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8.01x - Lect 6 - Newton's Laws — Transcript

by Lectures by Walter Lewin. They will make you ♥ Physics. · 6,073 words · 809 segments · language en · Watch on YouTube

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0:03Last time we discussed that an acceleration is caused
0:10by a push or by a pull.
0:14Today we will express this more qualitatively
0:17in three laws which are called Newton's Laws.
0:21The first law really goes back
0:23to the first part of the 17th century.
0:27It was Galileo who expressed
0:29what he called the law of inertia
0:32and I will read you his law.
0:37"A body at rest remains at rest
0:42"and a body in motion continues to move
0:45"at constant velocity along a straight line
0:49unless acted upon by an external force."
0:53And now I will read to you
0:55Newton's own words in his famous book, Principia.
1:01"Every body perseveres in its state of rest
1:06"or of uniform motion in a right line
1:10"unless it is compelled to change that state
1:15by forces impressed upon it."
1:20Now, Newton's First Law
1:22is clearly against our daily experiences.
1:25Things that move don't move along a straight line
1:28and don't continue to move, and the reason is, there's gravity.
1:31And there is another reason.
1:32Even if you remove gravity
1:35then there is friction, there's air drag.
1:38And so things will always come to a halt.
1:41But we believe, though, that in the absence of any forces
1:45indeed an object, if it had a certain velocity
1:48would continue along a straight line forever and ever and ever.
1:55Now, this law, this very fundamental law
1:59does not hold in all reference frames.
2:02For instance, it doesn't hold in a reference frame
2:06which itself is being accelerated.
2:09Imagine that I accelerate myself right here.
2:13Either I jump on my horse, or I take my bicycle
2:16or my motorcycle or my car
2:19and you see me being accelerated in this direction.
2:22And you sit there and you say, "Aha, his velocity is changing.
2:27"Therefore, according to the First Law,
2:29there must be a force on him."
2:31And you say, "Hey, there, do you feel that force?"
2:35And I said, "Yeah, I do!
2:36"I really feel that, I feel someone's pushing me."
2:39Consistent with the first law.
2:41Perfect, the First Law works for you.
2:43Now I'm here.
2:45I'm being accelerated in this direction
2:47and you all come towards me
2:49being accelerated in this direction.
2:50I say, "Aha, the First Law should work
2:53so these people should feel a push."
2:57I say, "Hey, there!
2:58Do you feel the push?"
2:59And you say, "I feel nothing.
3:01There is no push, there is no pull."
3:02Therefore, the First Law doesn't work from my frame of reference
3:06if I'm being accelerated towards you.
3:10So now comes the question, when does the First Law work?
3:15Well, the First Law works when the frame of reference
3:20is what we call an "inertial" frame of reference.
3:23And an inertial frame of reference would then be
3:25a frame in which there are no accelerations of any kind.
3:30Is that possible?
3:31Is 26.100... is this lecture hall
3:34an inertial reference frame?
3:36For one, the earth rotates about its own axis
3:40and 26.100 goes with it.
3:42That gives you a centripetal acceleration.
3:44Number two, the earth goes around the sun.
3:48That gives it a centripetal acceleration
3:50including the earth, including you, including 26.100.
3:54The sun goes around the Milky Way, and you can go on and on.
3:58So clearly 26.100 is not an inertial reference frame.
4:06We can try to make an estimate
4:09on how large these accelerations are
4:12that we experience here in 26.100
4:16and let's start with the one
4:18that is due to the earth's rotation.
4:20So here's the earth... rotating with angular velocity omega
4:27and here is the equator, and the earth has a certain radius.
4:34The radius of the earth... this is the symbol for earth.
4:38Now, I know that 26.100 is here
4:40but let's just take the worst case that you're on the equator.
4:44You're... (no audio )
4:46You go around like this and in order to do that
4:48you need a centripetal acceleration, a c
4:51which, as we have seen last time, equals omega squared R.
4:56How large is that one?
4:58Well, the period of rotation for the earth
5:01is 24 hours times 3,600 seconds
5:08so omega equals two pi divided by 24 times 3,600
5:15and that would then be in radians per second.
5:19And so you can calculate now what omega squared R earth is
5:23if you know that the radius of the earth
5:26is about 6,400 kilometers.
5:30Make sure you convert this to meters, of course.
5:34And you will find, then
5:36that the centripetal acceleration at the equator
5:38which is the worst case-- it's less here--
5:41is 0.034 meters per second squared.
5:46And this is way, way less-- this is 300 times smaller
5:51than the gravitational acceleration
5:53that you experience here on Earth.
5:56And if we take the motion of the earth around the sun
5:59then it is an additional factor of five times lower.
6:02In other words, these accelerations
6:04even though they're real and they can be measured easily
6:07with today's high-tech instrumentation--
6:10they are much, much lower than what we are used to
6:13which is the gravitational acceleration.
6:15And therefore, in spite of these accelerations
6:18we will accept this hall
6:21as a reasonably good inertial frame of reference
6:26in which the First Law then should hold.
6:30Can Newton's Law be proven?
6:33The answer is no, because it's impossible to be sure
6:39that your reference frame is without any accelerations.
6:42Do we believe in this?
6:44Yes, we do.
6:45We believe in it since it is consistent
6:47within the uncertainty of the measurements
6:50with all experiments that have been done.
6:55Now we come to the Second Law, Newton's Second Law.
7:02I have a spring...
7:09Forget gravity for now--
7:10you can do this somewhere in outer space.
7:11This is the relaxed length of the spring
7:14and I extend the spring.
7:17I extend it over a certain amount, a certain distance--
7:21unimportant how much.
7:23And I know that I when I do that that there will be a pull--
7:28non-negotiable.
7:31I put a mass, m1, here, and I measure the acceleration
7:37that this pull causes on this mass
7:39immediately after I release it.
7:40I can measure that.
7:41So I measure an acceleration, a1.
7:45Now I replace this object by mass m2
7:50but the extension is the same, so the pull must be same.
7:53The spring doesn't know what the mass is at the other end, right?
7:57So the pull is the same.
7:58I put m2 there, different mass
8:00and I measure the new acceleration, a2.
8:04It is now an experimental fact that m1 a1 equals m2 a2.
8:15And this product, ma, we call the force.
8:20That is our definition of force.
8:23So the same pull on a ten times larger mass
8:28would give a ten times lower acceleration.
8:33The Second Law I will read to you:
8:37"A force action on a body gives it an acceleration
8:42which is in the direction of the force..."
8:44That's also important--
8:46the acceleration is in the direction of the force.
8:49"And has a magnitude given by ma."
8:52ma is the magnitude
8:53and the direction is the direction of the force.
8:57And so now we will write this in all glorious detail.
9:02This is the Second Law by Newton
9:08perhaps the most important law in all of physics
9:13but certainly in all of 801:
9:16F equals ma.
9:20The units of this force
9:22are kilograms times meters per second squared.
9:28In honor of the great man, we call that "one newton."
9:35Like the First Law, the Second Law only holds
9:38in inertial reference frames.
9:42Can the Second Law be proven?
9:46No.
9:47Do we believe in it?
9:49Yes.
9:50Why do we believe in it?
9:52Because all experiments and all measurements
9:54within the uncertainty of the measurements
9:57are in agreement with the Second Law.
10:03Now you may object and you may say
10:07"This is strange, what you've been doing.
10:10"How can you ever determine a mass
10:12"if there is no force somewhere?
10:15"Because if you want to determine the mass
10:16"maybe you put it on a scale,
10:18"and when you put it on a scale to determine the mass
10:20"you made use of gravitational force
10:22"so isn't that some kind of a circular argument
10:24that you're using?"
10:25And your answer is "No."
10:29I can be somewhere in outer space
10:30where there is no gravity.
10:32I have two pieces of cheese; they are identical in size.
10:36This is cheese without holes, by the way.
10:38They are identical in size.
10:41The sum of the two has double the mass of one.
10:44Mass is determined by how many molecules--
10:46how many atoms I have.
10:47I don't need gravity to have a relative scale of masses
10:51so I can determine the relative scale of these masses
10:54without ever using the force.
10:56So this is a very legitimate way
10:59of checking up on the Second Law.
11:10Since all objects in this lecture hall and the earth
11:15fall with the constant acceleration, which is g
11:19we can write down that the gravitational force
11:24would be m times this acceleration, g.
11:28Normally I write an "a" for it, but I make an exception now
11:30because gravity, I call it "gravitational force."
11:35And so you see that the gravitational force
11:38due to the earth on a particular mass
11:41is linearly proportional with the mass.
11:44If the mass becomes ten times larger
11:46then the force due to gravity goes up by a factor of ten.
11:54Suppose I have here this softball in my hands.
11:59In the reference frame...
12:0126.100 we will accept to be an inertial reference frame.
12:05It's not being accelerated in our reference frame.
12:09That means the force on it must be zero.
12:13So here is that ball.
12:17And we know if it has mass, m--
12:19which in this case is about half a kilogram--
12:22that there must be a force here, mg
12:25which is about five newtons, or half a kilogram.
12:30But the net force is zero.
12:33Therefore it is very clear
12:36that I, Walter Lewin, must push up with a force
12:42from my hand onto the ball, which is about the same...
12:46which is exactly the same, five newtons.
12:48Only now is there no acceleration
12:52so I can write down that force of Walter Lewin
12:58plus the force of gravity equals zero.
13:03Because it's a one-dimensional problem
13:05you could say that the force of Walter Lewin equals minus mg.
13:14F equals ma.
13:17Notice that there is no statement made
13:21on velocity or speed.
13:24As long as you know F and as long as you know m
13:27a is uniquely specified.
13:29No information is needed on the speed.
13:32So that would mean, if we take gravity
13:35and an object was falling down with five meters per second
13:39that the law would hold.
13:41If it would fall down with 5,000 meters per second
13:47it would also hold.
13:49Will it always hold?
13:51No.
13:53Once your speed approaches the speed of light
13:57then Newtonian mechanics no longer works.
14:00Then you have to use Einstein's theory of special relativity.
14:03So this is only valid as long as we have speeds
14:07that are substantially smaller, say, than the speed of light.
14:13Now we come to Newton's Third Law:
14:19"If one object exerts a force on another
14:25"the other exerts the same force
14:27in opposite direction on the one."
14:31I'll read it again.
14:32"If one object exerts a force on another
14:38"the other exerts the same force
14:40in opposite direction on the one."
14:43And I normally summarize that as follows, the Third Law
14:52as "Action equals minus reaction."
15:00And the minus sign indicates, then, that it opposes
15:04so you sit on your seats
15:06and you are pulled down on your seats because of gravity
15:12and the seats will push back on you with the same force.
15:17Action equals minus reaction.
15:19I held the baseball in my hand.
15:23The baseball pushes on my hand with a certain force.
15:26I push on the baseball with the same force.
15:30I push against the wall with a certain force.
15:34The wall pushes back in the opposite direction
15:36with exactly the same force.
15:39The Third Law always holds.
15:41Whether the objects are moving or accelerated
15:45makes no difference.
15:46All moments in time, the force--
15:49we call it actually the "contact force" between two objects--
15:53one on the other is always the same as the other on one
15:56but in the opposite direction.
16:02Let us work out a very simple example.
16:07We have an object which has a mass, m1.
16:14We have object number one and m1 is five kilograms.
16:19And here, attached to it, is an object two
16:23and m2 equals 15 kilograms.
16:28There is a force
16:31and the force is coming in from this direction.
16:34This is the force--
16:36and the magnitude of the force is 20 newtons.
16:40What is the acceleration of this system?
16:43F equals ma.
16:48Clearly the mass is the sum of the two--
16:50this force acts on both--
16:52so we get m1 plus m2 times a.
16:58This is 20, this is 20
17:02so a equals one meters per second squared
17:06in the same direction as F.
17:09So the whole system is being accelerated
17:11with one meters per second squared.
17:14Now watch me closely.
17:15Now I single out this object--
17:19here it is... object number two.
17:25Object number one, while this acceleration takes place
17:30must be pushing on object number two.
17:32Otherwise object number two could never be accelerated.
17:35I call that force F12
17:40the force that one exerts on two.
17:44I know that number two has an acceleration of one.
17:47That's a given already.
17:51So here comes F equals ma.
17:54F12 equals m2 times a.
17:59We know a is one, we know m2 is 15
18:04so we see that the magnitude of the force 12 is 15 newtons.
18:12This force is 15.
18:17Now I'm going to isolate number one out.
18:23Here is number one.
18:27Number one experiences this force, F, which was the 20
18:34and it must experience a contact force from number two.
18:41Somehow, number two must be pushing on number one
18:45if one is pushing on number two.
18:47And I call that force "F21."
18:53I know that number one is being accelerated
18:56and I know the magnitude is one meter per second squared.
18:59That's non-negotiable,
19:01and so we have that F, this one, plus F21
19:09must be m1 times a.
19:13This is one, this is five, this is 20
19:17and so this one, you can already see, is minus 15.
19:23F21 is in this direction
19:26and the magnitude is exactly the same as F12.
19:31So you see?
19:33One is pushing on two with 15 newtons in this direction.
19:37Two is pushing back on one with 15 newtons
19:40and the whole system is being accelerated
19:44with one meter per second squared.
19:47Now, in these two examples--
19:49the one whereby I had the baseball on my hand--
19:53you saw that it was consistent with the Third Law.
19:57In this example, you also see
20:00that it's consistent with the Third Law.
20:02The contact force from one on the other
20:04is the same as from the other on one
20:05but in opposite signs.
20:06Is this a proof?
20:08No.
20:10Can the Third Law be proven?
20:12No.
20:13Do we believe in it?
20:15Yes.
20:16Why do we believe in it?
20:18Because all measurements, all experiments
20:21within the uncertainties are consistent with the Third Law.
20:29Action equals minus reaction.
20:32It is something that you experience every day.
20:36I remember I had a garden hose on the lawn
20:43and I would open the faucet
20:45and the garden hose would start to snake backwards.
20:47Why?
20:48Water squirts out.
20:50The garden hose pushes onto the water in this direction.
20:53The water pushes back onto the garden hose and it snakes back.
20:57Action equals minus reaction.
21:01You take a balloon.
21:04You take a balloon and you blow up the balloon
21:08and you let the air out.
21:10The balloon pushes onto the air.
21:12The air must push onto the balloon.
21:15And therefore, when you let it go
21:17the balloon will go in this direction
21:19which is the basic idea behind the rocket.
21:22(huffing and puffing)
21:26I love to play with balloons, don't you?
21:31So, if I do it like this, and I let it go
21:34the air will come out in this direction
21:36and so then it means the balloon
21:38is pushing on the air in this direction.
21:39the air must be pushing on the balloon in this direction.
21:42There it goes.
21:43(whistles)
21:44It didn't make it to the moon
21:45but you saw the idea of a rocket.
21:50Action equals minus reaction.
21:56If you fire a gun, the gun exerts a force on the bullet
22:03the bullet exerts an equal force on the gun
22:06which is called the recoil.
22:07You feel that in your hands and your shoulder.
22:13I have here a marvelous device
22:16which is a beautiful example of "action equals minus reaction."
22:20I show you from above what it looks like.
22:22You'll see more details later.
22:26This rotates about this axis rather freely--
22:29the axis is vertical--
22:31and we have here a reservoir of water, which we will heat up.
22:35It turns into steam
22:36and these are hollow tubes and the steam will squirt out.
22:39And so when the steam squirts out in this direction
22:45the tube exerts a force on the steam in this direction
22:50so the steam exerts an equal force in the opposite direction
22:54and so the thing will start to rotate like this.
22:59And I would like to demonstrate that.
23:12You can see it now there.
23:14With a little bit of luck, there you see it.
23:17So we're going to heat it.
23:21(torch hissing)
23:24Walking.
23:26When you walk, you push against the floor.
23:30The floor pushes back at you
23:33and if the floor wouldn't push back at you
23:36you couldn't even walk, you couldn't go forwards.
23:41If you walk on ice, very slippery--
23:43you can't go anywhere, because you can't push on the ice
23:47so the ice won't push back on you.
23:50That's another example where you see
23:52action equals minus reaction.
23:56This engine is called "Hero's engine."
24:00Hero, according to the Greek legend
24:04was a priestess of Aphrodite.
24:08Let's first look at it.
24:20She was a priestess of Aphrodite and her lover, Leander
24:27would swim across the Hellespont every night to be with her.
24:31And then one night the poor guy drowned
24:34and Hero threw herself into the sea.
24:38Very romantic thing to do
24:40but, of course, also not a very smart thing to do.
24:44On the other hand, it must have been a smart lady
24:47if she invented, really, this engine.
24:51Yesterday, I looked at the Web, "ask.com."
24:58It's wonderful-- you can ask any question.
25:00You can say, "How old am I?"
25:02Now, you may not get the right answer
25:03but you can ask any question.
25:05And I typed in, "Hero's engine."
25:08And out popped a very nice high- tech version of Hero's engine.
25:16A soda can-- you pop four holes in the soda can at the bottom.
25:22So here's your soda can.
25:25You pop four holes in here, but when you put a nail in there
25:28you bend every time the nail to the same side
25:30so the holes are slanted.
25:34You put it in water
25:35you lift it out of water and you have a Hero's engine.
25:39And I made it for you-- it took me only five minutes.
25:42I went to one of MIT's machines, got myself a soda
25:49put the holes in it, and here it is.
25:52It's in the water there.
25:54When I lift it out, you will see the water squirts.
25:56There it goes.
25:59High-tech version of Hero's engine.
26:05Also makes a bit of a mess, but okay.
26:09All right.
26:14Try to make one-- it's fun and it's very quick.
26:17It doesn't take much time at all.
26:27There are some bizarre consequences of these laws.
26:34Imagine that an object is falling towards the earth.
26:38An apple is falling towards the earth
26:41from a height, say, of, hmm, I'd say 100 meters.
26:47And let's calculate how long it takes
26:49for this apple to hit the earth
26:52which should for you be trivial, of course.
26:55So here's the earth...
27:00and the mass of the earth
27:04is about 6 times 10 to the 24 kilograms.
27:10And here at a distance, h--
27:12for which we will take 100 meters--
27:14is this apple, m, which, say, has a mass of half a kilogram.
27:21There's a force from the earth onto the apple
27:26and this is that force.
27:28And the magnitude of that force is mg and that is 5 newton.
27:36I make g ten and just round it off a little.
27:40Now, how long does it take this object to hit the earth?
27:47So, we know that 1/2 gt squared equals h.
27:54It doesn't start with any initial speed, so that is 100.
27:59g is 10, this is 5, so t squared is 20.
28:03So t is about 4½ seconds.
28:08So after 4½ seconds, it hits the earth-- so far, so good.
28:12But now, according to the Third Law
28:16the earth must experience
28:18exactly the same force as the apple does
28:20but in opposite direction.
28:23So therefore the earth will experience this same force, F--
28:305 newton, in this direction.
28:33What is the earth going to do?
28:35Well, the earth is going to fall towards the apple-- F equals ma.
28:42So the force on the earth is the mass of the earth
28:47times the acceleration of the earth.
28:50The force, we know, is 5.
28:52We know the mass, 6 times 10 to the 24
28:55so the acceleration will be 5 divided by 6 times 10 to the 24
29:02which is about 8 times 10
29:04to the minus 25 meters per second squared.
29:12How long will the earth fall?
29:14Well, the earth will fall roughly 4½ seconds
29:17before they collide.
29:20How far does the earth move in the 4½ seconds?
29:23Well, it moves one-half a earth t squared.
29:30That's the distance that it moves.
29:32We know a and we know t squared, which is 20.
29:36One-half times 20 is 10
29:39so that means this distance becomes that number times 10.
29:42It's about 8 times 10 to the minus 24 meters.
29:47The earth moves 8 times 10 to the minus 24 meters.
29:52That, of course, is impossible to measure.
29:57But just imagine what a wonderful concept this is!
30:03When this ball falls back to me
30:08the earth and you and I and MIT are falling towards the ball.
30:15Every time that the ball comes down
30:18we're falling towards the ball.
30:20Imagine the power I have over you and over the earth!
30:24But you may want to think about this--
30:27if I throw the ball up, going to be away from the earth
30:32I'll bet you anything
30:33that the earth will also go away from the ball.
30:36So as I do this, casually playing--
30:40believe me, man, what a glorious feeling it is--
30:42earth is going down, earth is coming towards the ball.
30:46The earth is going down and I'm part of the earth
30:48and I'm shaking this earth up and down
30:51by simply playing with this ball.
30:54That is the consequence of Newton's Third Law
30:58even though the amount by which the earth moves
31:01is, of course, too small to be measured.
31:07I now want to work out with you a rather detailed example
31:15of something in which we combine what we have learned today--
31:20a down-to-earth problem--
31:22the kind of a problem that you might see
31:25on an exam or on an assignment.
31:29We hang an object on two strings
31:37and one string makes an angle of 60 degrees with the vertical
31:46and the other makes an angle of 45 degrees with the vertical.
31:50So this is the one that makes an angle...
31:56oh, 60 degrees with the horizon, 30 degrees with the vertical
32:00and this one, 45 degrees.
32:05Let's assume that the strings have negligible mass.
32:09So they are attached here to the ceiling
32:12and I hang here an object, m.
32:16Well, if there's an object m
32:20for sure there will be a force mg, gravitational force.
32:28This object is hanging there, it's not being accelerated
32:31so the net acceleration must be zero.
32:35And so one string must be pulling in this direction
32:37and the other string must be pulling in this direction
32:40so that the net force on the system is zero.
32:45Let's call this pull, for now, "T1."
32:49We'll call that the tension in the string
32:51and we call the tension in this string "T2."
32:58And the question now is how large is T1 and how large is T2?
33:04There are various ways you can do this.
33:06One way that always works-- pretty safe--
33:10you call this the x direction.
33:13You may choose which direction you call "plus."
33:16I call this plus, I call this negative.
33:20And you could call this the y direction
33:23and you may call this plus and this negative.
33:27I know, from Newton's Second Law-- F equals ma--
33:36that there is no acceleration, so this must be zero
33:40so the sum of all forces on that mass must be zero.
33:46These three forces must eat each other up, so to speak.
33:51Well, if that's the case, then the sum of all forces
33:54in the x direction must also be zero
33:56because there's no acceleration in the x direction
33:58and the sum of all forces in the y direction must be zero.
34:02And so I am going to decompose them--
34:04something we have done before.
34:06I am going to decompose the forces
34:08into an x and into a y direction.
34:14So here comes the x component of T1
34:21and its magnitude is T1 times the cosine of 60 degrees.
34:37Now I want to know what this one is.
34:45This one is T1 times the sine of 60 degrees.
34:56This projection, T2, cosine 45 degrees
35:06and the y component, T2 times the sine of 45 degrees.
35:15So we go into the x direction.
35:18In the x direction I have T1 cosine 60 degrees
35:26minus T2 cosine 45 degrees equals zero--
35:34that's one equation.
35:36The cosine of 60 degrees is one-half
35:42and the cosine of 45 degrees is one-half square root two.
35:48Now I go to the y direction.
35:51This is plus, this is minus, so we get one component here
35:56which is T1 times the sine of 60 degrees
36:02plus T2 times the sine of 45 degrees minus mg.
36:09It's in the opposite direction-- must be zero.
36:13That's my second equation.
36:15The sine of 60 degrees equals one-half the square root three
36:26and the sine of 45 degrees
36:28is the same as the cosine one-half square root two.
36:33Notice I have two equations with two unknowns.
36:36If you tell me what m is
36:38I should be able to solve for T1 and for T2.
36:41In fact, if we add them up
36:43it's going to be very easy because we lose this
36:46because we have both one-half square root two.
36:49And so you see immediately here that one-half times T1
36:55plus one-half square root three times T1 equals mg
37:03and so you find that the tension 1 equals two mg
37:10divided by one plus the square root of three.
37:16I can go back now to this equation--
37:20T1 times one-half
37:23equals T2 times one-half square root of two.
37:29I lose my half
37:31and so T2 equals T1 divided by the square root of two.
37:37So the bottom line is, you tell me what m is
37:40I'll tell you what T1 is and I'll tell you what T2 is.
37:43Suppose we take a mass of four kilograms--
37:47m equals four kilograms, so mg is about 40
37:54if we make g ten for simplicity.
37:57Then T1, if you put in the numbers, is about 29.3
38:03and T2... 29.3 newtons
38:08and T2 is about 20.7 newtons, I believe.
38:16It's very difficult to rig this up as an experiment
38:21but I've tried that.
38:22I'll show you in a minute.
38:25I want you to know that there is another method
38:29which is perhaps even more elegant
38:33and which you may consider
38:36in which there is no decomposition
38:39in the two directions.
38:43Here is mg-- that's a given.
38:48And we know that the other directions are also given--
38:52this angle of 30 degrees here and this angle of 45 degrees.
38:59If these two forces must cancel out this one
39:02why don't I flip this one over?
39:05Here it comes.
39:08I flip it over.
39:11There it is.
39:13T1 and T2 now, together, must add up to this one.
39:17Then the problem is solved, then the net force is zero.
39:21Well, that's easy-- I do this.
39:29And now I have constructed
39:32a complete fair construction of T1 and of T2.
39:39No physics anymore now, it's all over.
39:41You know this angle here, 45 degrees, so this is 45 degrees.
39:45This is 30, this is 30.
39:46You know all the angles and you know this magnitude is mg
39:49so it's a high school problem.
39:51You have a triangle with all the angles and one side;
39:53you can calculate the other sides
39:55and you should find exactly the same answer, of course.
40:00We made an attempt to rig it up.
40:03How do we measure tension?
40:04Well, we put in these lines, scales, tension meters
40:09and that is problematic, believe me.
40:12We put in here a tension meter, we put in here a tension meter
40:16and the bottom one, we hang on a string with a tension meter
40:24and then here we put four kilograms.
40:27These scales are not massles.
40:30That's already problematic.
40:32The scales are not very accurate
40:34so we may not even come close to these numbers.
40:38For sure, if I put four kilograms here
40:41then I would like this one to read 40 newtons
40:44or somewhere in that neighborhood
40:46depending on how accurate my meters are.
40:50These are springs, and the springs extend
40:53and when the springs extend, you see a handle... a hand go.
40:58You can clearly see how that works
41:01because if there is a force on that bottom scale
41:07in this direction, which is mg, and it's not being accelerated
41:12then the string must pull upwards
41:15and so... in order to make the net force zero.
41:18And if you have a pull down here and you have a pull up here
41:21and you have in here a spring
41:24then you see you have a way of measuring that force.
41:26We often do that--
41:27we measure with springs the tension in strings.
41:31For whatever it's worth, I will show you what we rigged up.
41:36Now a measurement without knowledge of uncertainties
41:39is meaningless-- I told you that.
41:42So maybe this is meaningless, what I am going to do now.
41:44Let me do something meaningless for once.
41:47And remember, when I show it, you can always close your eyes
41:52so that you haven't seen it.
41:54So we have here something that approaches this 60 degrees
41:59and this approaches the 45 degrees
42:02and we're going to hang four kilograms at the bottom.
42:09There it is, and here it is.
42:12All right, this one-- it's not too far from 40.
42:15It's not an embarrassment.
42:17This one is not too far from 20.7.
42:20This one is a bit on the low side.
42:21Maybe I can push it up a little.
42:23I think that's close to 30; it's not bad.
42:25So you see, it's very difficult to get these angles right
42:29but it's not too far off.
42:32So let's remove this again
42:34because this will block your view.
42:37These scales were calibrated in newtons, as you could see.
42:46Now we come to something very delicate.
42:52Now I need your alertness and I need your help.
43:00I have a block-- you see it there--
43:02and that block weighs two kilograms.
43:06A red block.
43:08So here it is.
43:10It's red.
43:13And I have two strings.
43:16It's hanging from a black string here and a black string there.
43:20Ignore that red string, that is just a safety.
43:23But it's avery thin thread here and here.
43:27And they are as close as we can make them the same.
43:29They come from the same batch.
43:34This one has a mass of two kilograms
43:36and this string has no mass.
43:40This is two kilograms.
43:43So what will be the tension in the upper string
43:46which is string number one?
43:49This is string number two.
43:51Well, this string must be able to carry this two kilograms
43:54so the tension has to be 20 newtons.
43:57So you will find here the tension-- call it T1--
44:01which is about 20 newtons.
44:07So it's pulling up on this object.
44:12It's also pulling down from the ceiling, by the way.
44:14Think about it, it's pulling from the ceiling.
44:18The tension is here, 20 newtons.
44:21We could put in here one of these scales
44:24and you would see approximately 20 newtons.
44:26What is the tension here?
44:28Well, the tension here is very close to zero.
44:31There's nothing hanging on it and the string has no weight
44:35so there's no tension there-- you can see that.
44:40Now I am going to pull on here
44:46and I'm going to increase the tension on the bottom one
44:52until one of the two breaks.
44:56So this tension goes up and up
45:01and therefore, since this object is not being accelerated--
45:06we're going to get a force down now on this object--
45:10this tension must increase, right?
45:13You see that?
45:14If I have a force on this one...
45:18so there's a force here, and there is mg
45:23then, of course, this string must now be mg plus this force.
45:28So the tension will go up here and the tension will go up here.
45:33The strings are as identical as they can be.
45:37Which of the strings will break first?
45:40What do you think?
45:43LEWIN: Excuse me?
45:44(student answers unintelligibly)
45:46I can't hear you.
45:47STUDENT: The one on top.
45:48LEWIN: The one on top.
45:49Who is in favor of the one on top?
45:53Who says no, the bottom one?
45:56(Student answers unintelligibly)
45:59LEWIN: Who says they won't break at all?
46:03Okay, let's take a look at it.
46:06The one on top-- that's the most likely, right?
46:11Three, two, one, zero.
46:17The bottom one broke.
46:21My goodness.
46:22Newton's Second Law is at stake.
46:24Newton's Third Law is at stake.
46:26The whole world is at stake!
46:29Something is not working.
46:32I increased tension here, this one didn't break.
46:37This one's stronger, perhaps.
46:38No, I don't cheat on you; I'm not a magician.
46:40I want to teach you physics.
46:45Did we overlook something?
46:46You know, I'll give you a second chance.
46:48We'll do it again.
46:50Let's have another vote.
46:53So I'll give you a chance to change your minds.
46:55It's nothing wrong in life, changing your mind.
46:57It's one of the greatest things that you can do.
47:03What do you think will happen now?
47:06Who is in favor still of the top one?
47:08Seeing is believing.
47:09You still insist on the top one?
47:11Who is now in favor of the bottom one?
47:13Ah, many of you got converted, right?
47:17Okay, there we go.
47:19Three, two, one, zero.
47:24The top one broke.
47:26So some of you were right.
47:27Now I'm getting so confused.
47:30I can't believe it anymore.
47:31First we argued that the top one should break
47:34but it didn't-- the bottom one broke.
47:37Then we had another vote and then the top one broke.
47:41Is someone pulling our leg?
47:43I suggest we do it one more time.
47:46I suggest we do it one more time
47:47and whatever's going to happen, that's the winner.
47:51If the top one breaks, that's the winner.
47:55If the bottom one breaks, well, then, we have to accept that.
47:59But I want you to vote again.
48:02I want you to vote again on this decisive measurement
48:07whether the top one will break first or the bottom one?
48:12Who is in favor of the top one?
48:17Many of you are scared, right?
48:18You're notvoting anymore!
48:20(class laughs)
48:21LEWIN: I can tell, you're not voting.
48:23Who is in favor of the bottom one?
48:26Only ten people are voting.
48:28(class laughs)
48:31LEWIN: Let's do this in an undemocratic way.
48:34You may decide-- what's your name?
48:37Alicia?
48:39Georgia, close enough.
48:40(laughter)
48:42You may decide whether the top one
48:44or the bottom one will break.
48:46Isn't that great?
48:47Doesn't it give you a fantastic amount of power?
48:53The bottom one.
48:55The bottom one.
48:58You ready?
48:59Three, two, one, zero.
49:01The bottom one broke.
49:03You were right.
49:04You will pass this course.
49:05Thank you, and see you Wednesday.
49:09By the way, think about this, think about this.

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