner of the chicken should pay the wreckage.This effect is called resonance, and apparently already knew thousands of years ago.The mechanism is the same as making a sustained soprano note that broke the glass, as discussed below.A French physicist who also raised chickens called it the precept not necessary, since he never saw a chicken stick his head in a bowland break it with his crowing.Finally we see that the resonance was known at the dawn of civilization, where lawmakers typify in a manner somewhatstrange, based on a phenomenon that almost never happens, unless at that time was such great to typify problem.

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CHAPTER 2

RESONANCE EFFECTS TO SWIM

We have seen that when entering the resonance amplitude increases considerably, a small earthquake can knock down a house if both systems resonate.I have to confess I do not know exactly the consequences of enteringinto resonance with our own natural frequency of floating through our own momentum, but physically it is shown that the amplitude of the frequency increases without much knowledge of medicine believe that, mind, muscles , bones and blood circulation should be favored.Personally I've been doing two years and the results are spectacularin terms of my own fitness.I have seen many times at night on television these magical devices to lose weight, I have nothing against them but I think a hoax.Before going I want to say that if you feel bad any kind of exercisedescribed in this book or let you see a doctor, I just try to put atthe service of physical exercise physical tools we know.About two years ago an idea came out of the blue and that the human body can be compared to a table, the example is a bit macabre, butwhen someone puts it in a coffin we see that not much room left over.A coffin has a natural buoyancy frequency as in the case of a ship depends only on its sides.The Greeks knew that when soaking a body was experiencing an upward force equal to the weight of water displaced (Archimedes' principle), the other force acting on the floating body is the 3

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weight Swimmer Swimmer own.No I will demonstrate how to get to the formula of the natural frequency of a body submerged floating level, although any sophomorephysical should be able to find it taking into account the density of the liquid, but for a body square is T = 2 * pi (h / g)^1/2 where(h) the extent of the wings and (g) the gravity constant 9.81 m / s 2 on Earth's surface.If you do a weighted average of the cost of a human body is to be about 14 cm (wider in the thighs, ankles thinner and wider in the stomach)By measuring this variable in your body we can measure the time thatwe do a full stroke, in my case T = 2 * 3.14 * (0.14 / 9.81)^1/2= 0.84 seconds this is the time when I do a full stroke (since we started doing the back stroke and plunge us back up to be in the same position), thus we get the strength to develop produce this movement comes into resonance with the natural frequency of our own body to float.Summarizing my body to float up and down repeatedly oscillates within a certain time, this is called a simple harmonic motion, fromnow (m.a.s)For example when climbing a mountain the force of gravity is constant but swim down a little and plunging back up to the surface,the force of gravity and the upward thrust produced by the water displaced, make our floating body to climb up and down again following a (m.a.s).Two years ago I started doing this exercise in pools but I was afraid to walk the pool, now I think it's not worth worrying about moving too, and yes make the move to sink and rise to the surface without much progress, so it is can be done in a small pool.If you want to make progress, your body must be driven along a sine function of period you've found for your own body, the period in (more) is the time when the body repeat twice the same position, eg since begin the stroke immerse ourselves back up and being in the same position.

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I think it is best to dive and go forward a bit, it is better because it is precisely in the vertical movement of the body where it produces (more), more than the horizontal movement of progress where there occurs the (more). I have also tried in thesea, but the waves have their own natural frequency, and it is unlikely that two physical oscillators (wave frequency and the

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natural buoyancy of the body) in resonance with our own driving force.Displaying the formula you realize that people with a wider side must swim more slowly than thinner people.It does not matter how much you immerse us as a property models (mas) is that the frequency does not depend on the amplitude, if we stretch a spring two meters vibrate with the same frequency as if stretched 10 cm, obviously first case will move more faster because it has more room to go. As we shall see the natural frequency flotation our body is very similar to that of a T = 2 * pi (h / g)^ 1/2 pendulum changing (h) the width of our sides by (l) the length of a pendulum T = 2 * pi (l / g)^1/2Later we will use the formula of a physical pendulum for different parts of our body.Finally consider that the human body is not like a table but an approximation of 80% is good enough physical approximations used continuously in many cases and that greatly simplify the calculations and often are perfectly valid to solve the problems we face.

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CHAPTER 3

RESONANCE EFFECTS ON LEGS

The period of a pendulum is T = 2 * pi * (g / l) however half a pendulum has a dependency on the position on his speed, just see itsequation of motionx = A * sin (wt + O) where (A) the amplitude (w) and angular velocity (0), the phase angle.A pendulum when released does not start to decrease fast, stops, accelerates again and then follows a steady rate, ie no stutters, but follows a harmonious movement, so he called him harmonic movement to movement.To simplify the movement of legs and arms have to have a couple of thoughts on the pendulum at the ends of their movement speed is 0 and acceleration is maximum and its lowest point is maximum speed and acceleration is 0So in the coming years we should start the pendulum movement of our arms and legs in a way slowly, reaching its maximum speed at its lowest point and reduce its speed to the other end.All this time we will find below for each arm or leg especially everyone.I know it's a little complicated but with a little practice you can get.In fact for swimming exercise must also follow thisprocedure, starting with a slow immersion, increasing the speed at the midpoint of the dive and reducing the speed until it reaches thelowest point of the dive.The first exercise is the classic ballerina with one foot on the floor and leaning with one hand, keeping the other leg in the air

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freely for all purposes leg is in the air in a rigid movable like a physical pendulum anchored cylindrical hip.

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For all purposes in this leg acts as a physical pendulum motion of cylindrical shape of period T = 2 * pi * (I / mgh)^1/2, where (I) isthe moment of inertia of a cylinder (m) mass of the leg, in the human body despite the leg is 10% of total body weight (g) the acceleration of gravity at sea level 9.81 m / s^2 may be considered constant anywhere in the planet earth, unless you do the exercise inthe Himalayan peak (h) is the distance from the hip that acts as a point ofanchor the center of mass of the leg that goes on about the knee.The moment of inertia of a cylinder is I = ¼ MR^2 + 1/12 ML^2 being MR2 (M) the mass of the leg (R) cylinder radius, ie the radius of the leg if we consider it as a cylinder, this measurement may be made to the knee (L) the total length of the leg from hip to foot.I'll find the period of oscillation of my leg for this exercise, let's start with the moment of inertia of my leg.(M) the mass of my leg is about 7.1 Kg. 10% of my weight (R) the radius of my leg as the knee comes to about 0,025 m. (All measures must be put in an international system of measurement, kilograms, meters and seconds) (L) the length of my leg is about 0.7 meters.Thus the moment of inertia is I = 1/4 * 7.1 * 0.0252^2 + 1/12 * 7.1 * 0.72^2 = 0.29 kg * m^2Spending this amount to the period of oscillations formula we have: T = 2 * 3.14 * (0.29 / 7.1 * 0.35 * 9.81)^1/2 = 0.68 seconds.The other exercise I suggest is to have the thigh horizontal and move the leg that runs from the knee to the foot of a pendulum.

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Now for all purposes the part that goes from knee to foot acts as a physical pendulum anchored in the knee, in my experience it should be a little careful as you force your knee slightly in position to act as the knee anchor point in a slightly unsteady.

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I will find the corresponding period for my leg and start at the moment of inertia.(M) the mass of my leg from knee to foot is about 3.5 Kg. 5% of my weight (R) the radius of the bottom of my leg doing a weighted average between the knee and the foot comes to be 0.02 m. (All measures must be put in an international system of measurement, kilograms, meters and seconds) (L) the length from knee to foot is about 0.3 meters.Thus the moment of inertia is I = 1/4 * 3.5 * 0.02 ^ 2 + 1/12 * 3.5 * 0.3 ^ 2 = 0.026 kg * m^2Spending this amount to the period of oscillations formula we have: T = 2 * 3.14 * (0.02 / 3.5 * 0.3 * 9.81)^1/2 = 0.27 seconds.I'll describe it is a physical pendulum, but is physically a bit advanced not worry since the calculations for this exercise are verysimple, but this is a physics book and so I put it, if you're not interested in knowing works as a physical pendulum you can skip thispart.

The article is taken from Wikipedia.***A compound pendulum or physical pendulum is any rigid body can swingfreely in the gravitational field around a fixed horizontal axis that does not pass through its center of mass.The physical pendulum is a system with one degree of freedom; corresponding to the rotation about the fixed axis ZZ '(Figure 1).

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Physical pendulum position is determined, at any instant,by the angle 0 which is determined by the rotation axis (ZZ ') and the center of gravity (G) of the pendulum with the vertical plane passing through the rotation axis plane.Call h, the distance from the center of gravity (G) of the pendulum to the rotation axis ZZ '. When the pendulum is deflectedof its equilibrium (stable) an angle O, two forces acting upon it (mg and N) the resultant moment about the axis ZZ 'is a vector directed along the axis of rotation ZZ', in the negative direction of thereof; i.e.,M = -mgh * sin (O)If (I) the moment of inertia of the pendulum about the axis of suspension ZZ 'Ö and call the same angular acceleration, angular momentum theorem allows us to write the differential equation of rotational motion of the pendulum without -mgh * (0) = I * Ö we can write in the formÖ + mgh * sin (O) / I = 0which is a second order differential equation, the same type as that

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found for the simple pendulum.In the event that the angular amplitude of the oscillations is small, we can set sin (O) = O and the equation takes the formcorresponding to a simple harmonic motion.The oscillation period is T = 2 * pi * (I / mgh)^1/2

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I'll find the period of oscillation of my leg for this exercise, let's start with the moment of inertia of my leg.(M) the mass of my leg is about 7.1 Kg 10% of my weight (R) the radius of my leg as the knee comes to about 0,025 m (all measures must be put in international system of measurement, kilograms, meters and seconds) (L) the length of my leg is about 0.7 meters.Thus the moment of inertia is I = 1/4 * 7.1 * 0.0252 + 1/12 * 7.1 * 0.72 = 0.29 seconds kg * m2Spending this amount to the period of oscillations formula we have: T = 2 * 3.14 * (0.29 / 7.1 * 0.35 * 9.81)^1/2 = 0.68 seconds.Another exercise is the exercise bike, the pedal is a circle that will divide into two semicircles, when the pedal is in the middle position (neither up nor down).

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then begins to drop and turns on the rear middle position again. Part of our leg from the knee to the foot acts to all effects as a physical pendulum cylindrical, as in the previous section but now using only the portion of the knee to the foot, when the foot makes this movement and begins to rise the other leg back to doing the same motion made before the other leg begins to lower and raise semicircular like a pendulum, that is, by measuring the period of half pedaling so that the part of our leg from knee to foot between in resonance with the natural frequency of oscillation of our leg from knee to foot multiplied by two will know the time we have to doa full rotation of the pedal.I will find the corresponding period for my leg and start at the moment of inertia.(M) the mass of my leg from knee to foot is about 3.5 Kg 5% of my weight (R) the radius of my leg as the central thigh becomes 0.02 m ( all measures must be put in an international system of measurement, kilograms, meters and seconds) (L) the length from kneeto foot is about 0.3 meters.Thus the moment of inertia is I = 1/4 * 3.5 * 0.02 ^ 2 + 1/12 * 3.5 * 0.3 ^ 2 = 0.026 kg * m^2Spending this amount to the period of oscillations formula we have: T = 2 * 3.14 * (0.02 / 3.5 * 0.3 * 9.81)^ 1/2 = 0.27 seconds multiplied by two is 0 , 54 seconds, this is the time when we must make a complete rotation of the pedal so that our driving force in resonance with the natural frequency of our legs.In some cities there are devices where you support the foot and movethe leg rigidly it forward and backward movement is like making skiers, but the device is a bit off the ground, for all purposes have to use period encountered earlier in the example ofthe dancer to move a leg rigidly, but this way we can use both legs.Finally consider that we are making approximations for physical pendulum with small oscillations, which are sufficiently accurate,

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these approaches are perfectly valid for not more than 45 ° oscillation amplitude. you can use the general formula for a pendulum that will expose the end of the book but it is quite complicated and not think it's necessary for our calculations as we usually do we exceed this range, although it is true that some exercises can be overcome somewhat, who want to use it is perfectly right, I also give you some examples, but again stress that I consider quite complicated for our calculations.Summarizing the first exercise with stiff leg, hip acts as an anchorpoint and we have to consider the entire leg from the hip to the foot.The second exercise, the exercise bike, the anchor point of the physical pendulum is located in the knee and must consider the leg from knee to foot, to make our calculations.

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CHAPTER 4

HEAD AND NECK

Many times I have seen people move to relax the neck and head from left to right to relax the neck and head are a very important part of our body, indeed the latter allows us to think.In this chapter I will give you the formula so that you find the time to the natural frequency of torsional oscillation of the neck and head.The necks for all purposes is a torsion pendulum, the formula of theperiod is T = 2 * pi * (I / c)^1/2, can be made perfectly approximate the moment of inertia of the head and neck is a cylinderI = 1/12 MR^2 + 1/4 ML^2 the same as in the first paragraph of the preceding example, being (M) the mass of the head, the mass of the head of an adult being comes to about 8 Kg . owe this to add the weight of the neck, find no better way to weigh your head and neck to put them on a scale, I think you will not be very difficult (R) is the radius of the head neck system, or a little less than half ofthe skull as the radius of the neck is a little smaller, you can make a weighted average head system neck with a measure a little less than half of the skull, I believe that the approximation is good enough (L) is the length of the base of the neck at the highestpoint of your head.The equilibrium position is that we have facing forward, if for example found in your neck period is 0.5 seconds,movement have to do is since deviate neck slightly to the left or right you move to the other side and returns to the position where

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you had to start exercising, or spending twice the equilibrium position, basically like a pendulum , but in this case the torsion.My experience in laboratories, constant torque head neck system is relatively large, and otherwise difficult to measure due to the large number of muscles acting in that movement.I will use the method of measuring the constant typical torque in a laboratory, which is turning the head 15 °, 30 ° and 45 ° and find the time of the force, I have to say that although there are ways tofind the moment of force not have the necessary equipment to find itand I'll do exactly ojímetro, then I shall find the regression line with the 3 data and finally find the slope of the line to find a constant torque.The 3 data are:Moment exerted 15th 2N * 0.05 m.Moment exerted 30th 4N * 0.05 m.Moment exerted 6N 45 * 0.05 m.

Tabla de Datos

X Y X2 X*Y

15 0.1 225 1.5

30 0.2 900 6

45 0.3 2025 13.5

Totales 90 0.6 3150 21

Then:

b=((3 )(21)-(90)(0.6)) / ((3)(3150)-(90)2)

b=0.00666666666667

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a=(0.6 - 0.00666666666667*90)/3a=1.85037170771E-16

Therefore, the best-fit line is:

Y = 1.85037170771E-16 + 0.00666666666667x

The regression line is y = 0.007x + 0 so its slope is 0.007 and torsion constant "c" = 0.007 N * mI'll find the period of oscillation for my neck, but remember that the neck is very united head system, that is another exercise for a future issue, but should be able to find it and !.The moment of inertia of a cylinder, in the case of my neck is:20

I = ¼ MR^2+ 1/12 ML^2As (M) the mass of my neck about 2 Kg.(R) The radius of my neck about 4 cm.(L) The height of my neck about 7 cm.Therefore I = 1/4 * 2 * 0.04^2 + 1/12 * 2 * 0.07^2 = 0.0016 kg * m^2With these data we find the natural oscillation period of my neck, considering a torsion pendulum.T = 2 * pi * (I / c) 1/2 = 2 * 3.14 * (0.0016 / 0.007)^ 1/2 = 3 secondsThat although the calculations have been done in a very subjective way, we conclude that we should move the neck in a very slow manner.It's funny how the physical formulas are very similar and the body consists of pendulums (arms and legs) torsion pendulums (neck) or floating tables (when our body sinking a little nothing)19

Also keep in mind that these movements should be made as close as possible to the natural frequency when power our body in the same natural frequency of the body that the force tends to infinity and is a force to be advantage, besides not enter phase with the naturalfrequency which makes your exercise more effective, personally the results are quite spectacular and not have to do more than 5 or 10 minutes of exercise to be effective.In the graph of resonance we can see that the approach much to the natural frequency, not too broad peak tends to infinity, even if youdo the exercise with small deviation of the period of its natural frequency still rogue chub, but with a deviation occurs some tenths and you only get about 20% strength of resonance.

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CHAPTER 5

TONGUE

The tongue is an organ of the human body vibration of the first magnitude, basically you move from top to bottom, from inside to outside or arching upward to disturb the air and thus produce syllables and words.We will not take their normal motion that is when it is in its horizontal position resting on the lower jaw and where arches, raised or curve to produce words, this movement does not interest us, first because it is a complicated move to make calculations and moreover this movement is not a moreBut if we make a simple change of coordinates 90 degrees, magically and with the help of physics we are in a situation where the language is now for all purposes a physical pendulum tabular anchored in the front of the mouth .Or in the position in which the head down to savor an ice cream and tongue position is upright.

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We started doing the math and started calculating the moment of inertia of the tongue I = 1/3 * M * L2 (L) is the total length of the tongue, in an adult becomes 10 cm. if you want more accuracy youwill have to measure it with a ruler.

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(M) is the mass of the language, I searched on the internet and say the mass of the language of an adult is between 100-150 grams, I think it's a fairly realistic mass and will use a mass of 125 grams for our example. If you want more accuracy the only thing I can think of is to take a small glass of water and make a mark on the top where the water reaches scoring half of the tongue and make another mark, although the liquid displaced by half the language, iethe content between the two marks and multiply it by two to find thetotal mass of the tongue, we see again that Archimedes principle is again useful.Well we complete the calculations for the moment of inertia of the tongue and not forget spending all the measures the international system. I = 1/3 * 0.125 * 0.12 = 0.00042 (put units)Using this measure in the formula for the period T = 2 * pi * (I / mgh) 1/2 where m = mass of the tongue, g = acceleration of gravity on Earth, h = distance from the anchor point of the tongue its center of mass, as the tongue is a little narrower at its end and wehave agreed that the language comes to be 10 cm. We will consider your center of mass a few millimeters before its center about 4.8 cmWith these data we can find the period T = 2 * 3.14 * (0.00042 / 0.125 * 9.81 * 0.048) 1/2 = 0.53 seconds.I sometimes do this exercise and tongue thanks you, for all purposeswhen doing this exercise language resonates with the natural frequency of oscillation and strengthens the language concerned, I am convinced that when our muscles, such as the language, resonates with the natural frequency of oscillation also get a mental benefit for all purposes both our mind and body work better.For now let's calculations so you can find the time or period in which you must move your tongue in the upright position.I want to emphasize that when I speak of time, talk time for all purposes, in fact the period is measured in seconds, the physical idea of time is equal to what we usually understand as something repetitive, like when a woman has her period, or it periodically.Strictly physical period is the time when a simple harmonic motion

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makes a full motion, for example when moving a pendulum to the rightwe drop, goes to the left and returns to its initial position to theright, in physics the time when there was the whole movement is called period, if there were no friction the pendulum would be doingall the time that movement in time we call period.Let's start with the calculations, the formula is the classical period for a physical pendulum T = 2 * pi * (I / mgh)^1/2, now is the moment of inertia corresponding to a rectangular table anchored by one side, in this case one corresponding to the part of the tongue anchored to the inside face of the narrow sides.The moments of inertia are tabulated on any page of physical and online you can find them without major problems, the technical name for a rectangular plate aby sides thickness c is cuboid, I admit that I struggled a bit to find this moment of inertia, I finally found on a page of agronomists, and was surprised that the moment ofinertia of a parallelepiped to the plane formed by the AOC hand, It was according to this page I = 1/12 Mb ^ 2, I was surprised that the moment of inertia of the language on this plane only depended onthe length of the tongue, an adult length of the tongue is measured 10 cm . We came to this hair as much simplified calculations and if want more accuracy it is sticking out tongue with a ruler and measure it, be careful not eat it. But then still could not see as much simplified formula of this moment of inertia, and decided to figure it out on my own, in physics the worst you can do is calculate ojímetro things and when you have a question it is best to get to do the math with pencil andpaper following the physical and mathematical rules necessary. Also sometimes I fail my spatial vision but sometimes I work very well asthe idea for the book came to me when considering arms and legs as physical pendulums cylindrical, or to bring the body and language astwo parallelepipeds in the first if swimming and in this particular case to consider language as a parallelepiped and anchored in its vertical position in the plane with the language form the front of

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the mouth. As this is a physics book will put the development to find this moment of inertia, the calculations are not overly complicated, it just takes a little skill in basic integration, but for anyone without physical knowledge may obviate this problem without any explanation and applied for final calculation formula will be explained below.

*** moment of inertia of a square aby c thick sides respect one side*****

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We started using the classic formula of the density to find the massof the tongue p = M / V, for all purposes we consider the density of

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the tongue constant at every point, ie p is a constant. Thus the mass of the tongue is M = pV, as a differentiator use an infinitesimally thin plate length side db acy therefore this differential mass element is: dm = p * a * b * db moment of inertia of such differential AOC the plane element is dm ** Dica = b2 = b2 Dica ie * p * a * b * db and the entire body, with respect to same plane a distance b (b = c = 0)of the plane AOC Ica is the integral b2 ** = p * a * b * db integration between the limits 0 and c, the sides of the infinitesimal volume element, and care constants and density, so go outside the integral and is p * a * b ** Ica = b2 * db integrating between the limits defined above is I = 1/3 * p * a * b * c3, this is where I wanted to come here and see that the moment of inertia depends on 3 sides of the parallelepiped, but our dear friends agronomists had realized that the mass of the cuboid is M = p * a * b * c so we can simplify and I= 1/3 * M * c2, we again see that physics is magic and the moment ofinertia considered, ie the language in an upright position only depends on the length of the language and its mass.

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We started doing the math and started calculating the moment of inertia of the tongue I = 1/3 * M * L2 (L) is the total length of the tongue, in an adult becomes 10 cm. if you want more accuracy you will have to measure itwith a ruler. (M) is the mass of the language, I searched on the internet and say the mass of the language of an adult is between 100-150 grams, I think it's a fairly realistic mass and will use a mass of 125 grams for our example. If you want more accuracy the only thing I can think of is to take a small glass of water and make a mark on the top where the water reaches scoring 27

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half of the tongue and make another mark, although the liquid displaced by the middle of the tongue, or is the content between thetwo marks and multiply it by two to find the total mass of the tongue, we see again that Archimedes principle is again useful. Well we complete the calculations for the moment of inertia of the tongue and not forget spending all the measures the international system. I = 1/3 * 0.125 * 0.12 = 0.00042 Kg * m2 Using this measure in the formula for the period T = 2 * pi * (I / mgh) ^ 1/2 where m = mass of the tongue, g = acceleration of gravityon Earth, h = distance from the anchor point language to its center of mass, as the tongue is a little narrower at its end and we have agreed that the language comes to be 10 cm. We will consider your center of mass a few millimeters before its center about 4.8 cm With these data we can find the period T = 2 * 3.14 * (0.00042 / 0.125 * 9.81 * 0.048) 1/2 = 0.53 seconds. I draw much attention to the natural frequency of a human body floating we consider normal, about 0.16 cm. Side is about 1 second and the language of a measurement of 10 cm. and a mass of 0.125 is 0.5 seconds, do not know who the second considered as a universal measure of time, but fully hit as it is consistent with two important activities of our morphology, first swimming (experts say that all living things including us both come from water), do not disparage anyone because I have a physical defect but universal media tends to 1 second, and the language that we use to talk about the very important function that is 1/2 second half, or we see a 1 to 1/2 then compare with measurements of arms, legs and neck-head tosee if it remains simple relationship. We have made some progress and we know how fast we have to move our tongue to taste ice cream and not get into phase with its own natural frequency and therefore does not make unnecessary efforts, in addition to what we said at the beginning about the soprano is able to break a glass using her vocal cords and the language itself.

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CHAPTER 6

ARMS

With arms we can do two exercises in which we can find easily the period. Do consider the approximation of arms as physical pendulums shaped cylinder, the first exercise is to maintain rigid biceps and elbow without moving this part only move the forearm,

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if we do it without any dumbbells in hand the center of mass would be located just above the middle of the forearm as the part near elbow wrist outweighs, we see that the mass of the arminfluences not as in swimming where only matter measurement side, ifwe use a small weight for example 1 kg. would move the center of mass about the center of the forearm as the weights will be more more big move the center of mass of the arm to the hand. We also have to take into consideration that the weights must be small (a ball of 1 kg for example) because if they are of a large volume such as a dumbbell break apart the approximation of considering the forearm as a cylinder and hamper the calculations. The formula we use is the same as in the example of arms, or that ofa cylinder. We must also take into account the radius of the arm, or if we consider it as a cylinder have to do a weighted average of its radius would be half the width measured at the half forearm forearm.For example my forearm is 20 cm. and half its width becomes about 2 cm. as these data we can find the period of the motion would be ... Can you find you just ?. I have to remember that the more weight we use in hand over will move the center of mass of the forearm was the wrist. Although obviously the point of attachment of this physical pendulumis the elbow, where we are neglecting friction occurs. The other exercise is to keep the arm vertically aligned with the body, the formula would be the same as in the previous example, we still consider the whole arm as a cylinder and now the anchor point would be the shoulder, the center of mass approximately elbow heightand length of the physical pendulum would be to the whole arm. The motion would move the stiff arm from his position balance to no more than the horizontal with the ground, better not reach the horizontal position of the arm, always maintaining the rigid arm. The weighted average of the radius we can measure the elbow and comes to be a little higher than when we measure the forearm, in my

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case the weighted average radius of my arm is about 2.5 cm with thisdata and the length my arm we find the period of the motion with a fairly good approximation. For example for an arm length of 40 cm and 2.5 cm radio period would be ... (Try to do the math you), this is the time when we must move the arm from its vertical position to a position not above the horizontal with the ground and return to its equilibrium position. Here again has importance if we use hand weight or not, but we use the center of mass weight would be on the elbow, but if we use weight shift the center of mass of the wrist was the larger shape ismore weight. The position is similar to the ancient Egyptians made engravings.

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CHAPTER 7

ARMS AS A DOCK

We have seen how to swim and move the limbs of a pendulum can be made following a simple harmonic motion, now we will see another wayto move your arms in harmony, but along the lines of a spring. The boats are at sea harmonic movements, pendulum clocks pendulums used for a specific period of clock gears move in a given time and thus move clockwise so that they can measure time, well the springs are also harmonic motion and now we are going to find the time when we must move our two arms in unison, to enter into resonance with its natural oscillation in the Earth's gravity, explaining that if we were on the moon, the calculations needed replacing as it exerts less force on our body. Let us find the differential equation of a spring to prove it's the same that we used in the chapter on the fingers of one hand, in thatexample was Ö + (mgh / I) * O = 0 (where O angular displacement). The force exerted by a spring on a mass is shown that F = k * x (where k is the spring constant and x recovery displacement). We see two characteristics: 1) opposite to the displacement force is 2) k is the constant recovery feature beginning each spring and is closely related to the material and shape of pier construction.

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Since F = m * a (Newton's second law) m * a = k * xy as accelerationis the second derivative of displacement with respect to time, we can write: d2x / dt2 = - (k / m) * x is exactly like Ö = - (mgh / I) * changingthe constant O (k / m) by (mgh / I) The exercise itself is to illustrate in the following drawing.

For all purposes the two arms act as two springs of hanging the 34

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remaining mass of the body, perhaps hard to see that the arms are springs of hanging the remaining part of the body in that position, but it's really when they act as springs to raise and lower your arms slightly elevating and lowering them. The arms act as the exercise recovery spring constant "k" very large, practically as a fine wire spring is deformed very little, although the inner muscles themselves are deformed more and therefore can be likened more to a traditional spring. Initially we will use a very high value of k slightly deformable material. We will find the period in which we have to raise the arms an inch and re-download, I recall that in simple harmonic motion amplitude is independent of time, or can raise the arms 2 or 10 inches but we have to do in the period we have found, obviously if we got 10 centimeters have to do it faster than if you just went up 2 centimeters, but whatever the climb to do the entire move will be done in the period found for your own arm. Here we do not enter into consideration of energy, if you raise the arms 10 cm. the potential energy produced is much greater than if wejust got our body 2 cm. We are now focused only on the simple harmonic motion of our two arms regarding them as two springs of hanging the remaining mass of our body. An important consideration is that the mass of the body hanging fromour two arms, or two springs, and when a mass hanging from two springs calculating the elastic constant "k" changes considerably, in a similar way to what making the resistances in electrical circuits, if they are connected in series or in parallel.

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We see that there is a clear analogy to the calculation of the spring constant and the equivalent resistance when both are connected in series, seems to me a clear analogy between mechanics and electromagnetism, but in our case our arms, are connected in parallel and the analogy with electromagnetism continues as describein the following drawing.

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Again we see that there is a clear analogy between the two systems, one mechanical and one of electromagnetism. First let's find the constant equivalent of both arms as if they

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were hanging docks remaining mass of our body. I insist that this year so we work are the arms from wrist to shoulder, and we go up and down trying not torsionarlos arms or helping other muscles in our body, and we believe that exercise every part of our body arms than an inert mass, and arm movement up and down is to do in the period we find to our own body. We will find the spring constant equal to my arms and generally for anyone, arms are almost non-deformable, not as metal rods, but constantly very large elastic, this means as I have said before thatdeform very little weight when hanging them. A "k" realistic arms for our walks on the order of 4000 N / m (I've found using a regression line with the values that our arms are deformed to different weights and calculating the slope), and asboth arms are in parallel, the resulting spring constant is equal to: 1 / Kr = 1/4000 + 1/4000 2/4000 = therefore Kr = 2000 N / m The period of a spring is equal to T = 2 * pi * (m / k)^^1/2 Our two arms weigh 20% of our total weight, therefore we have to measure mass is about 80% of our weight, in my case as the mass 70 Kg weight I have to put in the formula is 70 Kg * 0.8 = 56 Kg (All measurements in the international system). Therefore, the period in which we have to raise and lower arms it is in my case: T = 2 * 3.14 * (56 Kg / 2000 N / m)^1/2 = 1.05 s. Can you now find the time for your own arms and your body mass ?. I have to admit that this exercise is one of my favorites, and strengthens the body in an exceptional way, is that this year we introduced a new variant, and is considering arms as two springs in parallel, a spring constant very large, or nearly indeformables. At this point we will introduce another peculiar year, and is considered the entire body like a pendulum.

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CHAPTER 8

WHOLE BODY AS A PENDULUM

In this exercise we will consider the whole body of the arms hanginglike a pendulum cuboid, or in table form, and we will find its natural oscillation period for your own body. It is true that there is a gap above the head and the arms, as you can see in the picture below, but the approach is quite good.

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The exercise is easy, is hung balancearnos our arms, as if our wholebody were a pendulum in a table.

As we saw in the exercise of the tongue, the moment of inertia of a parallelepiped is: I = 1/3 * M * L2 (L) is the total length of the body that is the total height with arms raised, or slightly more than normal height, in my case about 1.96 me. and that I measure 1.72 me. (M) is the total mass of my body 70 ka. The moment of inertia for my case is:

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I = 1/3 * 70 * 1.962 = 89.64 Kg * m^^2 With these data and the help of a physical pendulum formula I can already find the natural oscillation period of my body, considering a physical pendulum in the form of table. (ha) the center of mass, we consider it a little below the center (L) as there is a substantial gap between arms and head, approximately 1.17 me. suspension point. T = 2 * pi * (I / Ghana) 1/2 = 2 * 3.14 * (89.64 / 70 * 9.81 * 1.17)1/2 = 2.01 seconds This is the time when we must make a full swing, forward and go backinto that full time. It must be said that the exercise is not to resist hanging, but do rolling in time for your own body found sometimes without getting tired or exhausted.

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CHAPTER 9

FINGERS

This chapter is perhaps a bit technical and people who are not overly familiar with physics can spend a little tiptoe through the physical explanation, but I tried to make it as pleasant and simple as possible, in any way at the end will have a simple expression to make their own calculations. Previously we were busy measuring the period in which they had to push a swing to match its own natural frequency and so the two systems come into resonance, the swing were our arms, legs, tongue and body, and were the driving force ourselves through our mind led through our muscles. Now let's look at the problem from another point of view, now we want to know that speed is the physical pendulum, or our fingers or our legs, at any given time and therefore what position it is in a certain time after releasing. Solving this problem can help us to know that long a pianist has to hit the piano keys once starts down the finger, or the speed at which a player has to hit the leg with a ball that is falling, although Mr. Iniesta seems to

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know this quite well. The physical pendulum is a physical entity that performs periodic 41oscillations under the action of the gravitational force, as we haveseen. The motion of a physical pendulum is fully governed by the differential equation Ö + (mgh / I) * W = 0 which is a differential equation of second identical to that of a spring order, as in both cases the acceleration is proportional and sign opposite to the displacement, not go into considerations such as differential equations are solved, which will be familiar that it is a second order differential equation which is solved by the sine or cosine function, since the derivative the sine and cosine function is the second derivative is within a negative sign. The origin of our coordinate system is at pi and considering a movement of 45 degrees or 1/2 * pi can find the acceleration Angular who has his finger in that position. Taking the length of the finger as turning radius is easy to calculate the speed you would finger hitting the piano key and the time it would have to make that move.

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Hopefully these laborious calculations serve for a pianist to play some sublime melody, if possible, than they usually do. The same considerations could be made on how fast you need to move your leg a player to hit a ball in the air where you need to move rigidly leg 15 degrees, for example from its rest position. An important consideration is that the pianist must move their

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fingers according to the beat of the song, this is just a physics book that aims to give an idea, the idea would be to make a melody to the rhythm of the natural frequency of the fingers of the pianist, but this is another story. The same consideration should be on the player, it has to shoot as hard as possible to score a goal, but we deliver in your workouts.

Epilogue:

The force of gravity is the cause of That move like pendulums move, it is Important to get along comfortable with.When I was writing this book I Heard on the radio in an informative fitness program That Were NASA experiments showing without gravity That age before, This can only happen in space as a feature of gravity is that you 'can not shield, This Means That When present can not be overridden by any Means, anyone interested on the issue of aging due to lack of gravity Should consult Their Own, I have notbut I was curious Studied.That remark Finally we are taking advantage of a free force,Serves Un certain constraints under gravity to save energy, like theclassic problem of building a long tunnel into the earth to launch rockets to the Moon with a remarkable energy saving, the problem would be how to build it.