README.md

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2	4	0:00:02	0:00:04	9 of the clock.
31	37	0:00:31	0:00:37	And I think we are in a super important part of the class.
38	45	0:00:38	0:00:45	We have introduced, you know, we spent the first couple of weeks talking about what acceleration was and dealing with that factor.
45	50	0:00:45	0:00:50	And now we are doing forces for the F equals ma.
50	55	0:00:50	0:00:55	Last time we introduced, let us see, we did tension.
56	59	0:00:56	0:00:59	the normal force, a spring force, and gravity.
60	64	0:01:00	0:01:04	So we did four forces in the last lecture, the very first time we introduced it.
65	70	0:01:05	0:01:10	And today, we are going to dedicate an entire lecture to friction.
71	73	0:01:11	0:01:13	Friction is a super important force.
73	81	0:01:13	0:01:21	And also, as you will see, static friction, you just have to be super careful with static friction.
81	83	0:01:21	0:01:23	And so that is going to be our emphasis.
83	89	0:01:23	0:01:29	As always, I ask the brightest students in the country, what are the things we should be paying attention to?
91	95	0:01:31	0:01:35	Yeah, one said, are we ever going to give you numbers for the coefficient of friction?
95	96	0:01:35	0:01:36	And the answer is yes.
97	103	0:01:37	0:01:43	Basically, almost all the time, we will give you a number, like 0.6 or something like that for it.
103	105	0:01:43	0:01:45	So that will be maybe more satisfying.
112	118	0:01:52	0:01:58	like a specific type of problem that we do a lot in physics, and we are going to get lots of practice on that.
119	122	0:01:59	0:02:02	Yeah, and this was another comment along those lines.
123	126	0:02:03	0:02:06	What equations to use in what situations.
126	132	0:02:06	0:02:12	So again, pretty much we are going to be using f equals ma all the time now.
132	135	0:02:12	0:02:15	So hopefully that is not bad.
135	137	0:02:15	0:02:17	Oh, you do not need to memorize them.
137	139	0:02:17	0:02:19	So we give you a formula sheet.
140	145	0:02:20	0:02:25	You can see what that formula sheet is, but you will have it for your quizzes and your exams.
146	149	0:02:26	0:02:29	You can look on the course website and it will show you what that is.
152	154	0:02:32	0:02:34	So let us go ahead and get started.
154	163	0:02:34	0:02:43	I am going to ask you if you have a laptop or even a tablet, if you put it flat so that people behind you do not see it, then you can use that to take notes.
164	165	0:02:44	0:02:45	And then, yeah, that is it.
165	166	0:02:45	0:02:46	Just one thing.
167	168	0:02:47	0:02:48	Let us make it happen.
176	179	0:02:56	0:02:59	is such an important idea.
179	182	0:02:59	0:03:02	I mean, it kind of dominates our understanding of motion.
182	184	0:03:02	0:03:04	We tend to think that things do not want to move.
184	185	0:03:04	0:03:05	They want to stop.
186	189	0:03:06	0:03:09	So if I give this block a push, it stops.
191	199	0:03:11	0:03:19	And the reason it is stopping is because of the way we would understand it with Newton is the reason it stops is because there is a frictional force.
200	208	0:03:20	0:03:28	And this frictional force opposes the relative motion between the block and the table.
209	210	0:03:29	0:03:30	So it is pushing back on it.
212	216	0:03:32	0:03:36	That is called kinetic friction, kinetic because there is relative motion.
217	219	0:03:37	0:03:39	We also have static friction.
220	225	0:03:40	0:03:45	And static friction is talking about, like here, this is not moving.
225	229	0:03:45	0:03:49	But we know gravity should be pulling it down here.
229	234	0:03:49	0:03:54	And the reason that it does not accelerate down this ramp is because of the frictional force.
234	239	0:03:54	0:03:59	It is opposing a relative motion between these two things.
240	242	0:04:00	0:04:02	There is molecular models for it.
243	244	0:04:03	0:04:04	We talked about that.
245	247	0:04:05	0:04:07	We are not going to make a big deal about it.
248	251	0:04:08	0:04:11	But physically, there is something really happening.
251	255	0:04:11	0:04:15	It is a contact force at the microscopic level.
256	258	0:04:16	0:04:18	We are going to deal with it macroscopically.
258	262	0:04:18	0:04:22	And so the first thing we are going to talk about is kinetic friction.
263	265	0:04:23	0:04:25	Now, kinetic friction is your friend.
267	274	0:04:27	0:04:34	Whenever you see a friction problem, if it has kinetic friction, you should just go ahead and have your brain release a little dopamine.
275	276	0:04:35	0:04:36	This is good news.
276	277	0:04:36	0:04:37	This is a happy place.
278	283	0:04:38	0:04:43	Because kinetic friction, we actually know the magnitude of that frictional force.
284	289	0:04:44	0:04:49	It is always equal to this coefficient of friction times the normal force.
291	292	0:04:51	0:04:52	And we also know the direction.
292	295	0:04:52	0:04:55	So let us do some kinetic friction here.
295	298	0:04:55	0:04:58	So if I put this block on, it slides down.
300	303	0:05:00	0:05:03	And so I can figure out exactly what
304	308	0:05:04	0:05:08	the magnitude and direction of that kinetic friction is.
308	309	0:05:08	0:05:09	There is an equal to sign.
309	313	0:05:09	0:05:13	So the direction is it does not want it to slide.
313	315	0:05:13	0:05:15	It is opposing the relative motion.
315	319	0:05:15	0:05:19	So the direction is always parallel to the surface.
320	321	0:05:20	0:05:21	And then it is opposing the motion.
321	323	0:05:21	0:05:23	So it would be up the ramp there.
324	327	0:05:24	0:05:27	And as long as I know what the normal force is, I can calculate it.
328	330	0:05:28	0:05:30	So that is kinetic friction.
332	337	0:05:32	0:05:37	And we are going to start with that, and mostly you should always get these problems and grab these points.
341	345	0:05:41	0:05:45	Then we have our nemesis, static friction.
345	350	0:05:45	0:05:50	I do not know, static, it seems like it should be easier, but it is not.
351	355	0:05:51	0:05:55	This little, it does not say equals to, it says less than or equals to.
356	359	0:05:56	0:05:59	And that is going to be the thing that is going to trip us up.
361	363	0:06:01	0:06:03	And we do not know the direction.
363	368	0:06:03	0:06:08	It has to be parallel to the two surfaces that are in contact.
368	371	0:06:08	0:06:11	But there is a whole bunch of directions.
372	377	0:06:12	0:06:17	If I have something parallel to these two surfaces, there is this direction, that direction.
377	379	0:06:17	0:06:19	We do not know what direction it is.
380	382	0:06:20	0:06:22	And we do not know what magnitude it is.
382	384	0:06:22	0:06:24	We just know it is there.
384	389	0:06:24	0:06:29	What we do know is we do know the maximum value that it can have.
390	396	0:06:30	0:06:36	And so lots of questions will ask for the maximum value and then it is a bit easier because the maximum value is when you set this equal.
397	404	0:06:37	0:06:44	So whenever you see static friction, do not release the dopamine, release the adrenaline, right?
404	409	0:06:44	0:06:49	This is the fight or flight mode you want to be in when you see static friction because you need to be super alert.
413	413	0:06:53	0:06:53	Question or comment?
421	423	0:07:01	0:07:03	Yeah, so that is a question.
423	428	0:07:03	0:07:08	If static friction is less than or equal to, does that mean it will move?
428	436	0:07:08	0:07:16	So what it means is, so the difference between these two blocks I have here, one has sandpaper, so it has a large value of mu.
437	443	0:07:17	0:07:23	So the static frictional force is bigger than the gravitational force, so it stays down.
446	448	0:07:26	0:07:28	I am sorry, so it does not accelerate down.
448	459	0:07:28	0:07:39	This is smooth paper, and so the static frictional force, the gravitational force is greater than mu times the normal force, and so then it accelerates.
464	466	0:07:44	0:07:46	All right, let us give you guys a chance to first...
470	472	0:07:50	0:07:52	Make sure you introduce yourself.
473	474	0:07:53	0:07:54	Say your name.
474	481	0:07:54	0:08:01	Do not assume the person remembers your name because they had quite a late night last night and their brain is not functioning well.
482	490	0:08:02	0:08:10	So reintroduce yourselves and then go ahead and answer this question that is about kinetic friction.
490	496	0:08:10	0:08:16	Here are the magnitudes of the kinetic friction here.
508	511	0:08:28	0:08:31	Oh you guys got to make the two seat leap there.
511	512	0:08:31	0:08:32	You are going to have to shout.
512	513	0:08:32	0:08:33	Get your names.
520	524	0:08:40	0:08:44	Okay good and you have reached across the divide.
525	526	0:08:45	0:08:46	You guys are doing it.
526	528	0:08:46	0:08:48	Oh good you are just in time to introduce yourselves.
529	531	0:08:49	0:08:51	So make sure you get each other's names.
531	533	0:08:51	0:08:53	You guys are looking confident like you are just
535	540	0:08:55	0:09:00	taunting me to see if I will ask you, so I will trust you know each other's names.
540	542	0:09:00	0:09:02	All right, fantastic.
597	603	0:09:57	0:10:03	All right, let us go ahead and stop this in three, two, one.
605	614	0:10:05	0:10:14	All right, so first question, what general equation will I be using to try to solve this problem?
614	616	0:10:14	0:10:16	Somebody in this section, just raise your hand and tell me.
616	619	0:10:16	0:10:19	What equation am I going to be using to solve this?
619	619	0:10:19	0:10:19	Yeah, in the back.
623	625	0:10:23	0:10:25	I am going to need to use the kinetic friction one.
625	626	0:10:25	0:10:26	That is going to come up in a second.
627	631	0:10:27	0:10:31	There is even a broader one that is going to require me to use kinetic friction.
631	632	0:10:31	0:10:32	Go way in the back.
633	634	0:10:33	0:10:34	F equals ma.
636	640	0:10:36	0:10:40	And so we are going to start with F equals ma, and then we are exactly, no, we are going to use your equation.
640	641	0:10:40	0:10:41	You are right.
641	643	0:10:41	0:10:43	It is just, you know, that is the problem with Illinois students.
643	645	0:10:43	0:10:45	You are always three steps ahead of me.
647	650	0:10:47	0:10:50	So we are going to do the sum of the forces is equal to the mass times the acceleration.
651	656	0:10:51	0:10:56	And in order to keep track of these forces, I am going to draw something called the free body diagram.
657	658	0:10:57	0:10:58	Let me just do case one first.
658	660	0:10:58	0:11:00	I will have to do this twice once for each case.
661	662	0:11:01	0:11:02	So for case one,
663	665	0:11:03	0:11:05	I am going to do a free body diagram of what?
665	667	0:11:05	0:11:07	Well, there is just one block there.
667	670	0:11:07	0:11:10	So I am going to do it for block M. And I am going to go to this group here.
670	675	0:11:10	0:11:15	And I am going to say, quick, somebody give me one force that is on this object.
677	678	0:11:17	0:11:18	I have the frictional force.
678	679	0:11:18	0:11:19	And what direction?
679	680	0:11:19	0:11:20	Well, it is labeled there.
680	683	0:11:20	0:11:23	So we know the frictional force is kinetic friction.
683	685	0:11:23	0:11:25	It opposes the relative motion.
685	686	0:11:25	0:11:26	So it is right there.
687	689	0:11:27	0:11:29	Somebody in this section, give me another force on it.
691	703	0:11:31	0:11:43	Yeah, the force of gravity, which is, which, it is, and the direction is down, yep, exactly, it is the weight, so it is mg, and then somebody in this section, give me another force.
704	705	0:11:44	0:11:45	Going To go back.
708	712	0:11:48	0:11:52	Oh, you mean the ordinary force, like the everyday force, is that what normal means?
715	716	0:11:55	0:11:56	What direction is it?
716	722	0:11:56	0:12:02	Yeah, it is the normal force, which is the perpendicular force, meaning it is perpendicular to the surface.
722	725	0:12:02	0:12:05	And so now you can see how we have separated.
725	727	0:12:05	0:12:07	There is two things.
728	729	0:12:08	0:12:09	The table is touching.
730	734	0:12:10	0:12:14	this block here, and we break it down into two different forces.
734	742	0:12:14	0:12:22	The force that is perpendicular, the normal force that you just did, and the force that is parallel, which was the frictional force.
742	744	0:12:22	0:12:24	And I forget who that was offered here.
745	749	0:12:25	0:12:29	And so then I am back to this group, and I say, give me another force on this object.
754	755	0:12:34	0:12:35	There is no more forces?
757	759	0:12:37	0:12:39	Yeah, first you have got to say back off, Professor Stelzo.
763	763	0:12:43	0:12:43	So you are right.
764	765	0:12:44	0:12:45	And your explanation was exactly right.
766	776	0:12:46	0:12:56	The reason you know you have all the forces on it is we have gravity, and then we only have one thing touching it, and we got both the perpendicular and the parallel component.
777	779	0:12:57	0:12:59	That is the confidence you need to have.
780	781	0:13:00	0:13:01	You really need to know there is no more.
782	783	0:13:02	0:13:03	So now I can do this.
783	787	0:13:03	0:13:07	This is a vector equation, but let me do it in the x and y direction separately.
788	789	0:13:08	0:13:09	So I am going to start in the y direction.
791	796	0:13:11	0:13:16	I should label my coordinate axes, x and y. So in the y direction, I have the normal force pointing up.
797	798	0:13:17	0:13:18	I have the weight pointing down.
799	802	0:13:19	0:13:22	And it is not accelerating in the y direction, so that is 0.
803	805	0:13:23	0:13:25	So I get the normal force is equal to the weight.
806	808	0:13:26	0:13:28	Well, that did not help me at all with the friction.
808	810	0:13:28	0:13:30	So let us try it in the x direction.
811	815	0:13:31	0:13:35	is the mass times acceleration in the x direction.
815	823	0:13:35	0:13:43	In the x direction, I just have the frictional force, and it is to the left, so I put a minus sign in there, is the mass times acceleration.
825	826	0:13:45	0:13:46	They did not even ask for this.
826	830	0:13:46	0:13:50	They just asked for... Oh, you were right.
830	832	0:13:50	0:13:52	I wanted to use F equals ma.
832	834	0:13:52	0:13:54	I was so excited, but I did not even have to, did I?
835	837	0:13:55	0:13:57	No, I had to use it for the vertical direction.
837	839	0:13:57	0:13:59	OK, phew, safe.
840	843	0:14:00	0:14:03	But yeah, so all we want to know is this frictional force.
844	845	0:14:04	0:14:05	So we know the frictional force.
845	848	0:14:05	0:14:08	It is kinetic friction, so it is our friend.
849	855	0:14:09	0:14:15	So this frictional force is going to be equal to mu sub k times the normal force.
855	858	0:14:15	0:14:18	And then I needed to use F equals ma in the y direction.
859	862	0:14:19	0:14:22	So this was going to be mu sub k times mg.
863	865	0:14:23	0:14:25	So that is what the frictional force is.
872	879	0:14:32	0:14:39	So I would say that f sub 1 is going to be equal to mu sub k times mg.
879	883	0:14:39	0:14:43	f sub 2 is going to be equal to mu sub k times 2mg.
885	888	0:14:45	0:14:48	So f sub 1 is half as big as f sub 2.
889	891	0:14:49	0:14:51	So this is the correct answer.
898	902	0:14:58	0:15:02	Quiet, I just do not hear well, so quiet please, yeah.
906	908	0:15:06	0:15:08	Did you say why did we multiply by g?
909	910	0:15:09	0:15:10	Yeah, that is a good question.
910	917	0:15:10	0:15:17	So the force from gravity, when we are near the surface of the Earth, is always the mass times this number g, 9.8.
922	925	0:15:22	0:15:25	Yeah, the velocity did not show up anywhere.
925	926	0:15:25	0:15:26	Why did they do that to me?
926	928	0:15:26	0:15:28	I did not even need it.
928	929	0:15:28	0:15:29	Yeah, the velocity did not matter.
929	932	0:15:29	0:15:32	It mattered that there was a velocity, that it is not 0.
933	935	0:15:33	0:15:35	Because if it is at 0, then there would not be kinetic friction.
936	938	0:15:36	0:15:38	But otherwise, yeah, it did not matter at all.
939	939	0:15:39	0:15:39	Just that it is not 0.
940	943	0:15:40	0:15:43	Oh, that is annoying.
954	959	0:15:54	0:15:59	Do we know why static friction is generally bigger than kinetic friction?
963	963	0:16:03	0:16:03	Let us see.
964	966	0:16:04	0:16:06	So there are simple intuitive models.
966	968	0:16:06	0:16:08	I mean, the answer is yes.
968	972	0:16:08	0:16:12	And then there is multiple different levels of understanding.
972	975	0:16:12	0:16:15	For me, I think the simplest model is if you think about...
978	989	0:16:18	0:16:29	If you think about it sort of as bumps between the two things and the connections, then when it is just sitting there, it can kind of get in the rut, and it is harder to get it going.
990	994	0:16:30	0:16:34	And when you are going over it, it kind of does not fall all the way into those ruts.
994	999	0:16:34	0:16:39	It just kind of hits some.
999	1000	0:16:39	0:16:40	It is a model.
1000	1002	0:16:40	0:16:42	That is not exactly what is happening.
1002	1005	0:16:42	0:16:45	But then people do understand at the molecular level, too.
1007	1012	0:16:47	0:16:52	Okay, so this is just nice so that when I post it, you can see what is happening.
1015	1022	0:16:55	0:17:02	Um, so now, there is this pre-lecture question that we asked is, which one goes further?
1026	1027	0:17:06	0:17:07	And people really struggled with it.
1028	1031	0:17:08	0:17:11	So let us go ahead and just talk about this.
1032	1033	0:17:12	0:17:13	This is solved the same way.
1033	1034	0:17:13	0:17:14	It is the same problem.
1035	1037	0:17:15	0:17:17	As you pointed out, the velocity did not show up anywhere.
1038	1039	0:17:18	0:17:19	It did not matter.
1040	1047	0:17:20	0:17:27	But now, instead of asking which one has a bigger force, we are asking about the acceleration.
1049	1059	0:17:29	0:17:39	And the cool thing that happens, because the frictional force is proportional to the mass, these masses cancel out.
1060	1067	0:17:40	0:17:47	And so the acceleration due to kinetic friction does not depend on the mass.
1068	1074	0:17:48	0:17:54	And so if they start with the same initial velocity and they have the same acceleration, they will travel the same distance.
1075	1082	0:17:55	0:18:02	So that is how that one is supposed to go, but you can see only 43% got that right before.
1083	1087	0:18:03	0:18:07	So I did that quickly.
1087	1090	0:18:07	0:18:10	Are there questions about that?
1092	1096	0:18:12	0:18:16	I know in the pre-lecture things are happening fast, but we want you to be able to do this.
1097	1099	0:18:17	0:18:19	Well, I am going to give you some more practice.
1099	1101	0:18:19	0:18:21	Hey, let us talk about this truck.
1104	1107	0:18:24	0:18:27	Now we are going to talk about static friction.
1107	1115	0:18:27	0:18:35	Okay, so the idea with this truck is that, so instead of a truck, I have two books here.
1115	1123	0:18:35	0:18:43	So think of this bottom book as being like the truck, and the top book as being like the crate in the truck, and which way are we going to go.
1123	1128	0:18:43	0:18:48	And we start from rest, and then we accelerate faster and faster and faster this way.
1131	1136	0:18:51	0:18:56	That is static friction because there is no relative motion between these two books.
1137	1139	0:18:57	0:18:59	So it is static friction that is happening.
1140	1146	0:19:00	0:19:06	And your first job for you is to figure out what is the direction of that static friction
1148	1149	0:19:08	0:19:09	if there is any.
1149	1159	0:19:09	0:19:19	And a first question we should ask before we answer that is, are there any other horizontal forces?
1160	1162	0:19:20	0:19:22	So static friction is one horizontal force.
1163	1166	0:19:23	0:19:26	You can imagine me coming up to you and saying, OK, give me another force.
1167	1175	0:19:27	0:19:35	If I come up to you and say, give me another force, are you going to say, back off, Professor Stelzer, or are you going to tell me what the other horizontal force is acting on this?
1176	1187	0:19:36	0:19:47	So I am going to let you vote on this, chat with the person next to you, and then I am going to come up to somebody and I am going to ask them, back off, or are you going to give me the force?
1198	1200	0:19:58	0:20:00	Are you going to tell me to back off or are you going to give me a force?
1204	1206	0:20:04	0:20:06	How do you know?
1214	1216	0:20:14	0:20:16	How come you do not think there is any other forces?
1219	1221	0:20:19	0:20:21	So we agree there is friction.
1237	1238	0:20:37	0:20:38	There is another force.
1238	1241	0:20:38	0:20:41	There is gravity and there is the normal force.
1241	1243	0:20:41	0:20:43	You know it because there is nothing else touching it.
1245	1247	0:20:45	0:20:47	Nothing else touching it.
1248	1251	0:20:48	0:20:51	Alright I am going to go ahead and stop this in three, two, one.
1256	1263	0:20:56	0:21:03	And then I am going to go up to a student, and I am just going to cold call on them.
1263	1269	0:21:03	0:21:09	And they are going to have to either tell me the force or tell me to back off and just be thinking, oh, thank god.
1269	1271	0:21:09	0:21:11	You are all like, ha, ha, ha, he is on the other side.
1272	1273	0:21:12	0:21:13	But here we go.
1273	1277	0:21:13	0:21:17	I am going to go right here, and I am going to cold call on you.
1277	1278	0:21:17	0:21:18	What are you going to do?
1278	1279	0:21:18	0:21:19	Do you got a force for me, or should I back off?
1283	1284	0:21:23	0:21:24	Any other forces?
1284	1285	0:21:24	0:21:25	No other forces.
1285	1290	0:21:25	0:21:30	But he knows that I have a thin skin, so he is not willing to say back off.
1292	1293	0:21:32	0:21:33	There are no other forces.
1293	1295	0:21:33	0:21:35	How do we know there are no other forces?
1295	1297	0:21:35	0:21:37	I am going to try to see.
1297	1299	0:21:37	0:21:39	I am trying to get everybody a chance.
1299	1300	0:21:39	0:21:40	So yeah, go ahead.
1306	1308	0:21:46	0:21:48	Well, so we know that we have friction.
1308	1311	0:21:48	0:21:51	The question is, how do we know we do not have anything else?
1312	1313	0:21:52	0:21:53	We need friction.
1314	1315	0:21:54	0:21:55	And that is going to be the next question.
1315	1317	0:21:55	0:21:57	Again, you guys are always ahead of me.
1317	1319	0:21:57	0:21:59	How do I know that there is not an extra force?
1319	1320	0:21:59	0:22:00	Yeah, you are going to tell me?
1322	1323	0:22:02	0:22:03	How do I know there is not another force?
1327	1328	0:22:07	0:22:08	Tell me another force.
1335	1338	0:22:15	0:22:18	A force acting in the opposite direction of acceleration.
1338	1342	0:22:18	0:22:22	And so, yeah, it seems like there should be, because otherwise... ...
1358	1382	0:22:38	0:23:02	yeah yeah exactly so you bring up a really really good point and this is how most of the people answer this question is to say well there must be because i need it to stay at rest and i know i have static friction so i need to have another force but that is not the right way to think about it so that is what it looks like if you are on the truck if you are in the truck
1383	1390	0:23:03	0:23:10	what you will do is you will think there is another force, but that is because you are in an accelerating reference frame.
1390	1391	0:23:10	0:23:11	That truck is accelerating.
1392	1395	0:23:12	0:23:15	Do not do physics in the accelerating reference frame.
1395	1407	0:23:15	0:23:27	You need to be safely on the side of the road, and then you will identify that the only horizontal force is that frictional force because there is nothing else touching it.
1408	1416	0:23:28	0:23:36	Right, the only thing touching it, so we have gravity pulling down, the only thing touching it is the bottom of the truck.
1417	1420	0:23:37	0:23:40	So the bottom of the truck gives two forces.
1420	1431	0:23:40	0:23:51	It gives the perpendicular force to counteract gravity and then it gives the frictional force and that frictional force is somehow parallel here and it is whatever direction it needs to be in.
1432	1437	0:23:52	0:23:57	So the correct answer here is that there is only one horizontal force.
1437	1440	0:23:57	0:24:00	It is the static frictional force.
1444	1445	0:24:04	0:24:05	Other questions or comments?
1448	1449	0:24:08	0:24:09	So this is really hard.
1454	1459	0:24:14	0:24:19	Oh, there would still only be one horizontal force.
1459	1460	0:24:19	0:24:20	It would be friction.
1461	1463	0:24:21	0:24:23	The next question is, what is the direction of it?
1463	1465	0:24:23	0:24:25	And that is what we are going to do next.
1465	1471	0:24:25	0:24:31	So now that we know this is the only horizontal force, what direction is that horizontal force?
1474	1477	0:24:34	0:24:37	So friction is our only horizontal force.
1477	1479	0:24:37	0:24:39	That is the first thing we needed to know.
1482	1485	0:24:42	0:24:45	Now the question is, what direction is that force?
1500	1501	0:25:00	0:25:01	Yeah, that is exactly what we want you to do.
1504	1505	0:25:04	0:25:05	Now we are talking.
1515	1520	0:25:15	0:25:20	Yeah, the answer is that it is in the direction of the acceleration.
1522	1525	0:25:22	0:25:25	Because we know the block is accelerating.
1526	1528	0:25:26	0:25:28	We are standing safely by the side of the road.
1529	1537	0:25:29	0:25:37	And so what we see, if we just analyze this block and we stand by the side of the road, it is at rest and then it starts accelerating.
1538	1541	0:25:38	0:25:41	So there must be some force to cause that acceleration.
1541	1544	0:25:41	0:25:44	And that force is the static frictional force.
1546	1550	0:25:46	0:25:50	And it is static friction because it is not moving relative to the other thing.
1551	1558	0:25:51	0:25:58	And we know that if we accelerate, let us see, one question was, what happens if they start slowing down?
1558	1561	0:25:58	0:26:01	Well, if they slow down, then the acceleration is that way.
1561	1564	0:26:01	0:26:04	So the frictional force, the static frictional force is the other way.
1564	1567	0:26:04	0:26:07	And we know that if we accelerate too fast...
1568	1577	0:26:08	0:26:17	Well, the acceleration of this bottom one is too much, static friction will not be enough, and then the block would slide back.
1578	1580	0:26:18	0:26:20	All right, so let me show you an example of that.
1580	1586	0:26:20	0:26:26	Yeah, this is one of your answers.
1588	1591	0:26:28	0:26:31	So this is, lots of people explained it, where they were doing...
1592	1595	0:26:32	0:26:35	The physics on the truck, that is dangerous.
1595	1600	0:26:35	0:26:40	Get out of the truck, stand in an inertial reference frame, and do your analysis there.
1603	1603	0:26:43	0:26:43	There you go.
1605	1611	0:26:45	0:26:51	OK, so here is a nice example of somebody having a tough start to their morning.
1611	1614	0:26:51	0:26:54	I do not know if you have had days like this.
1615	1619	0:26:55	0:26:59	You will see it is going to restart here in a second.
1620	1627	0:27:00	0:27:07	Right, they put all the carts on the truck and their wheels and there was not static friction.
1627	1630	0:27:07	0:27:10	And so the cart stayed still and the truck accelerated away.
1631	1635	0:27:11	0:27:15	And the thing that I love the lesson here is watch this guy here.
1636	1638	0:27:16	0:27:18	This is who you want to be, right?
1638	1640	0:27:18	0:27:20	This awful thing happened.
1640	1641	0:27:20	0:27:21	This is a nightmare.
1641	1642	0:27:21	0:27:22	They are going to have so much work.
1642	1645	0:27:22	0:27:25	They have to get these back up and then onto the truck.
1646	1647	0:27:26	0:27:27	And this guy's great.
1647	1648	0:27:27	0:27:28	He just smiles and laughs.
1648	1649	0:27:28	0:27:29	And that is what you got to do.
1650	1655	0:27:30	0:27:35	When life throws you something like this, you just have to laugh at the irony of the whole thing.
1657	1661	0:27:37	0:27:41	So let us look at the following situation.
1661	1664	0:27:41	0:27:44	Now I am going to have a book.
1666	1669	0:27:46	0:27:49	Here, I have a block on a table.
1669	1672	0:27:49	0:27:52	I actually have a block here on a table.
1672	1673	0:27:52	0:27:53	And I am going to pull it.
1674	1675	0:27:54	0:27:55	And you will be able to see the force.
1675	1678	0:27:55	0:27:58	I am going to put a tension, and this is going to tell you what the force is.
1679	1684	0:27:59	0:28:04	And I want you to first tell me what is the magnitude of the net force on this block.
1685	1687	0:28:05	0:28:07	I am not going to pull it hard enough for it to move.
1687	1689	0:28:07	0:28:09	It is just going to stay there still.
1690	1694	0:28:10	0:28:14	What is the magnitude of the net force on this block?
1726	1727	0:28:46	0:28:47	All right, very good.
1734	1736	0:28:54	0:28:56	The correct answer is zero.
1737	1739	0:28:57	0:28:59	Somebody tell me why is that the correct answer?
1739	1741	0:28:59	0:29:01	Yeah, way in the back.
1756	1759	0:29:16	0:29:19	The key word you said there is that it is not moving.
1759	1763	0:29:19	0:29:23	The acceleration is zero, so the net force is zero.
1763	1768	0:29:23	0:29:28	Exactly right, which means the two are canceling each other out, which is the next step.
1768	1769	0:29:28	0:29:29	That is exactly correct.
1770	1773	0:29:30	0:29:33	All right, the acceleration is zero, so the net force must be zero.
1774	1776	0:29:34	0:29:36	So what is the static frictional force?
1805	1808	0:30:05	0:30:08	All right, let us go ahead and stop this.
1808	1812	0:30:08	0:30:12	You got it in three, two, one, we are done.
1815	1822	0:30:15	0:30:22	Yeah, it is whatever the tension is, because we know they have to add up to zero.
1822	1834	0:30:22	0:30:34	And so I am going to do the, an example here where I am going to use this spring scale so you can actually see what the tension is, how much tension I am pulling with.
1835	1839	0:30:35	0:30:39	So here, well first I am not pulling at all, so static friction is zero.
1839	1848	0:30:39	0:30:48	Now I am pulling with 40 newtons, so static friction, I am pulling to your right with 40 newtons, so static friction is to your left.
1849	1855	0:30:49	0:30:55	Now I am pulling to your right with 70 newtons, so static friction is to the left with 70 newtons.
1856	1859	0:30:56	0:30:59	It is not mu sub s times the normal force.
1859	1861	0:30:59	0:31:01	It is just counteracting tension.
1861	1865	0:31:01	0:31:05	And now I am starting to hurt my finger, and I am pulling with 90.
1866	1872	0:31:06	0:31:12	And so right around 90 was when I reached the maximum value that static friction can have.
1872	1891	0:31:12	0:31:31	now I have kinetic friction and you can see that it is just around 45 newtons so it is about half of what the static friction could be so so that is the important lesson there is that it really it really depends on how hard you are pulling and it can have any value
1893	1895	0:31:33	0:31:35	all the way up, in this case, to 90 Newtons.
1896	1906	0:31:36	0:31:46	And then once that happens, now we started accelerating, we started moving, and then no matter how hard I pull, it will be that kinetic frictional force, which is just a constant value.
1908	1909	0:31:48	0:31:49	Question or comment about that?
1911	1914	0:31:51	0:31:54	So super careful whenever we see static friction.
1917	1920	0:31:57	0:32:00	Now, the ramp problem, we have it right up here.
1920	1922	0:32:00	0:32:02	It shows up in physics a ton.
1923	1926	0:32:03	0:32:06	And there is one thing that is super annoying about it.
1926	1928	0:32:06	0:32:08	And you are going to ask yourself, you know what?
1929	1932	0:32:09	0:32:12	I wonder if I could still get a B if I just never got a ramp problem right.
1933	1943	0:32:13	0:32:23	And the answer is, I do not know, probably not, but it is always going to bug you, and your friends from Purdue are going to say, I can do the ramp problem.
1944	1946	0:32:24	0:32:26	So you want to be able to do it.
1946	1948	0:32:26	0:32:28	So let us go through this.
1949	1951	0:32:29	0:32:31	The thing that we are going to do,
1953	1962	0:32:33	0:32:42	is what we want to do is we only want the acceleration to be in, we want to choose our axis so the acceleration is along one of those axes.
1962	1964	0:32:42	0:32:44	So we do not have acceleration in two different directions.
1965	1976	0:32:45	0:32:56	And so I am going to put my axes, instead of being horizontal and vertical, I am going to make one direction going parallel the way the block is going to accelerate down the ramp.
1977	1979	0:32:57	0:32:59	And then the other one has to be perpendicular to that.
1980	1981	0:33:00	0:33:01	So those are my two directions.
1982	1983	0:33:02	0:33:03	That seems fine.
1983	1986	0:33:03	0:33:06	I am going to go ahead and draw my free body diagram.
1987	1993	0:33:07	0:33:13	And so the thing with my free body diagram is I am going to label gravity the weight of the object.
1994	1997	0:33:14	0:33:17	And this, I think here, we will just start with a frictionless ramp.
1997	1999	0:33:17	0:33:19	So the weight goes straight down.
2000	2005	0:33:20	0:33:25	And I used to love the weight, because when things are flat, it is just in the negative y direction, and it is great.
2006	2007	0:33:26	0:33:27	But that is not true anymore.
2008	2021	0:33:28	0:33:41	Now my x and y directions are funny, so I have to take this weight that is pointing straight down, and I need to break it down into two pieces, the part that is parallel to the ramp and the part that is perpendicular to the ramp.
2023	2025	0:33:43	0:33:45	So this is the technique that you need to learn.
2025	2031	0:33:45	0:33:51	And here is the annoying thing, I have this angle here, the angle of the ramp,
2032	2035	0:33:52	0:33:55	But I need to somehow get it with these angle of these forces here.
2036	2037	0:33:56	0:33:57	So I have to do some geometry.
2037	2039	0:33:57	0:33:59	And I am terrible at geometry.
2040	2049	0:34:00	0:34:09	In fact, the only thing I really know about geometry is that the sum of the angles of a triangle is 180 degrees, and the sum of the angles of a line is 180 degrees.
2050	2051	0:34:10	0:34:11	So I am going to try to get by with that.
2052	2054	0:34:12	0:34:14	So the first thing I notice is that the gravity
2055	2062	0:34:15	0:34:22	is straight down and this is flat here, the bottom of the ramp is flat, so that angle is 90 degrees.
2063	2067	0:34:23	0:34:27	The sum of these three things has to be 180, so the angle up here must be 90 minus theta.
2073	2074	0:34:33	0:34:34	Not quite where I need to be yet.
2075	2077	0:34:35	0:34:37	But now I need one more step.
2077	2084	0:34:37	0:34:44	And I know that here, remember, this blue line here is perpendicular to the ramp.
2084	2085	0:34:44	0:34:45	That is how I chose it.
2086	2087	0:34:46	0:34:47	So that is a 90-degree angle.
2088	2090	0:34:48	0:34:50	And now this straight line is 180 degrees.
2090	2092	0:34:50	0:34:52	So that angle there is theta.
2093	2096	0:34:53	0:34:56	Now I have the triangle that I need.
2096	2100	0:34:56	0:35:00	And then I need to remember, is it Sokota?
2101	2102	0:35:01	0:35:02	I do not know.
2102	2105	0:35:02	0:35:05	How did you guys learn your sines, cosines, and tangents?
2109	2117	0:35:09	0:35:17	Whatever, however you remember it, for me I know that the adjacent is the cosine and the opposite is the sine.
2118	2120	0:35:18	0:35:20	And so I put those in there.
2120	2122	0:35:20	0:35:22	And now I have, that is the hard part.
2123	2130	0:35:23	0:35:30	Now I have done the part for getting, you know, gravity and knowing which part is parallel and which part is perpendicular.
2132	2133	0:35:32	0:35:33	You just practice it a bunch.
2134	2135	0:35:34	0:35:35	It is always like this.
2135	2138	0:35:35	0:35:38	There is nothing strange that ever comes out of it.
2138	2139	0:35:38	0:35:39	It is just sine or cosine.
2140	2143	0:35:40	0:35:43	I always check by saying, well, what happens if I made it flat?
2143	2145	0:35:43	0:35:45	Because I understand flat.
2145	2146	0:35:45	0:35:46	That is when it is not a ramp.
2146	2152	0:35:46	0:35:52	And I know that there should be no horizontal component of gravity.
2152	2155	0:35:52	0:35:55	And you can just check that sine of 0 is 0.
2155	2156	0:35:55	0:35:56	So it works.
2157	2159	0:35:57	0:35:59	Okay, so you have seen it here.
2159	2162	0:35:59	0:36:02	I know you are like, geez, I wish I could practice this more.
2162	2163	0:36:02	0:36:03	And guess what?
2164	2166	0:36:04	0:36:06	I put a ton of these on the homework.
2166	2168	0:36:06	0:36:08	And so you are going to get to practice that.
2168	2169	0:36:08	0:36:09	And you will just get good at it.
2170	2172	0:36:10	0:36:12	And I will put it on the exam for you, too.
2173	2175	0:36:13	0:36:15	So, yeah, no need to thank me now.
2179	2184	0:36:19	0:36:24	So here was a question about the static frictional force acting on the box.
2185	2188	0:36:25	0:36:28	And here was the answer distribution.
2188	2195	0:36:28	0:36:35	Most people said the static frictional force on the box was mu sub s times the weight times the cosine of theta.
2196	2202	0:36:36	0:36:42	That is the correct answer if I asked for what is the maximum static frictional force that could be on the box.
2203	2205	0:36:43	0:36:45	But that is not what we asked.
2206	2209	0:36:46	0:36:49	It is static friction, so you have got to be careful.
2209	2222	0:36:49	0:37:02	We know that the net force is zero, and so basically what we know is that the static frictional force has to cancel out the component of the weight that is parallel.
2223	2228	0:37:03	0:37:08	So the correct answer here is actually mg sine theta.
2228	2232	0:37:08	0:37:12	Okay, so again, just do not get...
2233	2234	0:37:13	0:37:14	Just be really careful.
2234	2236	0:37:14	0:37:16	This is a common mistake.
2236	2239	0:37:16	0:37:19	And on multiple choice exams, it is one of the choices.
2240	2242	0:37:20	0:37:22	And so I do not want you to lose those points.
2243	2244	0:37:23	0:37:24	Any question or comment there?
2257	2269	0:37:37	0:37:49	Yeah, that is a good question, is if I was on the ramp and I was accelerating upward, is that what you want to do, is you want to have it accelerate upward?
2270	2277	0:37:50	0:37:57	Yeah, because, you know, the sum of them is equal to the acceleration, and so you would need more static frictional force.
2277	2284	0:37:57	0:38:04	And so if the truck is going up a ramp, things are, like, accelerating, it is more likely that it will slide.
2289	2290	0:38:09	0:38:10	Things going in a circle, oh yeah.
2290	2292	0:38:10	0:38:12	So this has a really important point here.
2295	2297	0:38:15	0:38:17	Wheels are super, super useful.
2297	2302	0:38:17	0:38:22	I mean, they just like, they were a really good idea, all right?
2303	2304	0:38:23	0:38:24	There is no denying it.
2305	2310	0:38:25	0:38:30	Now here is the thing about wheels that is counterintuitive when we talk about friction.
2311	2320	0:38:31	0:38:40	Because the wheel, in order for the wheel to work well, there is friction with the road or the surface that you are going across.
2321	2325	0:38:41	0:38:45	And you would think, well, it is moving, so it should be kinetic friction, right?
2327	2329	0:38:47	0:38:49	Here is the crazy thing, it is not.
2329	2336	0:38:49	0:38:56	If you look at a wheel, the magic of the wheel is the fact that the part that is in contact is standing still.
2336	2346	0:38:56	0:39:06	The exception to this is if you are on your bike and you just lock up the brakes and it skids, that is kinetic friction.
2346	2354	0:39:06	0:39:14	But when you are rolling without slipping, like I am here, the part in contact is actually static friction.
2355	2363	0:39:15	0:39:23	that is static friction because, and you can just come up here and see, the part that is touching the table is not moving.
2364	2368	0:39:24	0:39:28	And you can see the difference because if I hold on to this, now that would be kinetic friction.
2369	2371	0:39:29	0:39:31	Static friction is that it does not move.
2372	2378	0:39:32	0:39:38	And that is actually what makes the wheel so useful, the fact that you are using static friction there instead of kinetic friction.
2378	2383	0:39:38	0:39:43	And so it is that less than or equal to sign that you have there.
2384	2386	0:39:44	0:39:46	So it can be a small force.
2388	2400	0:39:48	0:40:00	So there is an example of this where you figure out if you are going around in a circular track, so we are not going to speed up or slow down, but we are changing direction, there is an acceleration associated with that.
2402	2408	0:40:02	0:40:08	And they did this in the, I mean this is from the, this is from your pre-lecture.
2410	2415	0:40:10	0:40:15	And you say, well, if there was no friction, you would not go around.
2415	2416	0:40:15	0:40:16	And this happens.
2416	2421	0:40:16	0:40:21	You will get to experience it this winter when there is ice on the road, and you try to turn, and there is not enough friction.
2421	2423	0:40:21	0:40:23	You just go in a straight line.
2424	2426	0:40:24	0:40:26	And so it is a frictional force.
2426	2427	0:40:26	0:40:27	It is towards the center.
2428	2435	0:40:28	0:40:35	And then one of the questions we can say is, so the frictional force has to be the thing getting you to accelerate to the center.
2437	2445	0:40:37	0:40:45	And if you want to know what is the fastest you can go around this turn, well, you know the frictional force has a maximum value.
2445	2447	0:40:45	0:40:47	Static friction has a maximum value.
2448	2452	0:40:48	0:40:52	The maximum value is equal to mu sub s times the normal force.
2454	2459	0:40:54	0:40:59	And so you can see, again, we are going to see the masses cancel out here, which is kind of nice.
2460	2462	0:41:00	0:41:02	So that is another example of this.
2465	2467	0:41:05	0:41:07	Yeah, any questions or comments there?
2468	2473	0:41:08	0:41:13	It is a little bit funny to do the drawing, but you will get lots of practice here.
2473	2478	0:41:13	0:41:18	I think that is all I want to say there right now.
2480	2488	0:41:20	0:41:28	Okay, now let us do, you know, blocks attached by strings, which creates great stress.
2489	2500	0:41:29	0:41:40	Here, it looks like it is the same problem, but case one is when we had a frictionless surface, so there is no friction between M2 and the tabletop.
2501	2504	0:41:41	0:41:44	And case two, we have friction.
2506	2507	0:41:46	0:41:47	Let us make it kinetic friction.
2507	2509	0:41:47	0:41:49	So in both cases, they are going to accelerate.
2510	2511	0:41:50	0:41:51	It is kinetic friction.
2512	2518	0:41:52	0:41:58	In which case, oh, well, the question we ask is, which case is the tension in the string biggest?
2518	2520	0:41:58	0:42:00	Let us ask you a simpler question here.
2521	2526	0:42:01	0:42:06	In which case is the acceleration of the blocks the biggest, the magnitude of the acceleration?
2526	2531	0:42:06	0:42:11	When do you get the biggest acceleration, when there is no friction or when you have friction?
2532	2533	0:42:12	0:42:13	So let us start there.
2533	2534	0:42:13	0:42:14	When do we get the biggest acceleration?
2540	2543	0:42:20	0:42:23	We are not going to spend a lot of time.
2543	2545	0:42:23	0:42:25	Your intuition is good here.
2558	2561	0:42:38	0:42:41	Go ahead, get your vote in in three, two, one.
2564	2568	0:42:44	0:42:48	Yeah, the acceleration is biggest when there is no friction.
2569	2573	0:42:49	0:42:53	Friction is, in case two here, is going to oppose the motion.
2574	2575	0:42:54	0:42:55	It is going to slow this down.
2575	2577	0:42:55	0:42:57	It is going to have it accelerate less.
2578	2581	0:42:58	0:43:01	In fact, if you make the frictional force large enough, it will just stop.
2582	2582	0:43:02	0:43:02	It will not accelerate.
2584	2586	0:43:04	0:43:06	So that is what is going on there.
2587	2590	0:43:07	0:43:10	This is a brilliant comment by one of you.
2590	2601	0:43:10	0:43:21	And I brought it up here just because this person is saying, well, it looks like it is supposed to accelerate with mg, like it is just supposed to be gravity pulling it down.
2602	2605	0:43:22	0:43:25	And if it accelerated with mg, would not that mean that the tension would be zero?
2606	2608	0:43:26	0:43:28	So the tension must always be zero here.
2608	2609	0:43:28	0:43:29	And it is just brilliant.
2610	2615	0:43:30	0:43:35	Of course, we know that everything they are saying here, the reasoning is perfect.
2616	2618	0:43:36	0:43:38	And then they say, so what is the resolution?
2619	2621	0:43:39	0:43:41	Well, the resolution is not that the tension is zero.
2621	2624	0:43:41	0:43:44	The resolution is that it does not accelerate with mg.
2624	2625	0:43:44	0:43:45	That is what is going on.
2626	2633	0:43:46	0:43:53	And so I just really wanted to do this as an example of a great way to make sure you are understanding physics, really asking yourself these questions.
2635	2641	0:43:55	0:44:01	So in which case is the tension in the spring in the string going to be the biggest?
2642	2647	0:44:02	0:44:07	And so let me go ahead and just ask you that, and then we will work through it.
2700	2702	0:45:00	0:45:02	Which case is it the biggest?
2707	2708	0:45:07	0:45:08	Or it is the same.
2710	2712	0:45:10	0:45:12	So the friction does not change the tension.
2723	2729	0:45:23	0:45:29	Let us go ahead and stop this in three, two, one.
2734	2739	0:45:34	0:45:39	Yeah, so it is a harder question here.
2742	2746	0:45:42	0:45:46	And so the way to figure it out is to do the free body diagram.
2747	2752	0:45:47	0:45:52	And so we could do it for block one, block two, or the combination of blocks one and two.
2753	2755	0:45:53	0:45:55	We will just look at block one here.
2755	2758	0:45:55	0:45:58	And so I already did the forces for you.
2758	2759	0:45:58	0:45:59	We have the tension up.
2760	2761	0:46:00	0:46:01	We have the weight down.
2761	2764	0:46:01	0:46:04	It is going to be the mass times the acceleration.
2765	2766	0:46:05	0:46:06	We have the same thing here.
2766	2770	0:46:06	0:46:10	I mean, they are identical as far as block one is concerned.
2772	2774	0:46:12	0:46:14	But the acceleration is different.
2774	2778	0:46:14	0:46:18	Remember, we have a bigger acceleration in case one than we have in case two.
2780	2782	0:46:20	0:46:22	The weight force is the same.
2783	2790	0:46:23	0:46:30	So if I want this one to accelerate faster, I need a smaller tension force.
2795	2796	0:46:35	0:46:36	So the weight is the same.
2797	2802	0:46:37	0:46:42	If I wanted to accelerate faster, the tension force has to be smaller.
2804	2807	0:46:44	0:46:47	And so case two is the correct answer.
2808	2810	0:46:48	0:46:50	But let us go ahead and just work through this problem.
2811	2813	0:46:51	0:46:53	It is a classic problem.
2816	2818	0:46:56	0:46:58	So let us go ahead and work it out.
2819	2821	0:46:59	0:47:01	What we need to do is draw free body diagrams.
2822	2825	0:47:02	0:47:05	And as always, I am running a little bit low on time.
2826	2830	0:47:06	0:47:10	So I am going to go ahead and take care of doing block one.
2831	2833	0:47:11	0:47:13	So it is just got this.
2833	2835	0:47:13	0:47:15	Block two, quick, give me a force.
2835	2837	0:47:15	0:47:17	Somebody here, give me a force on block two.
2839	2843	0:47:19	0:47:23	What is it, friction?
2844	2845	0:47:24	0:47:25	Tension, tension, great.
2846	2848	0:47:26	0:47:28	So tension is to the right.
2848	2849	0:47:28	0:47:29	It is pulling.
2849	2850	0:47:29	0:47:30	Give me another force on block two.
2852	2856	0:47:32	0:47:36	And this is kinetic friction, so I know its direction.
2857	2859	0:47:37	0:47:39	And then give me another force on block two.
2859	2860	0:47:39	0:47:40	Somebody over here?
2861	2863	0:47:41	0:47:43	The normal force, yeah, perpendicular to the table.
2864	2866	0:47:44	0:47:46	And then I am not even going to play the game here.
2866	2867	0:47:46	0:47:47	There is nothing else touching it.
2867	2869	0:47:47	0:47:49	We have the two forces from the table.
2870	2874	0:47:50	0:47:54	And so then we can write F equals MA for each one of these blocks.
2875	2876	0:47:55	0:47:56	Oh, we did not do gravity yet.
2876	2877	0:47:56	0:47:57	Oh, I am sorry.
2880	2881	0:48:00	0:48:01	M2G, yes, it has gravity.
2881	2883	0:48:01	0:48:03	I thought we got that one first.
2884	2884	0:48:04	0:48:04	That is better.
2887	2888	0:48:07	0:48:08	So here we can do...
2889	2895	0:48:09	0:48:15	that the sum of the forces in the y direction is mass times acceleration in the y direction.
2895	2900	0:48:15	0:48:20	T minus the weight is equal to a sub 1.
2903	2913	0:48:23	0:48:33	Here, in the y direction, we get the normal force minus the weight is going to be equal to 0, because it does not accelerate in the y direction.
2914	2926	0:48:34	0:48:46	In the x direction, we get the tension minus the kinetic frictional force is equal to the mass times the acceleration of block one in the x direction.
2927	2928	0:48:47	0:48:48	Sorry, block two.
2930	2938	0:48:50	0:48:58	And then what we have here now, we can finish this up.
2938	2943	0:48:58	0:49:03	It is kinetic friction, so this is just mu sub k times the normal force.
2944	2946	0:49:04	0:49:06	which is equal to the weight.
2948	2950	0:49:08	0:49:10	And so this is going to be m2 a2.
2953	2962	0:49:13	0:49:22	And now we have two equations and three unknowns.
2964	2969	0:49:24	0:49:29	We do not know the tension, we do not know acceleration 1, and we do not know acceleration 2.
2970	2971	0:49:30	0:49:31	So we need another equation.
2972	2988	0:49:32	0:49:48	And what we need to know is that because these are connected by a string, the acceleration of 2 in the positive direction is equal, positive y is equal to the acceleration of 1 in the positive x is y in the negative direction for 1.
2989	2992	0:49:49	0:49:52	So we need to write down that a2 is equal to minus a1.
2994	2997	0:49:54	0:49:57	And now we have three equations and three unknowns, and you can solve it.
2998	3002	0:49:58	0:50:02	And I am out of time, so I am going to leave it there.
3003	3008	0:50:03	0:50:08	Have a fantastic weekend, and we will see you next week.