1 00:00:00,098 --> 00:00:02,737 - Alright, let's tackle part b, now. 2 00:00:02,737 --> 00:00:06,916 Derive the magnitude of the acceleration of block 2. 3 00:00:06,916 --> 00:00:11,916 Express your answer in terms of m1, m2, and g. 4 00:00:12,176 --> 00:00:13,988 And like always, try to pause the video 5 00:00:13,988 --> 00:00:15,428 and see if you can work through it yourself. 6 00:00:15,428 --> 00:00:18,740 We already worked through part 1, or part a, 7 00:00:18,740 --> 00:00:21,480 I should say, based on this diagram above, 8 00:00:21,480 --> 00:00:22,326 and there's a previous video. 9 00:00:22,326 --> 00:00:24,857 So, now we're ready to do part b. 10 00:00:24,857 --> 00:00:27,959 And we've already drawn the free-body diagrams, 11 00:00:27,959 --> 00:00:31,709 which will help us determine the acceleration of block 2. 12 00:00:31,709 --> 00:00:34,170 Let's just think about what the acceleration is, first. 13 00:00:34,170 --> 00:00:36,198 We know it's going to accelerate downwards, 14 00:00:36,198 --> 00:00:38,706 because, there's a couple ways you can think about it, 15 00:00:38,706 --> 00:00:42,854 the weight of block 2 is larger than the weight of block 1, 16 00:00:42,854 --> 00:00:44,379 they're connected by the string, so we know we're gonna 17 00:00:44,379 --> 00:00:46,577 accelerate downwards on the right-hand side and 18 00:00:46,577 --> 00:00:48,655 upwards on the left-hand side. 19 00:00:48,655 --> 00:00:49,758 The other way to think about it, is 20 00:00:49,758 --> 00:00:52,358 the weight of block 2 is larger than the upward force of 21 00:00:52,358 --> 00:00:57,358 tension. And the weight of block 1 is less than the 22 00:00:57,583 --> 00:00:59,939 tension pulling upwards. 23 00:00:59,939 --> 00:01:01,094 So, you're gonna accelerate upwards on the 24 00:01:01,094 --> 00:01:03,976 left-hand side, accelerate downwards on the right-hand side 25 00:01:03,976 --> 00:01:05,423 and a key realization is, 26 00:01:05,423 --> 00:01:08,653 the magnitude of the acceleration is going to be the same 27 00:01:08,653 --> 00:01:11,667 because they're connected by that string. 28 00:01:11,667 --> 00:01:16,033 So, the acceleration, I'll just draw it a little bit away 29 00:01:16,033 --> 00:01:20,128 from the actual dot, so the acceleration, here, 30 00:01:20,128 --> 00:01:23,309 has a magnitude a, it's gonna go in the downward direction 31 00:01:23,309 --> 00:01:26,403 the acceleration on the left-hand side is gonna be the same 32 00:01:26,403 --> 00:01:30,199 magnitude, but it's gonna go in the upwards direction. 33 00:01:30,199 --> 00:01:32,266 So that just gives us a sense of things. 34 00:01:32,266 --> 00:01:36,471 They say, derive the magnitude of acceleration of block 2 35 00:01:36,471 --> 00:01:37,330 Alright, so this is, 36 00:01:37,330 --> 00:01:39,419 let me leave the labels up there, 37 00:01:39,419 --> 00:01:41,858 so this is block 2 up here. 38 00:01:41,858 --> 00:01:44,936 And we know, from Newton's 2nd Law, 39 00:01:44,936 --> 00:01:47,271 that if we pick a direction, and the direction that matters 40 00:01:47,271 --> 00:01:49,501 here, is the vertical direction. 41 00:01:49,501 --> 00:01:50,748 All the forces are acting in 42 00:01:50,748 --> 00:01:53,339 either the upwards or downwards direction. 43 00:01:53,339 --> 00:01:55,673 So, the magnitude of our net forces, 44 00:01:55,673 --> 00:01:58,516 we care about the vertical dimension here, 45 00:01:58,516 --> 00:02:01,209 is going to be equal to the mass times the acceleration, 46 00:02:01,209 --> 00:02:04,674 in that, in that dimension, in that direction, 47 00:02:04,674 --> 00:02:06,101 I guess you could say. 48 00:02:06,101 --> 00:02:09,038 And so let's just think about block 2. 49 00:02:09,038 --> 00:02:13,461 Block, Block 2. 50 00:02:13,461 --> 00:02:16,233 And since the acceleration, we know is downwards, 51 00:02:16,233 --> 00:02:18,799 and we wanna figure out what a is, 52 00:02:18,799 --> 00:02:21,284 let's just assume that positive, 53 00:02:21,284 --> 00:02:24,918 positive magnitude specifies downwards. 54 00:02:24,918 --> 00:02:27,169 So what are the net forces? 55 00:02:27,169 --> 00:02:29,990 Well, the net forces are going to be the force of weight 56 00:02:29,990 --> 00:02:32,675 minus the tension, and that's going to be positive. 57 00:02:32,675 --> 00:02:35,245 If we think about it in the downward direction. 58 00:02:35,245 --> 00:02:37,439 The downward direction being positive. 59 00:02:37,439 --> 00:02:40,469 So we're gonna have m2g, the weight, 60 00:02:40,469 --> 00:02:42,470 minus the tension, 61 00:02:42,470 --> 00:02:44,859 tension is going against the weight, 62 00:02:44,859 --> 00:02:49,009 minus the tension is going to be equal to the mass. 63 00:02:49,009 --> 00:02:54,009 Is going to be equal to m2 times, times our acceleration, 64 00:02:55,746 --> 00:02:57,468 and we need to figure out what that acceleration 65 00:02:57,468 --> 00:03:00,123 is going to be. 66 00:03:00,123 --> 00:03:02,401 So, we do that same blue color. 67 00:03:02,401 --> 00:03:06,059 Times the acceleration. 68 00:03:06,059 --> 00:03:08,353 Now, we could divide both sides by m2, 69 00:03:08,353 --> 00:03:10,559 but that's not going to help us too much, just yet. 70 00:03:10,559 --> 00:03:12,719 Because then, we would've solved for acceleration in terms 71 00:03:12,719 --> 00:03:17,719 of m2g and T. 72 00:03:17,990 --> 00:03:21,149 We don't have any m1s here, so we're not solving in terms of 73 00:03:21,149 --> 00:03:22,823 m1, m2, and g. 74 00:03:22,823 --> 00:03:26,411 We're solving in terms of m2, T, and g. 75 00:03:26,411 --> 00:03:28,886 So, somehow, we have to get rid of this T. 76 00:03:28,886 --> 00:03:31,089 And what we can do to get rid of the T, 77 00:03:31,089 --> 00:03:34,270 is set up a similar equation for block 1. 78 00:03:34,270 --> 00:03:39,270 Block 1, Block 1 79 00:03:39,703 --> 00:03:42,397 Here, since we're concerned with magnitude and 80 00:03:42,397 --> 00:03:44,218 especially the magnitude of acceleration, and 81 00:03:44,218 --> 00:03:46,275 here the acceleration is going in the upward direction, 82 00:03:46,275 --> 00:03:47,656 we could say that the upward direction 83 00:03:47,656 --> 00:03:49,327 is the positive direction. 84 00:03:49,327 --> 00:03:52,391 And so, we could say that T 85 00:03:52,391 --> 00:03:53,854 minus, we know that the tension is 86 00:03:53,854 --> 00:03:55,152 larger than the weight, 87 00:03:55,152 --> 00:04:00,152 T minus m1g is going to be equal to 88 00:04:01,272 --> 00:04:04,411 is equal to m1, 89 00:04:04,411 --> 00:04:09,411 is going to be equal to m1 times the magnitude of 90 00:04:11,071 --> 00:04:11,977 the acceleration. 91 00:04:11,977 --> 00:04:14,786 And to be clear, these magnitudes are the same. 92 00:04:14,786 --> 00:04:16,853 And we already know that the magnitudes of the tension 93 00:04:16,853 --> 00:04:18,013 are the same. 94 00:04:18,014 --> 00:04:19,938 And now we have two equations 95 00:04:19,938 --> 00:04:21,420 with two unknowns, 96 00:04:21,420 --> 00:04:23,440 and so, if we can eliminate the tension, 97 00:04:23,440 --> 00:04:25,714 we can solve for acceleration. 98 00:04:25,714 --> 00:04:27,596 And we can acutally do that by just 99 00:04:27,596 --> 00:04:29,361 adding the left-hand side to the left-hand side, 100 00:04:29,361 --> 00:04:32,159 and the right-hand side to the right-hand side. 101 00:04:32,159 --> 00:04:37,159 You learned this probably first in algebra 1. If, 102 00:04:37,471 --> 00:04:40,306 if this is equal to that and that is equal to that, 103 00:04:40,306 --> 00:04:42,315 if we add the left side to the left side 104 00:04:42,315 --> 00:04:43,438 and the right side to the right side, 105 00:04:43,438 --> 00:04:44,944 well, we're still gonna get two things that are 106 00:04:44,944 --> 00:04:46,313 equal to each other. 107 00:04:46,313 --> 00:04:48,971 So when you add the left-hand sides, 108 00:04:48,971 --> 00:04:50,272 what are you going to get? 109 00:04:50,272 --> 00:04:55,224 So, you're gonna get m2g, 110 00:04:55,224 --> 00:05:00,034 m2g minus m1g, 111 00:05:00,034 --> 00:05:03,402 minus m1g, 112 00:05:03,402 --> 00:05:06,675 and then you're gonna get T minus T. 113 00:05:06,675 --> 00:05:08,100 These two are going to cancel out. 114 00:05:08,100 --> 00:05:09,548 So, let me just cross them out. 115 00:05:09,548 --> 00:05:11,093 So that was convenient. 116 00:05:11,093 --> 00:05:13,323 Is going to be equal to 117 00:05:13,323 --> 00:05:18,323 m2a, is going to be equal to m2a 118 00:05:18,468 --> 00:05:20,755 plus m1a, 119 00:05:20,755 --> 00:05:25,755 plus m1 times a. 120 00:05:26,665 --> 00:05:28,418 And now, we just need to solve for a. 121 00:05:28,418 --> 00:05:29,811 And how do we do that? 122 00:05:29,811 --> 00:05:32,228 Well, we can factor out an a out of this right-hand side 123 00:05:32,228 --> 00:05:37,228 here, so this going to be m2g minus m1g 124 00:05:39,616 --> 00:05:41,090 is equal to 125 00:05:41,090 --> 00:05:42,681 let's factor out an a, 126 00:05:42,681 --> 00:05:47,681 a times, times m2 plus m1, 127 00:05:50,030 --> 00:05:50,870 and now to solve for a, 128 00:05:50,870 --> 00:05:54,225 we just divide both sides by m2 plus m1. 129 00:05:54,225 --> 00:05:56,303 m2 plus m1, 130 00:05:56,303 --> 00:06:01,179 m2 plus m1, 131 00:06:01,179 --> 00:06:02,498 and there you have it, we get a 132 00:06:02,498 --> 00:06:04,040 is equal to 133 00:06:04,040 --> 00:06:05,552 a is equal to this. 134 00:06:05,552 --> 00:06:06,413 And, notice, we have solved it, 135 00:06:06,413 --> 00:06:08,586 we have solved for a in terms of 136 00:06:08,586 --> 00:06:10,502 we have solved for a in terms of 137 00:06:10,502 --> 00:06:13,323 m1, m2, and g. 138 00:06:13,323 --> 00:06:15,889 And this is the magnitude of the acceleration of either 139 00:06:15,889 --> 00:06:18,548 block 1 or block 2. 140 00:06:18,548 --> 00:06:19,592 Now, some of you might be thinking, 141 00:06:19,592 --> 00:06:21,670 wait, there might be an easier way to think about this 142 00:06:21,670 --> 00:06:24,751 problem, and I went straight from the free-body diagrams 143 00:06:24,751 --> 00:06:27,197 which is, ya know, it's implied that this is 144 00:06:27,197 --> 00:06:29,960 the way to tackle it, using the tension. 145 00:06:29,960 --> 00:06:31,388 But, another way to tackle it, 146 00:06:31,388 --> 00:06:32,538 you could've said, well, 147 00:06:32,538 --> 00:06:34,256 this would be analogous, it's not the 148 00:06:34,256 --> 00:06:35,893 exact same thing, but it would be analogous 149 00:06:35,893 --> 00:06:39,968 to imagine, two, these two blocks floating in space. 150 00:06:39,968 --> 00:06:42,255 So, this is m2, here, and I'm not gonna 151 00:06:42,255 --> 00:06:44,914 use pulleys here, so that's m2. 152 00:06:44,914 --> 00:06:46,667 And it's connected by 153 00:06:46,667 --> 00:06:49,070 a massless string, 154 00:06:49,070 --> 00:06:54,070 to m1, which has a smaller mass, m1. 155 00:06:54,608 --> 00:06:58,788 And let's say that you are pulling on, pulling 156 00:06:58,788 --> 00:07:00,378 in the rightward direction, now we're just 157 00:07:00,378 --> 00:07:02,861 drifting in space. 158 00:07:02,861 --> 00:07:06,983 With a force of m2g, we're not, 159 00:07:06,983 --> 00:07:09,839 we just care about the magnitude here, and 160 00:07:09,839 --> 00:07:11,429 I know you might be saying, wait, okay, 161 00:07:11,429 --> 00:07:14,332 is this is the gravitational field or whatever else, 162 00:07:14,332 --> 00:07:15,655 but I'm saying, let's just say you're pulling 163 00:07:15,655 --> 00:07:17,164 in this direction with a force 164 00:07:17,164 --> 00:07:20,949 that happens to be equal to, that has a magnitude of m2g, 165 00:07:20,949 --> 00:07:24,762 and let's say you're pulling in this direction, 166 00:07:24,762 --> 00:07:29,762 with a force that has a magnitude of m1, m1g. 167 00:07:30,936 --> 00:07:33,916 Now, this isn't exactly the same as our, as where we started 168 00:07:33,916 --> 00:07:37,144 where we started, we saw what I have just kind of drawn, 169 00:07:37,144 --> 00:07:39,356 but I have it in the presence of a gravitational field, 170 00:07:39,356 --> 00:07:41,028 and I have it wrapped around those pulleys, 171 00:07:41,028 --> 00:07:44,093 and then the gravitational field is providing these forces. 172 00:07:44,093 --> 00:07:46,205 But let's just assume, just for simplicity that 173 00:07:46,205 --> 00:07:47,808 that you're drifting in space and you're 174 00:07:47,808 --> 00:07:50,014 pulling on m2 to the right with a force that's 175 00:07:50,014 --> 00:07:52,951 equivalent to m2g, and you are pulling to the left 176 00:07:52,951 --> 00:07:55,842 on m1, with a force of m1g. 177 00:07:55,842 --> 00:07:58,812 Well, you could just view this as one big, 178 00:07:58,812 --> 00:08:02,820 you could just view the m1, the string, and m2 as just 179 00:08:02,820 --> 00:08:05,397 one combined mass. 180 00:08:05,397 --> 00:08:07,881 You could just view this as one combined mass, 181 00:08:07,881 --> 00:08:12,346 of m1 plus m2, and you could say, alright, 182 00:08:12,346 --> 00:08:14,662 well from that one combined mass, 183 00:08:14,662 --> 00:08:15,940 I am, 184 00:08:15,940 --> 00:08:16,949 whoops, 185 00:08:16,949 --> 00:08:20,721 that one combined mass, I am 186 00:08:20,721 --> 00:08:24,537 I am pulling to the right with a magnitude of m2g, 187 00:08:24,537 --> 00:08:27,350 and I am pulling to the left 188 00:08:27,350 --> 00:08:29,154 let me make that a slightly different color 189 00:08:29,154 --> 00:08:30,562 so you can see it. 190 00:08:30,562 --> 00:08:35,562 And I am pulling to the left with a magnitude of m1g 191 00:08:37,043 --> 00:08:39,482 and now, this becomes a pretty straight-forward thing. 192 00:08:39,482 --> 00:08:40,880 What would be the acceleration here? 193 00:08:40,880 --> 00:08:42,842 Well, you could say the net, 194 00:08:42,842 --> 00:08:46,403 the net magnitude of the force, 195 00:08:46,403 --> 00:08:48,071 or the magnitude of the net force would be 196 00:08:48,071 --> 00:08:50,846 m2g, we'll take the rightward direction as 197 00:08:50,846 --> 00:08:55,846 the positive direction, m2g minus m1g 198 00:08:56,117 --> 00:09:00,064 minus m1g, and then, so that's the net force, 199 00:09:00,064 --> 00:09:01,504 in the right direction, 200 00:09:01,504 --> 00:09:03,826 the magnitude of the net force in the rightward direction 201 00:09:03,826 --> 00:09:05,081 and then you divide it by its mass, 202 00:09:05,081 --> 00:09:06,624 and you're gonna get acceleration. 203 00:09:06,624 --> 00:09:08,979 Force divided by mass gives you acceleration. 204 00:09:08,979 --> 00:09:13,979 So, you divide that by our mass, which is going to be 205 00:09:14,077 --> 00:09:18,083 which is going to be m1 plus m2, 206 00:09:18,083 --> 00:09:21,020 that's going to give you your acceleration. 207 00:09:21,020 --> 00:09:22,928 So, you could view this as a simpler way 208 00:09:22,928 --> 00:09:24,279 of thinking about it. But they, 209 00:09:24,279 --> 00:09:28,179 notice, either of them give you the exact same answer. 210 00:09:28,179 --> 00:09:30,630 That's one of the fun things about science, 211 00:09:30,630 --> 00:09:32,509 as long as you do logical things, 212 00:09:32,509 --> 00:00:00,000 you get to the same point.