1 00:00:01,820 --> 00:00:06,900 Let's consider the formula for the period of a simple 2 00:00:06,900 --> 00:00:11,472 pendulum of length L is formula is T period. 3 00:00:12,560 --> 00:00:14,080 Is equal to 2π? 4 00:00:15,180 --> 00:00:21,072 Times the square root of L over G&L is the length of 5 00:00:21,072 --> 00:00:22,054 the pendulum. 6 00:00:23,220 --> 00:00:27,114 Now on Earth, we tend to regard G's being fixed it. 7 00:00:27,114 --> 00:00:30,654 There is a little with altitude, but we tend to 8 00:00:30,654 --> 00:00:32,778 think of it as being fixed. 9 00:00:33,810 --> 00:00:39,998 Just supposing we had a pendulum of a fixed length L, and we took 10 00:00:39,998 --> 00:00:43,976 it somewhere else. Let's say the moon or Mars. 11 00:00:44,670 --> 00:00:45,970 Somewhere out of the solar 12 00:00:45,970 --> 00:00:50,810 system. Gravity there would not be the same and we might want to 13 00:00:50,810 --> 00:00:54,530 measure gravity, so one of the ways will be to take our 14 00:00:54,530 --> 00:00:55,770 pendulum, set it swinging. 15 00:00:56,780 --> 00:00:58,100 And measure the time. 16 00:00:59,060 --> 00:01:03,536 Once we measured the time, we could then use that to Calculate 17 00:01:03,536 --> 00:01:09,504 G, but before we could do it, we would need to know what G was in 18 00:01:09,504 --> 00:01:14,726 terms of the rest of the symbols in this formula. So we need to 19 00:01:14,726 --> 00:01:17,710 rearrange this formula so that it said G. 20 00:01:19,230 --> 00:01:20,210 Equals. 21 00:01:22,200 --> 00:01:23,559 Now it's what? 22 00:01:24,270 --> 00:01:28,518 Goes instead of this question mark that we're going to have a 23 00:01:28,518 --> 00:01:33,120 look at in this video. We're going to be looking at how we 24 00:01:33,120 --> 00:01:36,660 transform formally, how we move from an expression like this. 25 00:01:37,700 --> 00:01:39,218 To another expression. 26 00:01:40,080 --> 00:01:44,780 Involving exactly the same variables, but one that helps us 27 00:01:44,780 --> 00:01:49,950 answer the questions that were after in terms of a different 28 00:01:49,950 --> 00:01:52,508 variable. To do this? 29 00:01:53,410 --> 00:01:57,840 Transformation of formula will need all the techniques that we 30 00:01:57,840 --> 00:02:02,270 had in terms of solving equations, so the video solving 31 00:02:02,270 --> 00:02:07,586 linear equations in one variable might be very useful to have a 32 00:02:07,586 --> 00:02:12,016 look at, but let's just recap by having a look. 33 00:02:13,570 --> 00:02:15,990 A simple linear equation. 34 00:02:16,710 --> 00:02:21,336 So the one going to take three X +5. 35 00:02:22,860 --> 00:02:28,410 Equals 6 - 3 * 5 36 00:02:28,410 --> 00:02:31,185 - 2 X. 37 00:02:32,740 --> 00:02:37,206 Now we've got this equation we want to solve it for X, so our 38 00:02:37,206 --> 00:02:39,758 aim is to end up with X equals. 39 00:02:40,910 --> 00:02:45,206 So one of the first things we would do is multiply out 40 00:02:45,206 --> 00:02:45,922 the bracket. 41 00:02:48,110 --> 00:02:53,258 So we have minus three times by 5 is minus 15. 42 00:02:54,130 --> 00:03:00,439 Minus three times by minus 2X is plus 6X. 43 00:03:01,650 --> 00:03:09,030 Next, we can simplify this bit. Here three X +5 44 00:03:09,030 --> 00:03:16,410 equals 6X, and now six takeaway 15 is minus 9. 45 00:03:17,760 --> 00:03:24,368 Now, one of the things we would want to do is to try and get all 46 00:03:24,368 --> 00:03:30,150 the ex is together. So we were three X here under 6X there, so 47 00:03:30,150 --> 00:03:34,693 will take 3X away from each side, so that's three X 48 00:03:34,693 --> 00:03:38,823 takeaway. Three X +5 equals 6X takeaway, 3X minus 9. 49 00:03:39,630 --> 00:03:46,013 3X takeaway 3X the X is go. We've known left and then we've 50 00:03:46,013 --> 00:03:52,396 got this. Five is equal to six X takeaway 3X. That's three X 51 00:03:52,396 --> 00:03:53,869 again takeaway 9. 52 00:03:55,400 --> 00:04:00,524 Now we need to get the constant terms, the numbers together. So 53 00:04:00,524 --> 00:04:07,783 to do that we would add 9 to each side. So we 5 + 9 is equal 54 00:04:07,783 --> 00:04:10,772 to three X minus 9 + 9. 55 00:04:11,570 --> 00:04:17,078 So again, we're doing to the same thing to both sides. Here 56 00:04:17,078 --> 00:04:23,045 we took three X away from each side. Here we're adding nine to 57 00:04:23,045 --> 00:04:29,930 each side, so five and nine is 14 equals 3X, and we minus 9 + 58 00:04:29,930 --> 00:04:34,520 9.0. Now we need to divide both sides by three. 59 00:04:35,340 --> 00:04:41,724 So we have 3X over three an 14 over 3, dividing both sides by 60 00:04:41,724 --> 00:04:46,740 three, and so those three canceled and we're left with X 61 00:04:46,740 --> 00:04:51,756 equals 14 over 3. Now I've written this out very fully. 62 00:04:51,756 --> 00:04:56,772 I've written out every step that I've said, but we wouldn't 63 00:04:56,772 --> 00:05:00,876 normally expect to see all of that written down. 64 00:05:02,100 --> 00:05:03,720 Starting from here. 65 00:05:07,750 --> 00:05:11,974 The next line we'd expect to see is this one because 66 00:05:11,974 --> 00:05:16,198 we say Take 3X away from both sides and we'd expect 67 00:05:16,198 --> 00:05:18,502 to see that as the result. 68 00:05:19,540 --> 00:05:24,714 Then we'd say add 9 to both sides, and So what we would 69 00:05:24,714 --> 00:05:28,296 expect to see having done that would be that. 70 00:05:28,930 --> 00:05:31,990 And then we, say, divide both sides by three. 71 00:05:34,030 --> 00:05:38,398 And so we see that So what we would see written down in our 72 00:05:38,398 --> 00:05:41,830 exercise book or on a piece of paper that carried the 73 00:05:41,830 --> 00:05:44,638 solution to this equation will be something like this. 74 00:05:45,830 --> 00:05:49,810 OK, we've been through the steps now of solving an 75 00:05:49,810 --> 00:05:52,994 equation. These techniques, particularly idea of a balance 76 00:05:52,994 --> 00:05:57,770 of keeping the same on both sides by doing the same thing 77 00:05:57,770 --> 00:06:02,944 to both sides is what we're going to do when we look at 78 00:06:02,944 --> 00:06:04,536 the transformation of formally. 79 00:06:07,060 --> 00:06:14,460 So let's begin with our first formula V equals U 80 00:06:14,460 --> 00:06:20,155 plus AT. And the variable that we're going to try and 81 00:06:20,155 --> 00:06:21,310 find is TI. 82 00:06:22,320 --> 00:06:28,464 T is the variable we're going to find what is T expressed? 83 00:06:29,130 --> 00:06:33,486 In terms of the rest of the letters or symbols that there 84 00:06:33,486 --> 00:06:37,842 are in this formula well, in order to model this, what I'm 85 00:06:37,842 --> 00:06:42,198 going to do is I'm going to write down an equation which 86 00:06:42,198 --> 00:06:44,739 looks like this, but it just got 87 00:06:44,739 --> 00:06:51,595 numbers in. With a T there as the unknown. So what we might 88 00:06:51,595 --> 00:06:56,445 have is something like this. Seven for V5 for you. 89 00:06:57,040 --> 00:07:02,026 Let's say a tool for A and then T. 90 00:07:03,440 --> 00:07:08,432 Now what would we do to solve this equation? The first thing 91 00:07:08,432 --> 00:07:15,088 we try and do is get the TS on their own and so to do that, 92 00:07:15,088 --> 00:07:20,912 we take 5 away from each side. So this would be 2. Over here 93 00:07:20,912 --> 00:07:25,904 equals 2 T, so having done that there, let's do it here. 94 00:07:25,904 --> 00:07:29,648 V minus U taking you away from each side. 95 00:07:31,280 --> 00:07:36,190 Now we would divide both sides by two over here. 96 00:07:39,400 --> 00:07:46,600 In order to end up with just T on its own, so let's do the 97 00:07:46,600 --> 00:07:52,840 same. Here. A is what multiplies by T, so we need to divide 98 00:07:52,840 --> 00:07:59,080 everything on this side V minus U over a equals T. That's it. 99 00:07:59,080 --> 00:08:05,320 Notice everything here is over a, not just one part of it, but 100 00:08:05,320 --> 00:08:07,240 both the whole expression. 101 00:08:07,470 --> 00:08:10,070 The minus U over a. 102 00:08:11,800 --> 00:08:16,665 Let's take another one. V squared equals 103 00:08:16,665 --> 00:08:20,140 U squared plus 2A S. 104 00:08:21,190 --> 00:08:24,778 And this time, let's say the symbol that I'm going to try and 105 00:08:24,778 --> 00:08:28,366 find all the variables that I'm going to try and find in terms 106 00:08:28,366 --> 00:08:30,022 of all the others, is you. 107 00:08:31,990 --> 00:08:36,577 So that's the case. Again, I'm going to write down an 108 00:08:36,577 --> 00:08:39,913 equation over here which looks like this one. 109 00:08:41,090 --> 00:08:42,209 So let's have. 110 00:08:43,450 --> 00:08:44,620 25 111 00:08:46,060 --> 00:08:48,370 equals U squared. 112 00:08:49,360 --> 00:08:51,660 +9. 113 00:08:53,180 --> 00:08:58,262 Weather 9 is in place of this lump here of algebra. 114 00:08:58,850 --> 00:09:04,752 So what would we do here? Our first step would be to take 115 00:09:04,752 --> 00:09:10,654 nine away from both sides. So let's do that. 25 - 9 equals 116 00:09:10,654 --> 00:09:16,556 U squared and that gets us EU squared on its own. So let's 117 00:09:16,556 --> 00:09:21,550 do that over here. Let's take this lump away from both 118 00:09:21,550 --> 00:09:26,544 sides. 20 squared minus 2A S is equal to U squared. 119 00:09:28,150 --> 00:09:33,535 Now what I would want to do now is to take the square root of 120 00:09:33,535 --> 00:09:37,843 both sides 'cause I want just you and here I've got you 121 00:09:37,843 --> 00:09:43,228 squared. So I need the square root of 25 - 9 and I want the 122 00:09:43,228 --> 00:09:47,895 square root of all of that not square root of 25 minus the 123 00:09:47,895 --> 00:09:53,280 square root of 9. I want the square root of 25 - 9, so again 124 00:09:53,280 --> 00:09:58,306 I've got to do the same over here I want the square root of. 125 00:09:58,430 --> 00:10:00,620 All. Love it. 126 00:10:05,400 --> 00:10:10,847 Let's just check why I need the square root of all of this. 127 00:10:12,030 --> 00:10:18,675 25 - 9 is 16, so that 16 equals U squared. Take the square root 128 00:10:18,675 --> 00:10:24,434 of both sides. Four is equal to you and we're happy. That's the 129 00:10:24,434 --> 00:10:30,566 answer. But what if I do the square root of 25 minus the 130 00:10:30,566 --> 00:10:36,488 square root of nine? Well, that's 5 - 3 is 2, which is not 131 00:10:36,488 --> 00:10:41,564 the answer very definitely not the answer, so we need to take 132 00:10:41,564 --> 00:10:46,217 the square root of everything, not just each little piece. And 133 00:10:46,217 --> 00:10:52,139 so this over here has to be the square root of all of that. 134 00:10:55,720 --> 00:11:01,456 I've been working so far with the formula that are for uniform 135 00:11:01,456 --> 00:11:05,280 acceleration, so let's continue with that and take. 136 00:11:05,890 --> 00:11:13,534 Another one of them S equals UT plus 1/2 AT squared, and 137 00:11:13,534 --> 00:11:16,082 this time it's a. 138 00:11:17,630 --> 00:11:21,914 Then I'm going to be looking for can I rearrange this formula? 139 00:11:21,914 --> 00:11:24,770 Can I transform it so it says a 140 00:11:24,770 --> 00:11:32,063 equals? Again, let me model it with an equation. Let's say that 141 00:11:32,063 --> 00:11:39,945 S is 21, but U times by T is 15 + 1/2 of a 142 00:11:39,945 --> 00:11:41,634 Times by 9. 143 00:11:42,870 --> 00:11:49,240 And it's this a the time after first of all. Let's isolate the 144 00:11:49,240 --> 00:11:55,610 term with the unknown in it. So that's the ater. So let's take 145 00:11:55,610 --> 00:12:02,960 15 away from both sides. So I'm 21 - 15 is equal to 1/2 of 146 00:12:02,960 --> 00:12:09,820 a times by 9. So again, let's do that here. Let's take this lump 147 00:12:09,820 --> 00:12:12,760 away. S minus Utah equals 1/2. 148 00:12:12,870 --> 00:12:18,525 AT squared, so again, my first steps are to try and isolate the 149 00:12:18,525 --> 00:12:20,700 variable. The term that I'm 150 00:12:20,700 --> 00:12:25,922 looking for. Now this one looks a bit complicated. We 151 00:12:25,922 --> 00:12:31,434 got a half. There would be nice to get rid of the half, 152 00:12:31,434 --> 00:12:36,946 so let's multiply both sides by two. If we do it at this 153 00:12:36,946 --> 00:12:43,306 side, that's just a times by 9. We do it this side 2 * 21 154 00:12:43,306 --> 00:12:48,818 - 15 and it multiplies all of it. So let's do the same 155 00:12:48,818 --> 00:12:53,482 here. Multiply both sides by 2, two times S minus UT. 156 00:12:54,780 --> 00:12:58,338 Equals AT squared. 157 00:13:00,000 --> 00:13:06,104 Going back to the equation we got equals a Times by 9. I just 158 00:13:06,104 --> 00:13:12,208 want a on its own and so I must divide both sides by 9. 159 00:13:15,470 --> 00:13:18,935 Here the thing that's doing the multiplying it's T 160 00:13:18,935 --> 00:13:23,555 squared. So let's divide both sides by T squared and in the 161 00:13:23,555 --> 00:13:27,790 same way as I divided everything by 9, I've got to 162 00:13:27,790 --> 00:13:30,100 divide everything here by T squared. 163 00:13:37,740 --> 00:13:40,400 And that gives me a. 164 00:13:42,540 --> 00:13:47,300 Notice I haven't made any effort to cancel because T 165 00:13:47,300 --> 00:13:52,536 is not a common factor. It is not a common factor. 166 00:13:54,770 --> 00:14:00,590 No, we've been working with the equations that are to do with 167 00:14:00,590 --> 00:14:05,042 uniform acceleration. But there are some other kinds of 168 00:14:05,042 --> 00:14:09,434 equations that we need to gain practice at and in order to 169 00:14:09,434 --> 00:14:13,460 develop these skills, I'm going to start with some made up 170 00:14:13,460 --> 00:14:16,388 equations that don't have physical applications in the 171 00:14:16,388 --> 00:14:20,414 real world, but we will come back once we've developed those 172 00:14:20,414 --> 00:14:24,074 skills to looking at some real equations that do contain 173 00:14:24,074 --> 00:14:27,734 physical applications that we will be able to transform, but 174 00:14:27,734 --> 00:14:32,492 we need to develop some skills to begin with. So first of all. 175 00:14:32,670 --> 00:14:38,440 Let's take this expression, let's call it rather than a 176 00:14:38,440 --> 00:14:39,017 formula. 177 00:14:40,840 --> 00:14:46,990 So then we have Y times 2X plus one equals X Plus One and the 178 00:14:46,990 --> 00:14:52,320 thing that we're going to be after is X. Can we rearrange it 179 00:14:52,320 --> 00:14:54,370 so it says X equals? 180 00:14:55,030 --> 00:14:58,998 Well gain, let me try and model this. 181 00:15:00,600 --> 00:15:05,495 With an equation. So all I need to do to model 182 00:15:05,495 --> 00:15:09,945 this as an equation is replaced the Y by three. 183 00:15:11,470 --> 00:15:16,030 And if I was to solve this inequation, my first step would 184 00:15:16,030 --> 00:15:20,970 be to multiply out the bracket. So let's do that. 6X plus three 185 00:15:20,970 --> 00:15:23,250 is equal to X plus one. 186 00:15:23,970 --> 00:15:30,273 So let's do that over here. Multiply out this bracket. So 187 00:15:30,273 --> 00:15:33,138 that's two XY Plus Y. 188 00:15:33,850 --> 00:15:37,216 Is equal to X plus one. 189 00:15:38,730 --> 00:15:44,749 Our next step with the equation will be to get all the excess 190 00:15:44,749 --> 00:15:50,768 together. So I would take X away from both sides, so that would 191 00:15:50,768 --> 00:15:57,250 be 6X takeaway X +3, and taking the X away from this side just 192 00:15:57,250 --> 00:16:03,269 leaves me with one. So let's do that here. Let's take this X 193 00:16:03,269 --> 00:16:08,825 away from both sides, so have two XY minus X Plus Y. 194 00:16:09,060 --> 00:16:10,648 Is equal to 1. 195 00:16:12,030 --> 00:16:18,390 Now I would naturally want to combine these two in some way. 196 00:16:18,390 --> 00:16:21,570 6X minus X is just 5X. 197 00:16:22,790 --> 00:16:28,880 I can't really do that at this side, but what I can do is 198 00:16:28,880 --> 00:16:34,535 gather together the terms in X by taking out X as a common 199 00:16:34,535 --> 00:16:40,190 factor. So we take out X from this as a common factor. The 200 00:16:40,190 --> 00:16:42,800 other factor is 2 Y minus. 201 00:16:44,010 --> 00:16:46,946 And taking X out of there is one. 202 00:16:48,330 --> 00:16:52,490 Plus Y equals 1. 203 00:16:54,800 --> 00:16:59,935 What now? Well now I've got my ex is together over here. Let's 204 00:16:59,935 --> 00:17:05,860 write this as 5X plus 3 equals 1. My next step will be to leave 205 00:17:05,860 --> 00:17:10,995 the X term on its own by taking three away from each side. 206 00:17:13,190 --> 00:17:19,538 So let's do that here. Lead the Exterm on its own by 207 00:17:19,538 --> 00:17:24,828 taking the other term. That's why away from both sides. 208 00:17:30,200 --> 00:17:34,796 Now I'm just going to simplify this. Five X equals minus two, 209 00:17:34,796 --> 00:17:40,541 and in order to solve this to get a value of XI need to divide 210 00:17:40,541 --> 00:17:45,520 both sides by this number 5. So that's X equals minus two over 211 00:17:45,520 --> 00:17:51,265 5. So if I come back to this this term in the bracket, two Y 212 00:17:51,265 --> 00:17:56,244 minus one is the term that's multiplying the X, and so I need 213 00:17:56,244 --> 00:18:03,498 to divide. Both sides by so X is equal to 1 minus Y over 214 00:18:03,498 --> 00:18:09,372 2Y plus. Sorry made a mistake there. Two Y minus one. 215 00:18:12,020 --> 00:18:16,676 I just look again at what we've done here. We've mimic the 216 00:18:16,676 --> 00:18:20,556 solving of an equation. First, we multiplied out the brackets. 217 00:18:22,300 --> 00:18:25,606 Then we got the terms together 218 00:18:25,606 --> 00:18:29,198 in X. The variable that we were after. 219 00:18:31,310 --> 00:18:33,760 Having got those terms together. 220 00:18:34,540 --> 00:18:38,566 We then isolated them so that they were on their own. 221 00:18:39,970 --> 00:18:43,870 So we need to bear that in mind and follow it through. 222 00:18:47,190 --> 00:18:54,165 Let's take another expression Y over Y plus X. 223 00:18:55,770 --> 00:18:58,710 +5 is equal to X. 224 00:18:59,440 --> 00:19:04,480 This time. We're going to be finding why we're going to be 225 00:19:04,480 --> 00:19:08,562 getting Y equals, and we want some lump of numbers and X is 226 00:19:08,562 --> 00:19:09,504 over this sigh. 227 00:19:11,180 --> 00:19:19,064 So let's model this with an equation Y over Y plus 3 228 00:19:19,064 --> 00:19:23,006 + 5 is equal to 3. 229 00:19:24,580 --> 00:19:28,100 Look at this side. Here we've got an algebraic fraction. 230 00:19:28,950 --> 00:19:33,617 Y over Y plus 3Y plus three is the denominator. It's in the 231 00:19:33,617 --> 00:19:35,053 bottom of the fraction. 232 00:19:35,840 --> 00:19:41,885 So in order to get rid of that, we need to multiply everything 233 00:19:41,885 --> 00:19:46,535 in this equation by this denominator. So I'm going to 234 00:19:46,535 --> 00:19:53,045 write that out in full Y over Y, plus three. Put that in a 235 00:19:53,045 --> 00:19:55,370 bracket times Y plus 3. 236 00:19:56,680 --> 00:19:59,888 Plus five times Y. 237 00:20:00,290 --> 00:20:06,530 Three is equal to three times Y plus three. Let me just 238 00:20:06,530 --> 00:20:13,290 emphasize we have to do the same thing to both sides, so here 239 00:20:13,290 --> 00:20:19,530 I've had to multiply everything on both sides of the equation by 240 00:20:19,530 --> 00:20:22,130 this term Y plus 3. 241 00:20:23,230 --> 00:20:27,190 Here those will cancel out, just leaving me with why, so 242 00:20:27,190 --> 00:20:30,790 let's do that at this site. Let's multiply everything by 243 00:20:30,790 --> 00:20:35,470 this term Y plus X so we know this first one. When we've 244 00:20:35,470 --> 00:20:39,790 multiplied it by wiper sex is just going to give us Why. 245 00:20:41,390 --> 00:20:48,090 Plus five times by Y plus X is equal to 246 00:20:48,090 --> 00:20:52,110 X times by Y plus X. 247 00:20:54,000 --> 00:20:57,344 Coming back to the equation here, faced with a lot of 248 00:20:57,344 --> 00:21:00,688 brackets, we know the first thing we would do is multiply 249 00:21:00,688 --> 00:21:03,120 out the brackets. So let's do that, why? 250 00:21:04,350 --> 00:21:11,938 Plus five times by Y is 5 Y five times by three is 15 251 00:21:11,938 --> 00:21:19,526 is equal 2 three times by Y is 3 Y and three times by 252 00:21:19,526 --> 00:21:21,152 three is 9. 253 00:21:23,210 --> 00:21:25,670 Do the same here, why? 254 00:21:27,050 --> 00:21:34,699 +5 Y. Five times by X Plus 5X is 255 00:21:34,699 --> 00:21:41,940 equal to X times by Y is XY&X times by X is X. 256 00:21:42,600 --> 00:21:43,230 Square. 257 00:21:45,390 --> 00:21:49,688 On this side now we would like to get all of our wise together. 258 00:21:50,680 --> 00:21:53,944 So we've Y plus 259 00:21:53,944 --> 00:22:00,138 5Y6Y. Take away 3 Y so we've six. Why 260 00:22:00,138 --> 00:22:05,367 already and we're going to take away 3Y. Plus 261 00:22:05,367 --> 00:22:11,177 15 is equal to 9, so let's gather all the 262 00:22:11,177 --> 00:22:15,825 wise together Y +5. Y is 6 Y. 263 00:22:17,340 --> 00:22:23,332 I need to bring this term over, so I'll take XY away from both 264 00:22:23,332 --> 00:22:24,616 sides minus XY. 265 00:22:25,410 --> 00:22:29,100 And then let's just write down the other terms. 266 00:22:30,940 --> 00:22:37,002 Now here I simplify this. I'd have six Y takeaway 3 Y and that 267 00:22:37,002 --> 00:22:43,064 would just leave me with three Y plus 15 equals 9. I can't do 268 00:22:43,064 --> 00:22:49,126 that here, but the thing that I can do is bring them much closer 269 00:22:49,126 --> 00:22:51,291 together by taking out why. 270 00:22:52,290 --> 00:22:58,192 The thing that I'm after as a common factor. So let's take why 271 00:22:58,192 --> 00:23:04,094 out of these two terms. Why brackets? Why times by 6 Y means 272 00:23:04,094 --> 00:23:11,358 I must have a 6 in their minus XY means I must have a minus X 273 00:23:11,358 --> 00:23:14,082 there plus 5X equals X squared. 274 00:23:15,380 --> 00:23:22,335 Now over here I'd isolate this term in why by taking 15 away 275 00:23:22,335 --> 00:23:29,825 from both sides 3 Y is equal to. Now this is 9 - 15, 276 00:23:29,825 --> 00:23:36,780 so I've got to do the same here. Take 5X away from each 277 00:23:36,780 --> 00:23:43,735 side, isolating this term in YY times 6 minus X is equal to 278 00:23:43,735 --> 00:23:45,340 X squared minus. 279 00:23:45,480 --> 00:23:47,170 5X. 280 00:23:48,930 --> 00:23:54,824 This side now I've got 3 Yi, just need Y so I would divide 281 00:23:54,824 --> 00:23:59,876 everything on this side by three. So I've got to do the 282 00:23:59,876 --> 00:24:04,507 same with this. I've got to divide everything on this side 283 00:24:04,507 --> 00:24:09,980 by this term 6 minus X so we have X squared minus 5X. 284 00:24:10,640 --> 00:24:14,620 Divided by 6 minus X. 285 00:24:15,960 --> 00:24:21,144 Put it all in a bracket to keep it together. Notice I'm 286 00:24:21,144 --> 00:24:25,464 attempting no canceling 'cause there is no common factor in 287 00:24:25,464 --> 00:24:29,784 this numerator and this denominator. They do not share a 288 00:24:29,784 --> 00:24:34,536 common factor, so that's my answer and I finished. I've got 289 00:24:34,536 --> 00:24:39,288 Y equals a lump of algebra involving X is and numbers. 290 00:24:42,980 --> 00:24:48,656 We started with a real formula. We started with the formula for 291 00:24:48,656 --> 00:24:50,075 a simple pendulum. 292 00:24:50,880 --> 00:24:57,075 Of length Lt equals 2π square root. 293 00:24:57,630 --> 00:25:04,270 L over G and the problem that we posed was what was G in terms of 294 00:25:04,270 --> 00:25:06,900 T? So G. 295 00:25:07,990 --> 00:25:10,909 After G equals. 296 00:25:13,230 --> 00:25:17,778 Let's have a look at a simple equation. Let's say it's 10 297 00:25:17,778 --> 00:25:22,705 equals 2π and I'll keep the two Pike's. Part is just a number, 298 00:25:22,705 --> 00:25:24,979 so 2π is just a number. 299 00:25:26,360 --> 00:25:30,950 Square root of 3 over G. 300 00:25:32,200 --> 00:25:37,030 Now as an equation, this one is a bit tricky because the G is 301 00:25:37,030 --> 00:25:41,170 trapped inside this square root sign. We need to bring it out 302 00:25:41,170 --> 00:25:46,000 and the way to get a square root sign to disappear, so to speak, 303 00:25:46,000 --> 00:25:50,140 is to square both sides, reverse the operation if you like, so 304 00:25:50,140 --> 00:25:56,573 will square. Both sides of this equation so 10 multiplied by 305 00:25:56,573 --> 00:26:02,897 itself. I'm going to write it as 10 squared equals 2π multiplied 306 00:26:02,897 --> 00:26:09,221 by itself, 'cause we're squaring the whole of both sides times by 307 00:26:09,221 --> 00:26:11,329 a nice square this. 308 00:26:11,920 --> 00:26:14,830 Three over G. 309 00:26:16,450 --> 00:26:21,546 Now we need to do the same here, square the whole of both sides. 310 00:26:22,420 --> 00:26:25,306 So that will be T squared. 311 00:26:26,890 --> 00:26:33,208 2π all squared L over G. 312 00:26:35,040 --> 00:26:38,648 Coming back to the equation, jeez, in the denominator I don't 313 00:26:38,648 --> 00:26:43,568 want it there. I want it to say G equals I want G upstairs, so 314 00:26:43,568 --> 00:26:48,488 to speak, so I have to get rid of it out of the denominator and 315 00:26:48,488 --> 00:26:52,096 so to do that I must multiply both sides by G. 316 00:26:52,950 --> 00:27:00,066 So I've got 10 squared times. G is equal to 2π squared 317 00:27:00,066 --> 00:27:07,182 times three, so let's do that here. Multiply both sides by G. 318 00:27:07,182 --> 00:27:14,298 So I have T squared times by G is equal to 2π 319 00:27:14,298 --> 00:27:17,263 all squared times by L. 320 00:27:19,500 --> 00:27:26,700 Gee, I want on its own. I want to say G equal, so I must divide 321 00:27:26,700 --> 00:27:33,000 by this thing 10 squared. So G is equal to 2π all squared times 322 00:27:33,000 --> 00:27:35,250 3, all over 10 squared. 323 00:27:36,320 --> 00:27:43,376 And I can work that out. Let's do the same with this G equals 324 00:27:43,376 --> 00:27:49,928 2π all squared times by L, and I'm getting rid of this T 325 00:27:49,928 --> 00:27:53,456 squared by dividing everything by T squared. 326 00:27:55,800 --> 00:27:58,544 And we can leave it like that. 327 00:27:59,350 --> 00:28:05,426 We can make it look a little bit nicer if we want to buy 328 00:28:05,426 --> 00:28:09,332 spotting that we've got a square here under Square 329 00:28:09,332 --> 00:28:13,672 there, and it might be nice to combine those two. 330 00:28:13,672 --> 00:28:17,578 Squaring process is by writing 2π over T all 331 00:28:17,578 --> 00:28:19,314 squared times by L. 332 00:28:20,410 --> 00:28:23,785 Basically this is the answer that we finished with. 333 00:28:24,700 --> 00:28:27,890 Let's take another couple of. 334 00:28:29,280 --> 00:28:33,804 Examples of real formula that we might want to be able to 335 00:28:33,804 --> 00:28:38,705 manipulate. So let's have a look at the lens Formula One over F. 336 00:28:39,540 --> 00:28:41,460 Is equal to one over U. 337 00:28:43,030 --> 00:28:44,098 This one over V. 338 00:28:44,930 --> 00:28:46,578 Let's say it's you. 339 00:28:48,120 --> 00:28:50,706 That we're after we want you 340 00:28:50,706 --> 00:28:55,655 equals. Well, let me write down an equation. 341 00:28:58,550 --> 00:28:59,260 Similar. 342 00:29:00,720 --> 00:29:05,164 One over 3 equals 1 over U plus one over 5. 343 00:29:06,650 --> 00:29:12,161 I want to isolate this term first in one over you. 344 00:29:13,380 --> 00:29:18,983 To do that, I'm going to take this away from both sides. 1/3 345 00:29:18,983 --> 00:29:21,569 takeaway 150 is equal to one 346 00:29:21,569 --> 00:29:26,862 over you. So let's do that. Isolate this term in you. 347 00:29:29,000 --> 00:29:33,680 And so to do that, we take away one over V from both sides, so 348 00:29:33,680 --> 00:29:34,928 we want over F. 349 00:29:35,530 --> 00:29:39,523 Take away one over V is equal to one over you. 350 00:29:40,930 --> 00:29:45,493 Now, faced with problems like this, it's very, very 351 00:29:45,493 --> 00:29:49,042 tempting to simply turn everything upside down. 352 00:29:50,290 --> 00:29:56,800 Well, I just have a think about that. This says 1/3 - 1/5. Now a 353 00:29:56,800 --> 00:29:58,970 third is bigger than 1/5. 354 00:30:00,810 --> 00:30:05,220 So 1/3 - 1/5 is a positive number, so you at the very least 355 00:30:05,220 --> 00:30:06,795 has got to be positive. 356 00:30:07,860 --> 00:30:12,080 Watch what happens if I just turn everything upside down. 357 00:30:12,710 --> 00:30:19,670 3 - 5 equals U, which tells me that minus two is equal to you. 358 00:30:19,670 --> 00:30:25,702 But we just agreed that you had to be positive and not negative. 359 00:30:25,702 --> 00:30:27,558 You can't do that. 360 00:30:28,380 --> 00:30:32,020 These are fractions, and so because their fractions we have 361 00:30:32,020 --> 00:30:36,388 to combine them as fractions, which means we have to find a 362 00:30:36,388 --> 00:30:40,028 common denominator and a common denominator for three and five 363 00:30:40,028 --> 00:30:44,032 is a number that both three and five will divide into. 364 00:30:44,910 --> 00:30:48,627 Easiest number is 3 times by 5. 365 00:30:49,270 --> 00:30:55,500 So three Zing to three times by 5, which is 15 goes five times. 366 00:30:55,500 --> 00:31:02,175 So I found multiplied 3 by 5. I must multiply by this one by 5, 367 00:31:02,175 --> 00:31:03,955 so that's 5 minus. 368 00:31:04,600 --> 00:31:10,168 I've multiplied 5 by three, so I multiply this one by three 369 00:31:10,168 --> 00:31:17,128 equals one over. You know if I did that here, I've got to do it 370 00:31:17,128 --> 00:31:21,746 over here. So I want to common denominator. 371 00:31:23,300 --> 00:31:27,944 Here the common denominator I talk was three times by 5, so 372 00:31:27,944 --> 00:31:30,266 let's take F times by vis. 373 00:31:32,720 --> 00:31:34,688 I've multiplied F. 374 00:31:36,230 --> 00:31:38,960 By the so I must multiply the 375 00:31:38,960 --> 00:31:40,730 one. By faith. 376 00:31:43,890 --> 00:31:44,730 Minus. 377 00:31:45,830 --> 00:31:49,436 I've multiplied the V by F. 378 00:31:50,180 --> 00:31:53,436 So I must multiply the one by F. 379 00:31:58,700 --> 00:32:04,212 Now if I come back over here, let me just simplify this. This 380 00:32:04,212 --> 00:32:10,148 is 2 over 15 is one over you, and now I've got a complete 381 00:32:10,148 --> 00:32:15,236 fraction on both sides. I can turn it upside down and say 382 00:32:15,236 --> 00:32:16,932 that's what you is. 383 00:32:18,370 --> 00:32:23,430 Now there was a lot of calculation went on here. I 384 00:32:23,430 --> 00:32:28,490 can't do that calculation here, but I have got a complete 385 00:32:28,490 --> 00:32:31,710 fraction here so I can turn it. 386 00:32:34,740 --> 00:32:35,820 Upside down. 387 00:32:36,850 --> 00:32:38,198 To give me you. 388 00:32:39,670 --> 00:32:44,930 So let's take as our final example the time dilation 389 00:32:44,930 --> 00:32:46,508 formula from relativity. 390 00:32:47,480 --> 00:32:52,709 The formula is T equals T, not. 391 00:32:53,990 --> 00:32:55,440 Divided by. 392 00:32:56,630 --> 00:33:04,206 1. Minus. V squared over C squared to the 393 00:33:04,206 --> 00:33:07,296 power. 1/2 or square root. 394 00:33:08,880 --> 00:33:13,534 So let's take some numbers and in fact what we're going to be 395 00:33:13,534 --> 00:33:17,114 after is we're going to be looking for this expression 396 00:33:17,114 --> 00:33:21,410 here, V over C. So I just write that down there. That's 397 00:33:21,410 --> 00:33:26,422 what we're going to be looking for. Can we get V over C in 398 00:33:26,422 --> 00:33:27,854 terms of T&T Nord? 399 00:33:29,160 --> 00:33:34,552 So let's have 6 equals 5 over 1 400 00:33:34,552 --> 00:33:36,574 minus X squared. 401 00:33:39,170 --> 00:33:42,798 To the half. Now solving this. 402 00:33:43,410 --> 00:33:48,282 What would we do? Well, we've got a square root. Let's Square 403 00:33:48,282 --> 00:33:53,560 both sides in order to get rid of that square root. So that 404 00:33:53,560 --> 00:33:58,432 would be 6 squared equals 5 squared all over 1 minus X 405 00:33:58,432 --> 00:34:03,666 squared. So let's do that here. Square both sides. 406 00:34:04,320 --> 00:34:11,074 T squared. Equals TN squared all over 1 407 00:34:11,074 --> 00:34:15,898 minus V squared over C squared. 408 00:34:17,590 --> 00:34:24,394 Now, well here the term I want in X is in the denominator. I 409 00:34:24,394 --> 00:34:29,254 need it upstairs, so let's multiply both sides of this 410 00:34:29,254 --> 00:34:34,114 equation by one minus X squared. That would be 6 411 00:34:34,114 --> 00:34:39,460 squared times 1 minus X squared is equal to 5 squared. 412 00:34:40,740 --> 00:34:42,750 So let's do that here. 413 00:34:43,990 --> 00:34:49,438 T squared times, 1 minus V squared over 414 00:34:49,438 --> 00:34:54,886 C squared is equal to T Nord squared. 415 00:34:57,150 --> 00:35:00,566 Now I want the X squared bit. 416 00:35:01,720 --> 00:35:06,652 And it would be nice perhaps to multiply out the bracket, but 417 00:35:06,652 --> 00:35:11,584 there's a slightly quicker way. I can divide both sides by 6 418 00:35:11,584 --> 00:35:15,694 squared, and that's nice. 'cause it keeps the square bits 419 00:35:15,694 --> 00:35:21,448 together, so to speak. So let me do that one minus X squared is 420 00:35:21,448 --> 00:35:26,791 equal to 5 squared over 6 squared. So I'm going to do that 421 00:35:26,791 --> 00:35:31,723 here. Divide both sides by T squared, 1 minus V squared over 422 00:35:31,723 --> 00:35:38,808 C squared. Is equal to TN squared over T squared? 423 00:35:40,740 --> 00:35:41,190 Now. 424 00:35:42,430 --> 00:35:45,430 Running out of paper here. So what I'm going to do is 425 00:35:45,430 --> 00:35:48,680 turn over and write this one down at the top of the next 426 00:35:48,680 --> 00:35:50,180 page, and this one as well. 427 00:35:52,980 --> 00:36:00,042 So we've got 1 minus V squared over. C squared is 428 00:36:00,042 --> 00:36:05,178 equal to T not squared over T squared. 429 00:36:07,340 --> 00:36:13,676 And here, with one minus X squared is equal to 5 squared 430 00:36:13,676 --> 00:36:15,260 over 6 squared. 431 00:36:17,050 --> 00:36:23,836 It's The X squared that we're after so I can add X squared 432 00:36:23,836 --> 00:36:30,100 both sides, so this is one equals 5 squared over 6 squared 433 00:36:30,100 --> 00:36:37,408 plus X squared. So let me add V squared over C squared to each 434 00:36:37,408 --> 00:36:44,194 side. One is equal to T not squared over T squared plus B 435 00:36:44,194 --> 00:36:46,282 squared over C squared. 436 00:36:47,840 --> 00:36:50,207 Now I want the X squared on it so. 437 00:36:51,440 --> 00:36:58,556 So take this away from both sides. 1 - 5 squared over 438 00:36:58,556 --> 00:37:05,672 6 squared is equal to X squared, so 1 minus T, not 439 00:37:05,672 --> 00:37:12,788 squared over T squared is equal to V squared over C squared. 440 00:37:12,788 --> 00:37:15,753 Taking this away from both 441 00:37:15,753 --> 00:37:23,507 sides. Almost there now it's X that I want not X squared 442 00:37:23,507 --> 00:37:26,909 selects. Take the square root of 443 00:37:26,909 --> 00:37:33,923 both sides. So do the same here. The oversee that I 444 00:37:33,923 --> 00:37:39,593 want, so let's take the square root of both sides. 445 00:37:43,540 --> 00:37:47,968 And that's it. We have found the ratio of the velocity to 446 00:37:47,968 --> 00:37:51,658 the speed of light in terms of the two times. 447 00:37:52,680 --> 00:37:57,095 So. That concludes this video on transformation of formula thing 448 00:37:57,095 --> 00:38:02,065 to do is to treat it always as though it was an equation that 449 00:38:02,065 --> 00:38:03,840 you were trying to solve. 450 00:38:05,000 --> 00:38:10,603 In terms of the variable that you want, X equals G equals and 451 00:38:10,603 --> 00:38:15,344 you just perform the steps as though you were solving an 452 00:38:15,344 --> 00:38:20,516 equation. If you keep that in mind, then most of your problems 453 00:38:20,516 --> 00:38:22,671 should be taken care of.