1 00:00:00,030 --> 00:00:01,979 mirror equation problems can be 2 00:00:01,979 --> 00:00:03,480 intimidating when you first deal with 3 00:00:03,480 --> 00:00:05,609 them and that's not because the mirror 4 00:00:05,609 --> 00:00:07,379 equation is all that difficult it's kind 5 00:00:07,379 --> 00:00:09,389 of easy it's just a few fractions added 6 00:00:09,389 --> 00:00:11,309 together the place where it gets tricky 7 00:00:11,309 --> 00:00:12,960 is deciding whether these should be 8 00:00:12,960 --> 00:00:14,820 positive or negative so there's a bunch 9 00:00:14,820 --> 00:00:16,800 of positive and negative sign decisions 10 00:00:16,800 --> 00:00:18,900 that you have to make and if you make 11 00:00:18,900 --> 00:00:20,880 even one of those incorrectly you can 12 00:00:20,880 --> 00:00:22,710 get the wrong answer so let's do a few 13 00:00:22,710 --> 00:00:24,900 more equation problems and you can see 14 00:00:24,900 --> 00:00:27,000 how the signs work now I have to warn 15 00:00:27,000 --> 00:00:29,070 you everyone has their own sign 16 00:00:29,070 --> 00:00:30,929 convention there's a lot of different 17 00:00:30,929 --> 00:00:32,880 sign conventions when you deal with 18 00:00:32,880 --> 00:00:35,520 optics the one I'm going to use is the 19 00:00:35,520 --> 00:00:37,380 one that I feel like most textbooks are 20 00:00:37,380 --> 00:00:39,540 using these days and it's the one where 21 00:00:39,540 --> 00:00:42,540 anything on the front side of the mirror 22 00:00:42,540 --> 00:00:45,660 so by front side your eye should be over 23 00:00:45,660 --> 00:00:46,800 here somewhere so let's say your eyes 24 00:00:46,800 --> 00:00:50,010 over here right you're looking at some 25 00:00:50,010 --> 00:00:52,260 object maybe it's an arrow or a crown 26 00:00:52,260 --> 00:00:54,300 right here a blue crayon you're holding 27 00:00:54,300 --> 00:00:56,489 in front of this mirror right here so 28 00:00:56,489 --> 00:00:58,590 this is the mirror right here and the 29 00:00:58,590 --> 00:01:00,449 convention I'm using is that anything on 30 00:01:00,449 --> 00:01:02,940 this side of the mirror is going to be 31 00:01:02,940 --> 00:01:05,549 counted as positive so if it's a focal 32 00:01:05,549 --> 00:01:08,250 point on that side positive if it's an 33 00:01:08,250 --> 00:01:10,380 object distance on this side its 34 00:01:10,380 --> 00:01:12,630 positive and if the image distance comes 35 00:01:12,630 --> 00:01:14,460 out positive you'll know that it's also 36 00:01:14,460 --> 00:01:16,200 on this front side of the mirror 37 00:01:16,200 --> 00:01:18,360 anything that comes out negative so if 38 00:01:18,360 --> 00:01:19,890 we get a negative image distance after 39 00:01:19,890 --> 00:01:22,170 we do our calculation we'll know that 40 00:01:22,170 --> 00:01:24,000 that thing is behind the mirror that 41 00:01:24,000 --> 00:01:26,340 kind of makes sense negative like behind 42 00:01:26,340 --> 00:01:29,880 positive like in front so the side over 43 00:01:29,880 --> 00:01:32,520 here is in front of the mirror positive 44 00:01:32,520 --> 00:01:35,040 and back here would be behind the mirror 45 00:01:35,040 --> 00:01:36,840 and that would be negative so I've got 46 00:01:36,840 --> 00:01:38,430 some numbers in here already let's just 47 00:01:38,430 --> 00:01:40,470 solve this one so what do we do we're 48 00:01:40,470 --> 00:01:42,270 going to use this mirror equation we're 49 00:01:42,270 --> 00:01:44,040 going to say that one over the focal 50 00:01:44,040 --> 00:01:46,259 length and already we have to decide on 51 00:01:46,259 --> 00:01:48,060 a positive or negative sign so this 52 00:01:48,060 --> 00:01:49,740 mirror the way it's shaped right here 53 00:01:49,740 --> 00:01:51,149 based on how we're looking at it is a 54 00:01:51,149 --> 00:01:53,280 concave mirror and with the sign 55 00:01:53,280 --> 00:01:55,170 conventions we just discussed and the 56 00:01:55,170 --> 00:01:57,689 signs I'm using in this formula concave 57 00:01:57,689 --> 00:02:00,149 mirrors always have a positive focal 58 00:02:00,149 --> 00:02:01,710 length so in other words since this 59 00:02:01,710 --> 00:02:04,439 focal point is 4 centimeters from the 60 00:02:04,439 --> 00:02:06,090 center of the mirror I'm gonna have to 61 00:02:06,090 --> 00:02:08,580 plug in the focal length as positive 4 62 00:02:08,580 --> 00:02:11,430 centimeters notice I'm not converting 63 00:02:11,430 --> 00:02:13,380 that's okay if you leave everything in 64 00:02:13,380 --> 00:02:14,970 centimetres you'll just get an answer in 65 00:02:14,970 --> 00:02:16,710 centimeters so it's okay you don't have 66 00:02:16,710 --> 00:02:18,090 to convert as long as everything's in 67 00:02:18,090 --> 00:02:19,770 the same units and we'll set that equal 68 00:02:19,770 --> 00:02:23,250 to one over the object distance just one 69 00:02:23,250 --> 00:02:25,220 other warning sometimes instead of do 70 00:02:25,220 --> 00:02:28,590 you'll see this as so4 object distance 71 00:02:28,590 --> 00:02:31,860 or you might see di as SI for image 72 00:02:31,860 --> 00:02:34,320 distance is the same thing it's just a 73 00:02:34,320 --> 00:02:36,300 different letter so the object distance 74 00:02:36,300 --> 00:02:39,330 again is on this side in front of the 75 00:02:39,330 --> 00:02:41,490 mirror so it's going to be positive 12 76 00:02:41,490 --> 00:02:43,710 centimeters since it's located 12 77 00:02:43,710 --> 00:02:45,300 centimeters from the center of the 78 00:02:45,300 --> 00:02:46,830 mirror so there's also going to be 79 00:02:46,830 --> 00:02:48,810 positive 12 centimeters and now we're 80 00:02:48,810 --> 00:02:51,690 going to add to that one over the image 81 00:02:51,690 --> 00:02:53,670 distance the image distance we don't 82 00:02:53,670 --> 00:02:55,230 know I haven't drawn the image on here 83 00:02:55,230 --> 00:02:56,610 it's going to be a surprise we don't 84 00:02:56,610 --> 00:02:57,840 know what this is going to be but we can 85 00:02:57,840 --> 00:02:59,670 solve for it so we can solve for image 86 00:02:59,670 --> 00:03:02,280 distance I'll subtract 1 over 12 from 87 00:03:02,280 --> 00:03:04,410 both sides which will give us 1 over 4 88 00:03:04,410 --> 00:03:07,620 centimeters minus 1 over 12 centimeters 89 00:03:07,620 --> 00:03:09,900 and that's going to have to equal 1 over 90 00:03:09,900 --> 00:03:13,020 the image distance so 1/4 you could 91 00:03:13,020 --> 00:03:16,410 rewrite as three twelfths so three 92 00:03:16,410 --> 00:03:19,200 twelfths minus 1/12 is just going to be 93 00:03:19,200 --> 00:03:21,330 two twelfths and that's going to equal 94 00:03:21,330 --> 00:03:23,640 one over the image distance but two 95 00:03:23,640 --> 00:03:27,240 twelfths is just 1/6 so 1 over the image 96 00:03:27,240 --> 00:03:29,610 distance is just going to be 1/6 but 97 00:03:29,610 --> 00:03:31,170 that's what 1 over the image distance is 98 00:03:31,170 --> 00:03:33,540 sometimes people forget to flip this at 99 00:03:33,540 --> 00:03:35,850 the end we don't want one over the image 100 00:03:35,850 --> 00:03:38,280 distance we want the image distance so 101 00:03:38,280 --> 00:03:40,620 finally you take one over each side and 102 00:03:40,620 --> 00:03:42,690 we solve and we get that the image 103 00:03:42,690 --> 00:03:44,730 distance is going to be 6 centimeters 104 00:03:44,730 --> 00:03:46,350 and it came out positive that's 105 00:03:46,350 --> 00:03:49,500 important this came out to be positive 6 106 00:03:49,500 --> 00:03:51,120 centimeters so where's our image going 107 00:03:51,120 --> 00:03:53,280 to be so since this image distance came 108 00:03:53,280 --> 00:03:55,260 out to be positive our image is going to 109 00:03:55,260 --> 00:03:57,330 be in front of the mirror so it's going 110 00:03:57,330 --> 00:03:58,560 to be over here it's going to be 6 111 00:03:58,560 --> 00:04:00,930 centimeters from the mirror somewhere 112 00:04:00,930 --> 00:04:03,450 around here so at this point right here 113 00:04:03,450 --> 00:04:05,580 is where our image is going to be but we 114 00:04:05,580 --> 00:04:06,959 don't know how big it's going to be or 115 00:04:06,959 --> 00:04:08,340 whether it's right side up or upside 116 00:04:08,340 --> 00:04:10,440 down to figure that out we have to use a 117 00:04:10,440 --> 00:04:12,090 different equation and that equation is 118 00:04:12,090 --> 00:04:14,250 called the magnification equation says 119 00:04:14,250 --> 00:04:16,738 that the magnification is equal to the 120 00:04:16,738 --> 00:04:18,450 height of the image divided by the 121 00:04:18,450 --> 00:04:20,579 height of the object and that's equal to 122 00:04:20,579 --> 00:04:24,210 negative the image distance divided by 123 00:04:24,210 --> 00:04:26,729 the object distance so it turns out this 124 00:04:26,729 --> 00:04:27,240 race 125 00:04:27,240 --> 00:04:29,220 vo of negative image distance over 126 00:04:29,220 --> 00:04:31,620 object distance is always equal to the 127 00:04:31,620 --> 00:04:34,199 ratio of the height of the image over 128 00:04:34,199 --> 00:04:35,669 the height of the object so what's the 129 00:04:35,669 --> 00:04:36,930 height of our image going to be let's 130 00:04:36,930 --> 00:04:38,580 just solve for it if we solve for the 131 00:04:38,580 --> 00:04:40,590 height of our image we get that the 132 00:04:40,590 --> 00:04:41,660 height of the image is going to be 133 00:04:41,660 --> 00:04:44,310 multiplied both sides by H Oh we'll have 134 00:04:44,310 --> 00:04:46,199 the negative signs already here so 135 00:04:46,199 --> 00:04:49,410 negative height of the object times this 136 00:04:49,410 --> 00:04:52,259 ratio of the image distance over the 137 00:04:52,259 --> 00:04:54,180 object distance and now we can just plug 138 00:04:54,180 --> 00:04:56,039 in numbers the height of the object it 139 00:04:56,039 --> 00:04:57,900 says that's three centimeters tall right 140 00:04:57,900 --> 00:04:59,729 here so it's three centimeters so 141 00:04:59,729 --> 00:05:02,280 negative three centimeters times the 142 00:05:02,280 --> 00:05:05,880 ratio of the image distance was six the 143 00:05:05,880 --> 00:05:08,190 object distance was 12 and so if you 144 00:05:08,190 --> 00:05:08,819 solve this 145 00:05:08,819 --> 00:05:11,159 you'll get 1/2 of negative 3 which is 146 00:05:11,159 --> 00:05:15,000 negative 1.5 centimeters the negative 147 00:05:15,000 --> 00:05:17,430 sign means that this image got inverted 148 00:05:17,430 --> 00:05:19,080 so got flipped over it's going to be 149 00:05:19,080 --> 00:05:20,490 upside down compared to what it was 150 00:05:20,490 --> 00:05:23,069 before and the 1.5 is how tall is going 151 00:05:23,069 --> 00:05:26,190 to be so what we end up with is an image 152 00:05:26,190 --> 00:05:28,949 6 centimeters from the mirror and it's 153 00:05:28,949 --> 00:05:31,139 going to be have a height half as tall 154 00:05:31,139 --> 00:05:33,360 so 1.5 centimeters and it's going to be 155 00:05:33,360 --> 00:05:35,610 upside down because of this negative 156 00:05:35,610 --> 00:05:37,710 sign so it's going to be 1.5 centimeters 157 00:05:37,710 --> 00:05:40,020 tall and upside down that's what you're 158 00:05:40,020 --> 00:05:41,550 going to see if you look into this 159 00:05:41,550 --> 00:05:43,919 mirror it's like a funhouse mirror it's 160 00:05:43,919 --> 00:05:45,870 a weird curved mirror you'd see an 161 00:05:45,870 --> 00:05:48,120 upside-down image right here it might 162 00:05:48,120 --> 00:05:49,650 look like you could reach out and grab 163 00:05:49,650 --> 00:05:51,030 it but it's going to be an optical 164 00:05:51,030 --> 00:05:52,349 illusion there's going to be no object 165 00:05:52,349 --> 00:05:53,580 there it's just going to be the image of 166 00:05:53,580 --> 00:05:56,159 this object here so that's an example 167 00:05:56,159 --> 00:05:58,919 using a concave mirror what would change 168 00:05:58,919 --> 00:06:01,050 what if we did this what if we took our 169 00:06:01,050 --> 00:06:03,060 object so say we don't put it here 170 00:06:03,060 --> 00:06:05,880 anymore we move it inside here so 171 00:06:05,880 --> 00:06:08,430 instead of being at 12 centimeters we 172 00:06:08,430 --> 00:06:11,460 moved it to like 3 centimeters what 173 00:06:11,460 --> 00:06:13,469 would we do differently everything would 174 00:06:13,469 --> 00:06:14,969 be the same we just would plug in 175 00:06:14,969 --> 00:06:17,280 instead of positive 12 down here we'd 176 00:06:17,280 --> 00:06:19,949 plug in positive 3 so the mirror 177 00:06:19,949 --> 00:06:21,569 equation works exactly the same you 178 00:06:21,569 --> 00:06:23,340 still plug in whatever that object 179 00:06:23,340 --> 00:06:25,650 distance is you solve for your image 180 00:06:25,650 --> 00:06:27,180 distance we're of course going to get a 181 00:06:27,180 --> 00:06:29,069 different image distance but whatever 182 00:06:29,069 --> 00:06:30,300 you get that would tell you where the 183 00:06:30,300 --> 00:06:31,800 image is and then you would take that 184 00:06:31,800 --> 00:06:34,349 plug it into the magnification equation 185 00:06:34,349 --> 00:06:36,389 if you wanted to decide how big the 186 00:06:36,389 --> 00:06:37,590 image is going to be 187 00:06:37,590 --> 00:06:38,700 whether it's going to be right side up 188 00:06:38,700 --> 00:06:40,560 or upside down so these numbers going to 189 00:06:40,560 --> 00:06:42,270 be different but you would use this 190 00:06:42,270 --> 00:06:44,460 equation the exact same way now what 191 00:06:44,460 --> 00:06:45,960 would happen if instead of using a 192 00:06:45,960 --> 00:06:47,310 concave mirror 193 00:06:47,310 --> 00:06:50,550 we used a convex mirror let's say we use 194 00:06:50,550 --> 00:06:52,680 the mirror shaped like this so imagine 195 00:06:52,680 --> 00:06:55,020 our eye again is over here looking at 196 00:06:55,020 --> 00:06:57,240 this object inside of the mirror and 197 00:06:57,240 --> 00:06:58,350 we're going to see an image of the 198 00:06:58,350 --> 00:07:00,030 object we're going to see the object 199 00:07:00,030 --> 00:07:01,200 right here but we're also going to see 200 00:07:01,200 --> 00:07:03,870 the image of the object this mirror this 201 00:07:03,870 --> 00:07:06,450 time instead of concave this is a convex 202 00:07:06,450 --> 00:07:09,420 mirror so its focal length is behind the 203 00:07:09,420 --> 00:07:11,100 mirror so what do we do now to figure 204 00:07:11,100 --> 00:07:13,620 out where the images we again use the 205 00:07:13,620 --> 00:07:15,060 mirror equation we're going to use the 206 00:07:15,060 --> 00:07:17,130 same equation we're going to have one 207 00:07:17,130 --> 00:07:19,350 over the focal length and again I 208 00:07:19,350 --> 00:07:21,150 immediately have to make a decision on 209 00:07:21,150 --> 00:07:23,070 the sign with the convention that I'm 210 00:07:23,070 --> 00:07:25,350 using this focal length is behind the 211 00:07:25,350 --> 00:07:27,300 mirror so this focal length for a convex 212 00:07:27,300 --> 00:07:28,830 mirror is going to be negative so this 213 00:07:28,830 --> 00:07:30,630 would be negative four centimeters and 214 00:07:30,630 --> 00:07:32,790 that's going to equal one over the 215 00:07:32,790 --> 00:07:35,070 object distance well again the object is 216 00:07:35,070 --> 00:07:36,600 in front of the mirror twelve 217 00:07:36,600 --> 00:07:38,760 centimeters so this object distance is 218 00:07:38,760 --> 00:07:40,770 going to be positive twelve centimeters 219 00:07:40,770 --> 00:07:42,780 and then we add to that the image 220 00:07:42,780 --> 00:07:44,610 distance which we don't know this is 221 00:07:44,610 --> 00:07:46,050 what we're going to solve for so this 222 00:07:46,050 --> 00:07:47,340 time if we solve we're going to have 1 223 00:07:47,340 --> 00:07:49,290 over negative 4 and again we have to 224 00:07:49,290 --> 00:07:51,930 subtract 1 over 12 and that's all going 225 00:07:51,930 --> 00:07:54,360 to have to be equal to 1 over the image 226 00:07:54,360 --> 00:07:56,850 distance so now on the left hand side we 227 00:07:56,850 --> 00:07:59,670 have negative 1/4 but that's the same as 228 00:07:59,670 --> 00:08:02,400 negative 3 twelfths so negative 3 229 00:08:02,400 --> 00:08:05,190 twelfths minus 1/12 is the same as 230 00:08:05,190 --> 00:08:08,370 negative 4 twelfths and that's got to 231 00:08:08,370 --> 00:08:10,800 equal 1 over the image distance but 232 00:08:10,800 --> 00:08:12,840 negative 4 twelfths is the same as 233 00:08:12,840 --> 00:08:16,080 negative 1/3 so 1 over the image 234 00:08:16,080 --> 00:08:18,630 distance has to be equal to negative 1/3 235 00:08:18,630 --> 00:08:21,450 and if we flip that over we get that the 236 00:08:21,450 --> 00:08:23,370 image distance finally is going to be 237 00:08:23,370 --> 00:08:26,700 negative 3 centimeters so in other words 238 00:08:26,700 --> 00:08:30,210 this here is negative 1/3 so when you 239 00:08:30,210 --> 00:08:31,860 flip that over you get that the image 240 00:08:31,860 --> 00:08:34,140 distance is negative 3 centimeters where 241 00:08:34,140 --> 00:08:35,940 is this image going to be well since it 242 00:08:35,940 --> 00:08:38,100 came out negative that means it's behind 243 00:08:38,100 --> 00:08:39,900 the mirror because that's the sign 244 00:08:39,900 --> 00:08:41,700 convention we're using so it's going to 245 00:08:41,700 --> 00:08:43,799 be 3 centimeters behind the mirror so 246 00:08:43,799 --> 00:08:45,930 it's going to be like over here about at 247 00:08:45,930 --> 00:08:48,000 this point right around here somewhere 248 00:08:48,000 --> 00:08:50,040 three centimeters behind 249 00:08:50,040 --> 00:08:51,600 this mere and again if we want to figure 250 00:08:51,600 --> 00:08:53,490 out how tall it's going to be whether 251 00:08:53,490 --> 00:08:55,110 it's right side up or upside down we're 252 00:08:55,110 --> 00:08:56,550 going to have to use the magnification 253 00:08:56,550 --> 00:08:58,740 equation and that magnification equation 254 00:08:58,740 --> 00:09:00,810 looked like this it said that the height 255 00:09:00,810 --> 00:09:02,070 of the image over the height of the 256 00:09:02,070 --> 00:09:04,560 object had to be negative image distance 257 00:09:04,560 --> 00:09:06,540 over object distance so let's just see 258 00:09:06,540 --> 00:09:07,769 what this right-hand side is going to be 259 00:09:07,769 --> 00:09:09,870 if we just set this equal to it's going 260 00:09:09,870 --> 00:09:12,329 to be negative of negative 3 is the 261 00:09:12,329 --> 00:09:13,889 image distance so I have to plug in the 262 00:09:13,889 --> 00:09:15,720 negative 3 and you you keep that 263 00:09:15,720 --> 00:09:17,190 negative in there this negative out here 264 00:09:17,190 --> 00:09:19,560 comes along always but now we have 265 00:09:19,560 --> 00:09:21,600 another negative inside of this image 266 00:09:21,600 --> 00:09:23,850 distance of negative 3 and the object 267 00:09:23,850 --> 00:09:26,279 distance again was 12 centimeters so 268 00:09:26,279 --> 00:09:28,139 what are we left with we get negative of 269 00:09:28,139 --> 00:09:30,959 negative 3 which is positive 3 over 12 270 00:09:30,959 --> 00:09:33,870 which is positive 1/4 and the 271 00:09:33,870 --> 00:09:35,759 centimeters cancel so what this ratio 272 00:09:35,759 --> 00:09:38,430 tells you which is the magnification is 273 00:09:38,430 --> 00:09:40,139 that the image is not going to be 274 00:09:40,139 --> 00:09:42,000 inverted this positive means it's going 275 00:09:42,000 --> 00:09:44,819 to be right side up and the 1/4 ratio 276 00:09:44,819 --> 00:09:47,399 means it's going to be 1/4 as large the 277 00:09:47,399 --> 00:09:49,259 image I should say is going to be 1/4 as 278 00:09:49,259 --> 00:09:51,300 large as the object is going to be and 279 00:09:51,300 --> 00:09:53,430 the reason is this ratio is equal to 280 00:09:53,430 --> 00:09:54,720 height of the image over the height of 281 00:09:54,720 --> 00:09:56,190 the object so in other words if I can 282 00:09:56,190 --> 00:09:57,690 multiply if I want to multiply both 283 00:09:57,690 --> 00:09:59,880 sides I can say that height of the image 284 00:09:59,880 --> 00:10:03,000 is equal to positive 1/4 times the 285 00:10:03,000 --> 00:10:05,310 height of the object well the height of 286 00:10:05,310 --> 00:10:07,139 our object was 3 centimeters in this 287 00:10:07,139 --> 00:10:08,970 case but whatever your height of the 288 00:10:08,970 --> 00:10:11,220 object is you multiply it by this ratio 289 00:10:11,220 --> 00:10:12,870 of negative DRDO 290 00:10:12,870 --> 00:10:14,190 and you get what the height of the image 291 00:10:14,190 --> 00:10:15,630 is going to be so we're going to get 1 4 292 00:10:15,630 --> 00:10:20,550 3 to get positive 3/4 of a centimeter so 293 00:10:20,550 --> 00:10:21,899 going to be it's going to be tiny it's 294 00:10:21,899 --> 00:10:23,670 gonna be a little little image that's 295 00:10:23,670 --> 00:10:25,350 going to be right side up so it's going 296 00:10:25,350 --> 00:10:27,180 to be right here but it's going to be 297 00:10:27,180 --> 00:10:29,870 teeny it's only going to be like this 298 00:10:29,870 --> 00:10:32,610 3/4 of a centimeter tall that's what our 299 00:10:32,610 --> 00:10:34,740 image is going to look like so recapping 300 00:10:34,740 --> 00:10:37,199 you can use the mirror equation to 301 00:10:37,199 --> 00:10:39,180 figure out where the images are going to 302 00:10:39,180 --> 00:10:41,069 be located the sign convention we're 303 00:10:41,069 --> 00:10:43,470 using is that objects images and focal 304 00:10:43,470 --> 00:10:44,939 lengths in front of the mirror are going 305 00:10:44,939 --> 00:10:47,100 to be positive anything behind the 306 00:10:47,100 --> 00:10:48,509 mirror is going to be negative and you 307 00:10:48,509 --> 00:10:50,160 could use the magnification equation to 308 00:10:50,160 --> 00:10:51,600 figure out how tall the image is going 309 00:10:51,600 --> 00:10:54,029 to be relative to the object by taking 310 00:10:54,029 --> 00:10:56,850 negative the image distance over the 311 00:10:56,850 --> 00:00:00,000 object distance