physics-6

Transcriber:	Julia Lien
Brief Bio:	a soon-to-be sernior in nehs.
Date finished:	7/10/05
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Last time, we discussed that an acceleration is caused by push or by pull. Today we will express this more qualitatively in three laws which are called Newton’s Laws. The first law really goes back to the first part of the 17^th century which was Galileo who expressed what he called the law of Inertia. And I will read you his law: a body at rest remains at rest, and a body in motion continues to move at constant velocity along a straight line unless acted upon by an external force. Now I will read to you Newton’s own words in his famous book Principia: "Every body perseveres in a state of rest or of uniform motion in a right line unless it is compelled to change that state by forces impressed upon it."

Now Newton’s First Law is clearly against our daily experiences. Things that move don’t move along a straight line and don’t continue to move, and the reason is there’s gravity. And there's another reason; even if you remove gravity, then there’s friction, there’s air drags, so things will always come to a halt. We believe though, that in the absence of any forces, indeed an object, if it had a certain velocity, will continue along a straight line forever and ever and ever.

Now this law, this very fundamental law, does not hold in all reference frames. For instance, it doesn't hold in a reference frame which itself is being accelerated. Imagine that I accelerate myself right here, either jump my horse, or take my bicycle, or my motorcycle, my car, and you see me being accelerated in this direction. And you sit there and you say, "Aha! His velocity is changing! Therefore, according to the first law, there must be a force on him. And you say, "Hey there! Do you feel that force?" And I say, "Yeah! I do! I really feel that... i feel someone is pushing me!" Consistent with the first law. Perfect. First law works for you.

Now I'm here. I'm being accelerated in this direction, and you all come towards me, being accelerated in this direction. And I say, "Aha! The first law should work, so these people should feel a push!" I say, "Hey there! Do you feel a push?" And you say, "I feel nothing." There is no push. There is no pull. Therefore, the first law doesn't work for my frame of reference if I'm being accelerated towards you.

So now comes the question: When does the first law work? Well, the first law works when the frame of reference is what we called an inertial frame of reference. An inertial frame of reference would then be a frame in which there are no accelerations of any kinds. Is that possible? Is 26-100, this lecture hall, an inertial refernce frame? For one, the earth rotates about its own axis, and 26-100 goes with it. That gives you a scientifical acceleration. Number two, the earth goes around the sun. That gives it scientifical acceleration, including earth, including you, including 26-100. The sun goes around the milky way, and you can go on and on. So, clearly, 26-100 is not an inertial reference frame.

We can try to make an estimate on how large these acceleration are, that we experience here in 26-100. And let's start with the one that is due to the earth's rotation. So here's the earth rotating with angular velocity, w (omega). And here's the equator, and earth has a certain radius. Radius of the earth . This is the symbol for earth. Now I know 26-100 is here, but let's just take the worst case, that you're on the equator. You go around like this. In order to do that, you need the scientifical acceleration, ac, which as we have seen last time, equals w²R. How large is that one? Well, the period of rotation for the earth is 24 hours times 3600 seconds. So w = pi /(24 * 3600), and that would then be in radiance per second, and so you can calculate now what w²R earth is if you know that the radius of the earth is about 6400 km. Make sure you convert these to meters, of course. And you will find then that the scientific acceleration at the equator, which is the worse case since it's less here, is 0.034 m/s². And this is way, way less; this is 300 times smaller than the gravitational acceleration that you experience here on earth. And if we take the motion of the earth around the sun, then it is an additional factor of 5 times lower. Another words, these acceleration, even though they're real and can be measured easily with today's high tech instrumentation, they're much much lower than what we're used to, which is the gravitational acceleration. And therefore, in spite these acceleration, we will accept these whole as a reasonably good inertial frame of reference in which the first law then should hold.

[6:30]

Can Newton's Law be proven? The answer is "no" because it's imppossible to be sure that your referece frame is without any acceleration. Do we believe in this? Yes, we do. We believe in it since it is consistent within the uncertainty of the measurements which all experiments that have been done. Now we come to the second law. Newton's second law.

I have a spring. Forget gravity for now you can do this one in outer space. This is the relax lane of the spring, and I extend the spring. I extend it over a certain amount, a certain distance, unimportant how much. And I know that when I do that, that there will be a pull, not negotiable. I pull the mass one here, and I measure the acceleration that this pull causes on this mass immediately after I release it. I can measure that. So I measure the acceleration a₁. Now I replace this object by mass m₂, but the extension is the same. So the pull must be the same. The spring doesn't know what the mass is at the other end, right? So the pull is the same. I put m₂ there, different mass, and I measure new acceleration a₂. It is now an experiemntal fact that m₁a₁= m₂a₂. And this product ma, we call, the force. That is our definition of force. So the same pull on a ten times larger mass will give a ten times lower acceleration.

The second law I will read to you. A force action on a body gives it an acceleration which is in the direction of the force (that's also important) and has an magnitude given by ma. ma is the magnitude, and the direction is the direction of the force. So now we will write this in all glories. Detailed as this is, the second law by Newton, perhaps the most important law in all of physics and certainly in all of 801. F = ma. The units of this force are kg * m / s² in honor of the grave man, we call that one Newton. Like the first law, the second law only holds in inertial reference frames. Can the second law be proven? No. Do we believe in it? Yes. Why do we believe in it? Because all experiments and all measurements within the uncertainty of the measurements are in agreement with the second law. Now you may object, and you may say, this is strange what you've been doing. How can you ever determine a mass if there's no force somewhere? Because if you want to determine the mass maybe you put it on a scale, and when you put it on a scale to determine the mass you make use of the gravitational force, so isn't that some kind of circular argument that you're using? And your answer is no. I can be somewhere in outer space where there is no gravity. I have two pieces of cheese. They are identical in size. This is cheese without holes, by the way. They are identical in size. The sum of the two has double the mass of one. Mass is determined by how many molecules, how many atoms I have. I don't need gravity to have a relative scale of masses, so I can determine the relative scale of these masses without ever using the force. So, this is a very legitimate way of checking up on the second law.

(11:08) Since all objects, in this lecture hall, on the earth, fall with the constant acceleration, which is g, we can write down that the gravitational force would be m times this acceleration g, normally I write an a for it, but I make an exception now because gravity I call it gravitational force. So you see that the gravitational force due to the earth on a particular mass is linearly proportional with the mass. If the mass becomes ten times larger, then the force due to gravity goes up by a factor of ten.

This ball that I have here, this softball in my hand, in the reference frame 26-100 we will accept to be an inertial reference frame. It's not being accelerated in the reference frame. That means the force on it must be zero. So here is the ball, and we know that it has mass and, which is in this case about half a kilogram, that there must be a force here, mg, which is about 5 Newton for half a kilogram. But the the net force is zero. Therefore, it is very clear that I, Walter Lewin, must push up with a force from my hand onto the ball, which is about the same, exactly the same, 5 Newtons, only now there's no accleration. So I can write down that force of Walter Lewin plus force of gravity equals zero. Because it is a one dimensional problem, you could say that the force of Walter Lewin equals -mg.

(13:13) F= ma. Notice that there's no statement made on velocity or speed. As long as you know f and as long as you know m, a is uniquely specified. No information is needed on the speed. But that would mean if we take gravity of an object that is falling down with 5 m/s that the law will hold, if it would fall down with 5000 m/s it would also hold. Would it always hold? No. Once you speed approaches the speed of light, then the Newtonian mechanics no longer works, then you would have to use Einstein's theory of special relativity. So it is only valid as long as you have speeds that are substantially smaller, say, than the speed of light.

Now we come to Newton's third law. If one object exerts a force on another, the other exerts the same force in the opposite direction on the one. I read it again. If one object exerts a force on another, the other exerts the same force in opposite direction on the one. And I normally summarize that as follows: the third law as action = -reaction. And the minus sign indicates that it opposes.

So you sit on your seat and you pull down on your seat because of gravity, and the seat will push back on you with the same force. Action = -reaction. I held a baseball in my hand. The baseball pushes on my hand, I push on the baseball with the same force. I push against the wall with a certain force, the wall pushes back in the opposite direction with exactly same force. The third law always holds. Whether the objects are moving or accelerated makes no difference. All moments in time, the force, we call it the contact force between two objects, one on the other is exactly the same as the other one, but in the opposite direction.

Let us work out a very simple example. We have an object. This has the mass m₁. It is object number 1, and m1 is 5 kg. And here, attached to it, is object 2, and m₂ equals 15 kg. There is a force, and the force is coming in in this direction. This is the force, and the magnitude of the force is 20 N. What is the acceleration of this system? F = ma. Clearly, the mass is the sum of the two. This force acts on both. So we get (m₁ + m₂)a. This is twenty, this is twenty, so a equals 1 m/s² in the same direction as F. So the whole system is being accelerated with 1 m/s².

Now watch me closely. Now I single out this object. Here it is, object 2. Object 1, while this acceleration takes place, must be pushing on object 2, otherwise object 2 could never be accelerated. I call that force F₁₂. Force at 1 exerts on 2. I know that number 2 has an acceleration of 1; that's a given already. So here comes F = ma. F₁₂ equals m₂ times a. We know F is 1, we know F₂ is 50. So we see the magnitude of the force ₁₂ is 15 N. This force is 15 N. Now I'm going to isolate number 1 out. Here is number 1. Number 1 experiences this force, F, which was 20, and it must experience a contact force from number 2. Somehow number 2 must be pushing on number 1 if 1 is pushing on number 2. And I call that force F₂₁. I know that number 1 is being accelerated and I know the magnitude is 1 m/s²; it's not negotiable. So we have that F, this one, plus F₂₁ must be m₁ times a. This is 1, this is 5, this is 20; so this one, you can already see, -15. F₂₁, in this direction, and the magnitude is exactly the same as F₁₂. Well you see, 1 is pushing on 2 with 15 N in this direction; 2 is pushing back on 1 with 15 N, and the whole system is being accelerated with 1 m/s².

In these two examples, the one whereby I had the baseball on my hand, you saw that it was consistent with the third law. In this example, you also see that it is consistent with the third law. The contact force from one on the other is the same as from the other one but with opposite signs. Is this approved? No. Can the third law be proven? No. Do we believe in it? Yes. Why do we believe in it? Because all measurements, all experiments within the uncertainty are consistent with the third law. Action equals minus reaction. It is something that you experience everyday. I remember I had a garden hose in the lawn, and I would open the faucet, and the garden hose would start to snake backwards. Why? Water squirts out; the garden hose pushes onto the water in this direction, the water pushes back on the garden hose, and it snakes back. Action equals minus reaction.

You take a balloon. You take a balloon and you blow at the balloon. And you let the air out. Balloon pushes onto the air, the air must push onto the balloon, and therefore, when you let it go, the balloon will go in this direction, which is the basic idea behind a rocket. _{I love to play with balloons, don't you? :)} So, if I do it like this and I let it go, the air will come out in this direction, and so then, it means the balloon is pushing on the air in this direction, the air must be pushing on the balloon in this direction. There it goes! Didn't make it to the moon, but you saw the idea of a rocket. Action equals minus reaction.

If you fire a gun, the gun exerts a force on the bullet, the bullet exerts equal force on the gun, which is called a recoil. You feel it. Your hands, your shoulder. I have here a marvelous device, which is a beautiful example of action equals minus reaction. I show you from above what it looks like, you will see more details later. It rotates about this axis, while really the axis is vertical. And we have here a vial of water which we will heat up, turn into steam, and these are hollow tubes, and the steam will squirt out. So when the steam squirts out in this direction, the tube exerts a force on the steam in this direction, the steam exerts an equal force in the opposite direction, and so, this thing will start to rotate, like this. And I would like to demonstrate that. See it now there? With a little bit of luck there you'll see it. So, I'm gonna heat it. Walking! When you walk, you push against the floor. The floor pushes back at you, and if the floor doesn't push back at you, you couldn't even walk, you couldn't go forward. If you walk on ice, very slippery, you can't go anywhere. Because you can't push on the ice, so the ice can't push back on you. That's another example where you see action equals minus reaction.

This engine is called Hero's engine. Hero, according to the Greek legend, was a priestess of Aphrodite. She was a priestess of Aphrodite, and her lover, Leander, would swim across the Herring Pond every night to be with her. And one night, the poor guy drowned, and Hero threw herself to the sea. A very romantic thing to do, but of course, not a very smart thing to do. On the other hand, she must have been a really smart lady, if, she invented, really, this engine.

Yesterday, I looked at the web....Ask[dot]com. It's wonderful. You can ask any question. You can say "how old am I?" Now, you may not get the right answer, but you can ask any question. And I typed in "Hero's engine," and out popped a very nice, high-tech version of Hero's engine. The soda can, you pop it, put a hole in the soda can at the bottom. So here's your soda can, you pop four holes in here, but when you put the nail in there, you bend, everytime, the nail to the same side so the holes are slanted. You put it in water, you lift it out of water, and you have an Hero's engine. And I made it for you; took me only five minutes. I went through one of MIT's machines, got myself a soda, put the holes in it, and here it is. It's in the water there, and when I lift it out, you'll see the water squirts. There it goes---- high-tech version of Hero's engine. Also makes a bit of a mess, but it's okay. All right. Try to make one; it's fun, and it's very quick. Doesn't take much time at all.

There are some bizarre consequences of these loss. Imagine that an object is falling towards the earth...an apple is falling towards the earth from a height, say, of 100 m. Now let's calculate how long it takes for this apple to hit the earth, make sure you'll be thrilled, of course. So here's the earth, and the mass of the earth is about 6 * 10²⁴ kg. And here are the distance h, for which we'll take 100m. This is apple m, which say, have a mass of 0.5 kg. There's a force from the earth onto the apple. Now this is that force, and the magnitude of that force is mg, and that is 5 N. I take g 10. Now, how long does it take this object to hit the Earth? So we know that 1/2 gt² = h. Doesn't start with any initial speed, so that is 100. This is 10, this is 5, so t² = 20. So t is about 4 .5 seconds. So after 4.5 seconds, it hits the earth. So far, so good. But now, according to the third law, the Earth must experience exactly the same force as the apple does, but in the opposite direction. So therefore, the earth will experience the same force, F, 5N, in this direction. What is the Earth going to do? Well, the Earth is going to fall towards the apple. F=ma. So the force of the earth is the mass of the earth times the acceleration of the earth. The force, we know, is 5. We know the mass (6*10²⁴). So the acceleration will be 5/(6*10²⁴), which is about 8*10^-25 m/s². How long will the earth fall? Well, the earth will fall roughly 4 1/2 s before they collide. How far did the Earth move in 4.5 s? Well, it moves 1/2 a_eartht², the distance that it moves. We know a and we know t² which is 20. 1/2 * 20 is 10. So that means the distance, which is that number times 10, which is about 8 * 10^-24 m. The Earth moves about 8 * 10^-24 m. That, of course, is impossible to measure.

But just imagine, what a wonderful concept this is. When this ball falls back to me, the Earth and you and I and MIT are falling towards the ball. Everytime that the ball comes down, we're falling towards the ball. Imagine the power I have over you, over the Earth. But you may want to think about this. If I throw the ball up, it's going to be away from the Earth. I'll bet you anything that the Earth will also go away from the ball. So as I do this, casually, playing, believe me, man, what a gorgeous feeling it is. Earth is going down, Earth is coming towards the ball. The Earth is going down, and I'm part of the Earth. I'm shaking the Earth up and down by simply playing this ball. That is the consequence of Newton's third law even though the amount by which Earth moves is, of course, too small to be measured.

I now want to work out with you a rather detailed example of something with which we combine what we have learned today: a down-to-Earth problem, the kind of problem that you might see on an exam or on an assignment. We hang an object on two strings. And one string makes an angle of 60 degrees with the vertical, and the other makes an angle of 45 degrees with the vertical. So this is the one that makes the angle of 60 degrees with the horizon, 30 degrees with the vertical. And this one, 45 degrees. Let's assume that the string has negligible mass. So they are attached here to a ceiling, and I hang here object M. Well, if there's an object M, for sure there will be a force mg, gravitational force. This object, hanging there, is not being accelerated, so the net acceleration must be 0. So one string must be pulling in this direction and the other string must be pulling in this direction, so that the net force on the system is 0. Let's call this pull for now t₁, we call that the tension in this string. And we call the tension in this string t₂. And the question, always, how large is t₁, how large is t_2. In various ways you can do this. One way that always work, pretty safe. You call this the x direction. You may choose which direction to call plus. I call this plus, call this negative. And you could call this the y direction, and you may call this plus, and this negative. I know from Newton's second law, F = ma, that there's no acceleration, so this must be 0. So the sum of all forces on that mass must be 0. These three forces must eat each other up, so to speak. Well, if that's the case, then the sum of all forces in the x direction must also be 0 because there's no acceleration in the x direction. And the sum of all forces in the y direction must be 0. And so I'm going to decompose them, something we have done before. I'm going to decompose the forces into an x and into a y direction. So, here comes the x component of t₁ and its magnitude is t₁(cos 60). Now I want to know what this one is. This one is t1(sin 60). This projection t2(cos 45), and the y component t2(sin 45). So, we going to the x direction. In the x direction, I have t1(cos 60)- t2(cos 45)=0. That's one equation. The cos 60 is 1/2, and the cos 45 is 1/2 square-root 2. This is plus, this is minus, so we get one component here, which is t1(sin 60) + t2(sin 45) - mg, mg is in the opposite direction, must be 0. This is my second equation. sin 60 = 1/2 square-root 3. And sin 45 is the same as the cos, 1/2 square-root 2. Notice, I have two equations with two unknowns. If you tell me what m is, I should be able to solve for t1 and for t2. In fact, if we add them up, it's going to be very easy because we lose this, because we have both 1/2 square-root 2. And so you see, immediately here, that 1/2 t1 + 1/2 square-root 3 * t1 = mg. And so you find that the tension 1 equals 2mg/(1 + square-root 3). I can go back now to this equation, t1*1/2 = t2 *1/2 square-root 2. I lose my 1/2, so t2 = t1/square-root 2. So the bottom line is you tell me what m is, I'll tell you what t1 is, and I'll tell you what t2 is. Suppose we take a mass of 4 kg. m = 4 kg, and mg is around 40, we make g 10 for simplicity. Then, t1, if you put it into numbers it's about 29.3, and t2, 29.3 N, and t2 is about 20.7 N, I believe. It's very difficult to rig this up as an experiment. I've tried that. Saw it on the internet.

I want you to know that there's another method which is perhaps even more elegant, and which you may consider in which there's no decomposition in the two directions. Here is mg. That's a given. And we know that the other direction is also given. This angle is 30 degrees here, and this angle 45 degrees. If these two forces must cancel out this one, why don't I flip this one over? Here is comes. I flip it over. There it is. t1 and t2 now together must add up to this one. Then the problem is solved. Then the net force is 0. Well, that's easy. I do this. And now I have constructed a complete fair construction of t1 and of t2. No physics anymore now; it's over. You know this angle here,45 degrees so this is 45, this is 30, this is 30. You know all the angles and you know this magnitude is mg, so it's high school problem. You have a triangle, with all the angles on one side, you can calculate all the sides, and you should find exactly the same answer of course.

We made an attempt to rig it up. How do we mention tension? Well, we put in these lines scales tension meters, and that's problematic, believe me. We put in ehre a tesion meter, we put in here a tension meter, and the bottom one, we hang on a string with a tension meter, and here, we put 4 kg. These scales are not massless; that's already problematic. The scales are not very accurate. So, we may not even come close to these numbers. For sure, if I put 4kg here, then I would like this one to read 40 degrees or somewhere in that neighborhood, depending on how accurate my meters are. And these are springs, and springs expand, and when the springs expand, you see the handle, the hand go. You can clearly see how that works, because if there is, the force on that bottom scale in this direction, which is mg, is not being accelerated, then the string must pull upwards. And so in order to make the net force 0, and if you have a pull down here and you have a pull up here, and you have in here a string, then you see, you have a way of measuring that force. We often do that, we measure with springs the tension in strings. For whatever it's worth, I will show you what we rigged up. Now, a measurement without a knowledge of uncertainty is meaningless. I told you that. So maybe this is meaningless, what I'm going to do now. Let me do something meaningless for once. And remember, when I show it, you can always close your eyes. So you don't have to see it. So we have here, something that approaches 60 degrees, and this approaches 45 degrees, and we're going to hang 4 kg at the bottom. This and here it is. All right, this one is not too far from 40, it's not an embarrassment. This one is not too far from 20.7. This one is a bit on the low side. Maybe I can push it up a little. See, it gets close to 30, it's not bad. So you see, it's very difficult to get these angles right. But it's not too far off. So let's remove these again. Don't want these to block your view. These scales were calibrated in Newtons, as you can see.

Now we come to something very delicate. Now, I need your alertness and I need your help. I have a block, you see there, and the block weighs 2 kg. Red block. So here it is. It's red, and I have two strings it's hanging. There's a black string there and a red string there. Ignore that red string; that is just a safety. But they're very thin threads; here and here. And they are as close as we can make them the same; they are from the same batch. This one has a mass of 2kg and this string has no mass. This is 2 kg. So what will be the tension of the string which is string number one? This is string number two. Well this string must be able to carry 2 kg, so the tension must be 20 N. So you will find here tension of T1, which is about 20 N. So it's pulling up on this object. It's also pulling down from the ceiling by the way you think about it, it's pulling us. The tension is here, 20 N. We could put in here one of these scales here and you would see a quantity of 20 N. What is the tension here? Well, the tension here is very close to zero. Nothing hanging on it and the string has no weight so there's no tension there. You can see that. Now I'm going to pull on here and I'm going to increase the tension on the bottom one until one of the two breaks. So this tension goes up and up, and therefore, since this object is not being accelerated, you're gonna get a force down on this object. This tension must increase, right? You see that. If I have a force on this one...so there's a force here, and there is mg, then of course, this string now must equal to mg plus this force. So the tension will go up here, and the tension will go up here. The strings are as identical as they can be. Which of the strings will break first? What do you think?

[asks the students] Who is in favor of the one on top? Who says no to the bottom one? Who says they won't break at all? Okay. Let's take a look at it. The one on top. That's the most likely right? Three, two, one, zero. The bottom one broke. My goodness. Newton's second law is at stake, Newton's third law is at stake, the whole world is at stake. Something is not working. I increased the tension here. This one didn't break. This one is stronger perhaps? No, I don't cheat on you. I'm not a magician. I wanna teach you physics. Did we overlook something? You know? I'll give you a second chance. We'll do it again. Let's have another vote. So I'll giving you a chance to change your mind. There's nothing wrong about changing your mind. It's one of the greatest thing that you can do. What do you think will happen now? Who's in favor still of the top one? Seeing is believing. Still insists on the top one. Who is now in favor of the bottom one? Ah~ many of you got converted, right? Okay. There we go. Three, two, one, zero. The top one broke. So some of you were right. Now I'm getting so confused. I can't believe it anymore. First we argued that the top one would break, but it didn't; the bottom one broke. Then we had another vote, and the top one broke. Is someone pulling a rag? I suggest we do it one more time. I suggest we do it one more time, and whatever is going to happen that's the winner. If the top one breaks, that's the winner. If the bottom one breaks, well, we'll have to accept that. But I want you to vote again. I want you to vote again on this decisive measurement, whether the top one will break first or the bottom one. Who is in favor of the top one? Many of you are scared. You're not voting anymore! I can tell! You're not voting! Who is in favor of the bottom one? Only ten people are voting! Let's do this in an undemocratic way. You may decide. What's your name? Allisa? Georgia. Close enough. You may decide whether the top one or the bottom one will break. Isn't that great? Doesn't that give you a fantastic amount of power? The bottom one. The bottom one. You ready? Three, two, one, zero. The bottom one broke. You were right. You will pass this course. Thank you and see you Wednesday. By the way, think about this. Think about this.