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Leyes de Newton (Universo Mecánico 6) — Transcript

by Ciencias TV · 2,760 words · 424 segments · language en · Watch on YouTube

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0:01Force equals mass times acceleration.
0:05Mechanics, the science of motion can be
0:08summed up in that one equation. F = M*A
0:12. It's the core of classical mechanics.
0:19Newton's laws. In 1543, Copernicus
0:26wrote this little book and the
0:29Aristotelian world began to fall apart.
0:33A century and a half later it was still
0:36in ruins and there was nothing to
0:38replace it. Of course, some progress
0:41had been made. Kepler had already
0:43discovered his three laws. Falling
0:46bodies and inertia were understood, but
0:49there was no unifying principle and the
0:51seamless perfection of the Aristotelian
0:53world had turned into a tremendous mess
0:55. So in one imposing book Isaac Newton,
1:01the physical ideal that we know today,
1:04restoring order to the heavens and the
1:07earth. When you try to describe the
1:18basic machinery of the world, the
1:20question naturally arises as to where
1:23to begin. Isaac Newton began with three
1:27fundamental principles. Newton's laws.
1:31With the incredible speed and elegance
1:34of an unbridled genius, Newton
1:36explained the motion of almost
1:38everything in and on the surface of the
1:41Earth. Newton's three laws, refined to
1:47their core, are an enormously powerful
1:50equation. Force equals mass times
1:53acceleration. Understanding this
1:56equation well, what it means, and how
1:59to use it is the only way to understand
2:02the mechanical universe. At the same
2:06time, understanding its enormous
2:08complexity may lie in the fact that it
2:10seems pure and simple. For example, if
2:16we look at f = a m*a, it immediately
2:20reveals two complications. First, f = m
2:30*a is a vector equation. Both force and
2:37acceleration are vectors, in other
2:40words, they have definite directions.
2:45In f = a m times a they must have the
2:48same direction. Second. The second
2:53complication arises with the symbol A
2:56for acceleration. Remember,
2:58acceleration is not the position of
3:00something, nor is it the rate at which
3:02something changes position.
3:05Acceleration is the rate at which
3:07something's speed changes. Thus,
3:12acceleration is the derivative of a
3:15derivative, and Newton's simple
3:17equation F = m*a is a vector equation
3:20over the derivative of a derivative.
3:26For example, when a body or mass falls,
3:30gravity exerts a force F, a downward
3:33force, of course, and the result is
3:37acceleration. There it is. F = M*A.
3:56Acceleration and force in the same
3:58direction. Beautifully complicated in
4:02its simplicity, this is the well-known
4:05equation. Now, in the case of a falling
4:08body, what is known as acceleration? It
4:14is constant for one thing and for
4:16another. It is the same for all falling
4:19bodies, and what is more, it is called
4:26G. The force of gravity on any falling
4:35body is equal to its mass times the
4:41acceleration G in the downward
4:44direction. But this equation also
4:49describes motion in a direction other
4:52than downward, because gravity exerts
4:55the same force on a body regardless of
4:58the direction in which the body is
5:01moving. Once an object is in motion,
5:10apart from air resistance, the only
5:13force acting on it is the force of
5:16gravity. Whether it is falling from a
5:21tower, being thrown off a mountain, or
5:25being dropped by a tall man. All moving
5:34bodies fall under the influence of
5:36gravity. An equation that explains the
5:44trajectory of a projectile on Earth is
5:47undoubtedly very powerful, and to see
5:52how it really works, it is necessary to
5:55go back to the beginning. With his
6:07first law, Newton adopted the idea of
6:10inertia. He wrote, "Every body
6:14continues in its state of rest or of
6:17uniform rectilinear motion unless
6:20compelled to change that state by
6:24forces exerted on it." Inertia was an
6:28idea that Newton inherited from Galileo
6:32. It was an extraordinary idea which,
6:34illustrated with modern images, goes
6:37something like this. Once a body is in
6:45motion, it naturally continues in a
6:47straight line unless acted upon by some
6:50force. Newton's second law states
6:54exactly how a force changes the motion
6:56of an object. As Newton himself
7:04explained, the change in motion is
7:07proportional to the force applied. It
7:10is in the direction of the line in
7:12which the force is applied. Newton used
7:23the word motion when he meant momentum,
7:26the velocity of a body multiplied by
7:30its mass. Newton expressed himself in
7:34classical geometry using geometric
7:36figures and ratios of quantities. He
7:39knew that his laws could be expressed
7:42as differential equations.
7:46Newton's second law states that the
7:48applied force is equal to the rate of
7:51change of momentum. For a body of
8:01constant mass, this differential
8:03equation reduces to force equals mass
8:07times acceleration. F = a m times a.
8:20For every action there is always an
8:24equal reaction. That is, the mutual
8:29actions of two bodies are always equal
8:32and directed in opposite directions.
8:37That is Newton's third law. A body
8:44cannot touch another body without being
8:47touched at the same time. Or to put it
8:50another way, bodies not only act, they
8:53always interact with each other.
8:57Newton's three laws act and interact
8:59simultaneously throughout the physical
9:02world, and the trajectory of a
9:04projectile demonstrates in a powerful
9:07and exciting way the consequences of
9:09Newton's three laws. When an object is
9:13thrown and allowed to move freely, what
9:16is the nature of its trajectory? It
9:24does not matter what kind of projectile
9:27or its purpose. Could it be that all
9:52trajectories are essentially the same?
9:55Understanding the trajectory of a
9:57projectile arises from an insight into
9:59the laws governing force and motion.
10:02Someone credited in ancient times with
10:05such insight and firsthand knowledge of
10:08projectiles was David, best remembered
10:11for his legendary showdown with Goliath
10:14. Despite considerable odds, young
10:20David demonstrated the relationship.
10:22between force and momentum. In fact,
10:25perhaps David and Goliath were not as
10:27unequal as legend makes them out to be.
10:30After all, is this the model of
10:32overwhelming talent? In any case, David
10:37was able to demonstrate force and
10:39momentum with great skill, and without
10:42the help of Aristotle, much less Newton
10:44. Goliath understood it and he
10:48immediately obeyed the law of gravity.
10:56Another phenomenon that was already
10:58acting perfectly long before it was
10:59understood. Newton's laws did not
11:03change the way the world works, but
11:06they changed forever the way we
11:08understand it. And whether it is on a
11:11battlefield, a sports field or the
11:14green pastures of England, the
11:16trajectory of a projectile can always
11:19be described in the field of
11:22mathematics. In Newton's own words,
11:26because from these forces I deduced the
11:30motion of the planets, comets, the moon
11:33and the sea. Newton was not the first
11:37to study the trajectory of an object
11:39flying on the earth or through the sky,
11:41but he was the only one who understood
11:43that the trajectory of both was
11:45governed by the same laws. Long before
11:50Newton, Galileo Galilei had described
11:52the motion of a projectile perfectly.
11:55He realized that bodies can fall
11:58vertically while simultaneously moving
12:00horizontally. According to Galileo's
12:04view, which spanned everything from the
12:06heavens to the paradisiacal gardens of
12:08the Renaissance, the motion of a body
12:11has two components completely
12:13independent of each other. Galileo's
12:17extraordinary insight is explained in
12:19Isaac Newton's extraordinary equation.
12:30The vertical component of the vector is
12:34mg downward or-mg. In the horizontal
12:40direction no force acts at all. Only
12:45the vertical component of acceleration
12:48is-g. In the horizontal direction,
12:52where there is no force, the
12:54acceleration is zero. Acceleration is
12:57the rate of change of velocity. Since
13:00the horizontal velocity does not change
13:03, it must be constant. Constant
13:07velocity in the horizontal direction,
13:10constant acceleration downward, both
13:14acting independently and simultaneously
13:17. These are the elements of Galileo's
13:24trajectories and likewise the result of
13:27Newton's equation. f = m*a. But to
13:33understand the progressive meaning of f
13:36= a m times a, we must go back to the
13:38times before that scholars would have
13:41help in trying to explain the phenomena
13:43of the world. In ancient Greece,
13:53scholars believed that everything in
13:55nature returned to a state of rest,
13:58that returning to a state of rest was
14:01the nature of all things in motion.
14:04According to the Aristotelians, all
14:07objects in motion are propelled by a
14:09mover. When the mover could not be seen
14:12, as in the case of the motion of a
14:14projectile, Aristotle argued that the
14:16air itself was responsible for such
14:17motion. Much later, although Europe in
14:20the Middle Ages maintained the
14:22Aristotelian view of the world, this
14:24explanation was not entirely
14:26satisfactory. To explain the isolated
14:32motion of projectiles, such as spears,
14:35arrows, or cannonballs, scholars
14:37arrived at the idea of impetus. When a
14:45projectile was launched, it had to be
14:47saturated with a finite amount of
14:49impetus that would give the object its
14:51motion. When its impetus was exhausted,
14:58the object would fall rapidly to the
15:00ground. The idea of impetus was not bad
15:03, but fell short of its goal. The
15:05medieval idea of momentum fell short of
15:08the principle of inertia, which did not
15:10catch on until the Renaissance. When
15:17Galileo discovered the true parabolic
15:20trajectory of a projectile, Galileo
15:22said, "It has been observed that
15:24bullets and projectiles describe a
15:26curved path of a certain type. However,
15:30no one has pointed out the fact that
15:34this path is a parabola." With such
15:38insight, Galileo lived to see the 2,000
15:41-year reign of the Aristotelian
15:43worldview come crashing down, and he
15:50was not alone. Around this time,
15:53Johannes Kepler, Christian Heigens in
15:57the Netherlands, René de Car in France
16:01, and others were also beginning to see
16:04the universe with new eyes. But as
16:10extraordinary as this collection of
16:13scientists was, their individual views
16:16missed something, a synthesis, a
16:19principle that organized the physical
16:22world as a whole. To explain the
16:28universe, special circumstances would
16:31have been needed, requiring the right
16:33person, the right place, and the right
16:36time. The right time was the year 1665
16:49and the right person was Isaac Newton.
16:54At the tender age of 23, Newton
16:56conceived the discoveries that would
16:59forever alter our understanding of the
17:01universe. With just three fundamental
17:05laws, Newton gave a cause to motion and
17:08in doing so, his dynamic principle
17:10completed Galileo's mathematical
17:12description of motion. In other words,
17:16just as Galileo's kinematics described
17:19motion, Newton's dynamics explained it.
17:24But Newton's laws also explain how a
17:26body moves. By combining his
17:28calculations with his mechanics, the
17:30motion of a moving object can be
17:32perfectly described. The motion of a
17:44projectile has an acceleration equal to
17:47-g in the vertical direction and a
17:49constant velocity in the horizontal
17:52direction.
17:54What is the meaning of this constant?
17:58The force that propels this object
18:00comes from the push of the athlete's
18:02torso. This force lasts only a moment
18:05and determines the initial velocity of
18:07the projectile. When it leaves the hand
18:10, there is a component of velocity in
18:13the X direction, which is the constant
18:15value that the horizontal velocity has
18:18and that it will continue to have
18:20throughout its journey until it hits
18:22the ground. However, the vertical
18:26component of velocity is changing at a
18:30rate equal to-g. The derivative of
18:36vertical velocity with respect to time
18:39is-g, no. What function does-g serve as
18:47the derivative? GT is an answer, but-gt
18:55plus any constant is also an answer,
18:58since the derivative of a constant is
19:02zero. What is the meaning of the
19:06constant? That constant is the vertical
19:18velocity with which the object starts
19:20even if it starts with a shot. Let's
19:27imagine for a moment that there were no
19:30gravity here. Then any projectile would
19:36follow a straight line trajectory with
19:38the same horizontal and vertical
19:39velocities that it had at the beginning
19:41. But gravity shapes the motion of the
19:47projectile and how it does so is
19:49expressed in Galileo's law of falling
19:52bodies. What actually happens is that a
19:56projectile falls continuously below
19:59that imaginary straight line trajectory
20:02at a speed equal to 1/io of g*t²ad,
20:07exactly as if it had been dropped from
20:12rest. All these meritorious opinions of
20:16Galileo are the simple and direct
20:18consequence of a concise law of Newton.
20:22summarized in the equation f = a m
20:27times a. One more version of David and
20:33Goliath. I am David, of course, and
20:37there is Goliath hanging from that tree
20:39. Although this time, instead of a wave
20:43, I am going to take him down with a
20:45sophisticated projectile. Okay, but
20:49there is a problem. I know that the
20:52moment I pull the trigger, Goliath will
20:55get scared, leave the tree he is in and
20:57fall vertically. The question in this
21:00case is, where should I aim? At what
21:03point should I shoot to hit him? I
21:07already have the answer. I will do it
21:09like a good hunter. I will write the
21:10equations of the projectile. We have to
21:14know Z and X. We know, however, that dx
21:18times dt, because that is exactly v sub
21:23z, is equal to gt + v sub z0; and dx
21:27times dt, because that is exactly v sub
21:31x, is ig to v sub x0 not -1/2 of g
22:00squared. Now, without the effect of
22:06gravity, these equations would be those
22:08of a straight line, a straight line
22:10that would go from the barrel to the
22:13point that we have chosen as the target
22:15. The effect of gravity is to make the
22:18bullet fall below that straight line
22:21with a value equal to 1/io of GT². The
22:25actual trajectory would be this, the
22:29exact same distance traveled by the
22:31monkey when falling. Assuming that it
22:33is headed towards where the monkey was
22:34in the first place. It will fall
22:38exactly the same one half of a squared
22:40and will hit the bullet in mid-air.
22:43What do we want to achieve? The
22:45solution to the problem. Aim the bullet
22:47directly at the monkey. And that is
22:50what I am going to do. I am going to
22:52shoot that monkey. If anyone wants to
22:55know why I am going to shoot a monkey,
22:58don't do it. This is not an ethics
22:59class, it is a physics class. The laser
23:03beam, as you can see, passes through
23:05the sights of the weapon, here and here
23:08, and hits the monkey. The weapon is
23:10pointed directly at the monkey. Now I
23:14load the projectile launcher and we are
23:18ready to shoot. Okay, I have to tell
23:22you that I've practiced this many, many
23:24times, and in the hundreds of times
23:26I've done this test, not once have I
23:28managed to beat the monkey. That's why
23:31he volunteers. Okay, let's go. Are you
23:34ready? There we go. This is the first
23:47time this monkey has been defeated by
23:50an equation. But which equation? If you
23:55were to ask an average physicist on the
23:57street, what is the most important
23:58simple equation in all of physics? They
24:01would probably answer f = m*a. There's
24:09something even more mysterious about
24:10that equation. It consists of three
24:13quantities: force, mass, and
24:15acceleration. We have a pretty good
24:19understanding of what acceleration
24:21means, but what is force? And what do
24:24we mean by the word mass? One way to
24:28find out is to use the equation to tell
24:31us. That's what we've done today in the
24:35case of gravity. We haven't discovered
24:37the nature of gravity or why it acts
24:39the way it does. We've used this
24:42equation to find out what force is
24:44acting on a body. It doesn't tell us
24:46anything about gravity or how it acts.
24:49If we know the mass of a body and
24:51observe its acceleration, we can find
24:53the force acting on it. But without
24:56this equation we do not know the mass
24:58of a body. And if we do not know the
25:00mass of a body, the equation means
25:02nothing at all. Possibly the most
25:05important equation in all of physics is
25:07a meaningless logical absurdity. And
25:10that is represented in words. This
25:13problem has been debated by
25:15philosophers for over 300 years. And so
25:19far one thing is clear. Before Newton
25:22wrote this equation, the world was a
25:25world full of confusion. After he wrote
25:29it, the world became orderly and
25:32comprehensible and predictable. So
25:35whatever it is, this equation is not
25:38meaningless. The only way to understand
25:42what this equation is about is to use
25:45it, and that is what the rest of our
25:48story will be about. Force equals mass
25:54times acceleration. Mechanics, the
25:58science of motion, can be summed up in
26:00that one equation. F = m times A.

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