1 00:00:00,442 --> 00:00:02,601 - [Voiceover] What does period and frequency mean? 2 00:00:02,601 --> 00:00:04,316 The period is the number of seconds 3 00:00:04,316 --> 00:00:06,111 it takes for a process to complete 4 00:00:06,111 --> 00:00:09,127 an entire cycle, circle, or revolution. 5 00:00:09,127 --> 00:00:10,844 So, if there's some repeating process, 6 00:00:10,844 --> 00:00:14,788 the time it takes that process to reset is the period, 7 00:00:14,788 --> 00:00:16,297 and it's measured in seconds. 8 00:00:16,297 --> 00:00:18,457 The frequency is the number of cycles, 9 00:00:18,457 --> 00:00:22,012 or circles, or revolutions completed in one second. 10 00:00:22,012 --> 00:00:24,117 So, if there's some process that's repeating, 11 00:00:24,117 --> 00:00:26,287 the number of times the process repeats 12 00:00:26,287 --> 00:00:28,906 in one second would be the frequency. 13 00:00:28,906 --> 00:00:31,596 This means it has units of one over second, 14 00:00:31,596 --> 00:00:33,755 which is just called, the hertz. 15 00:00:33,755 --> 00:00:35,959 And because the period and frequency are defined 16 00:00:35,959 --> 00:00:38,677 in this inverse way as seconds per cycle 17 00:00:38,677 --> 00:00:40,714 or cycles per second, each one 18 00:00:40,714 --> 00:00:42,474 is just the inverse of the other. 19 00:00:42,474 --> 00:00:44,354 In other words, the period is 20 00:00:44,354 --> 00:00:45,929 just one over the frequency, and 21 00:00:45,929 --> 00:00:48,527 the frequency is equal to one over the period. 22 00:00:48,527 --> 00:00:50,540 One example of a repeating process 23 00:00:50,540 --> 00:00:53,456 is an object going in a circle at a constant speed. 24 00:00:53,456 --> 00:00:55,433 If this is the case, you can relate 25 00:00:55,433 --> 00:00:57,745 the speed, the radius of the circle, 26 00:00:57,745 --> 00:00:59,837 and the period of the motion since speed 27 00:00:59,837 --> 00:01:02,730 is just distance per time, and the distance 28 00:01:02,730 --> 00:01:06,118 the object travels in one cycle is two pi R 29 00:01:06,118 --> 00:01:08,087 the circumference, the speed would 30 00:01:08,087 --> 00:01:11,146 just be two pi R per the period, 31 00:01:11,146 --> 00:01:13,603 or since one over the period is the frequency, 32 00:01:13,603 --> 00:01:16,933 you could write the speed as two pi R times the frequency. 33 00:01:16,933 --> 00:01:18,858 Since time is not a vector these quantities 34 00:01:18,858 --> 00:01:21,676 are not vectors and they cannot be negative. 35 00:01:21,676 --> 00:01:23,193 So, what's an example involving period 36 00:01:23,193 --> 00:01:24,619 and frequency look like? 37 00:01:24,619 --> 00:01:26,428 Let's say moon travels around a planet 38 00:01:26,428 --> 00:01:30,234 in a circular orbit of radius R at a constant speed S. 39 00:01:30,234 --> 00:01:32,349 And we wanna know what the period and frequency are 40 00:01:32,349 --> 00:01:34,229 in terms of given quantities and 41 00:01:34,229 --> 00:01:36,394 fundamental constants, so we'll use the relationship 42 00:01:36,394 --> 00:01:38,811 between the speed, the period, and the frequency. 43 00:01:38,811 --> 00:01:40,717 We know that for object in circular motion 44 00:01:40,717 --> 00:01:43,183 the speed is two pi R over the period. 45 00:01:43,183 --> 00:01:44,506 And that means the period here 46 00:01:44,506 --> 00:01:47,277 would be equal to two pi R over the speed. 47 00:01:47,277 --> 00:01:49,513 And since frequency is one over the period, 48 00:01:49,513 --> 00:01:51,734 if we take one over this quantity 49 00:01:51,734 --> 00:01:53,164 we just flip the top and bottom 50 00:01:53,164 --> 00:01:56,187 and we get that this is the speed over two pi R. 51 00:01:56,187 --> 00:01:58,176 But we can't leave our answer in terms of V. 52 00:01:58,176 --> 00:02:00,497 We had to express this in terms of given quantities. 53 00:02:00,497 --> 00:02:02,576 We were given S, so our answer for the period 54 00:02:02,576 --> 00:02:05,689 has to be two pi R over S, and for frequency 55 00:02:05,689 --> 00:02:08,957 it would be S over two pi R, which is C. 56 00:02:08,957 --> 00:02:10,827 What is centripetal acceleration? 57 00:02:10,827 --> 00:02:12,493 The centripetal acceleration of an object 58 00:02:12,493 --> 00:02:14,446 is the acceleration that's causing 59 00:02:14,446 --> 00:02:16,499 that object to go in a circle. 60 00:02:16,499 --> 00:02:17,749 And it's important to note that 61 00:02:17,749 --> 00:02:20,015 this centripetal acceleration always points 62 00:02:20,015 --> 00:02:22,014 toward the center of the circle. 63 00:02:22,014 --> 00:02:24,305 The formula to find the centripetal acceleration 64 00:02:24,305 --> 00:02:27,153 is speed squared divided by the radius 65 00:02:27,153 --> 00:02:29,914 of the circle the object is travelling in. 66 00:02:29,914 --> 00:02:32,381 Even though this has a bit of an exotic formula 67 00:02:32,381 --> 00:02:35,091 for the acceleration, it's still an acceleration, 68 00:02:35,091 --> 00:02:37,870 so it still has units of meters per second squared, 69 00:02:37,870 --> 00:02:39,793 and it is a vector, which means it does 70 00:02:39,793 --> 00:02:43,091 have a direction, i.e toward the center of the circle. 71 00:02:43,091 --> 00:02:44,556 But this centripetal acceleration 72 00:02:44,556 --> 00:02:47,957 does not cause the object to speed up or slow down. 73 00:02:47,957 --> 00:02:50,424 This centripetal acceleration is only changing 74 00:02:50,424 --> 00:02:52,402 the direction of the velocity. 75 00:02:52,402 --> 00:02:54,533 If the object going in the circle 76 00:02:54,533 --> 00:02:57,082 is also speeding up or slowing down, 77 00:02:57,082 --> 00:02:58,331 there's also gotta be a component 78 00:02:58,331 --> 00:03:01,221 of the acceleration that's tangential to the circle, 79 00:03:01,221 --> 00:03:02,426 in other words, if the object 80 00:03:02,426 --> 00:03:04,189 is going in a circle and speeding up, 81 00:03:04,189 --> 00:03:05,766 there's gotta be a component of acceleration 82 00:03:05,766 --> 00:03:07,798 in direction of the velocity, 83 00:03:07,798 --> 00:03:09,225 and if the object is slowing down, 84 00:03:09,225 --> 00:03:10,913 there's gotta be a component of acceleration 85 00:03:10,913 --> 00:03:13,343 in the opposite direction to the velocity. 86 00:03:13,343 --> 00:03:15,618 So, centripetal acceleration changes the direction 87 00:03:15,618 --> 00:03:18,488 of the velocity, and tangential acceleration changes 88 00:03:18,488 --> 00:03:21,451 the magnitude or size of the velocity. 89 00:03:21,451 --> 00:03:23,805 But this formula of V squared over R 90 00:03:23,805 --> 00:03:25,283 is only giving you the magnitude 91 00:03:25,283 --> 00:03:27,254 of the centripetal acceleration. 92 00:03:27,254 --> 00:03:30,436 This does not account for any tangential acceleration. 93 00:03:30,436 --> 00:03:31,569 So, what's an example problem 94 00:03:31,569 --> 00:03:34,129 involving centripetal acceleration look like? 95 00:03:34,129 --> 00:03:36,271 Let's say particle A is travelling in a circle 96 00:03:36,271 --> 00:03:38,816 with a constant speed S and a radius R. 97 00:03:38,816 --> 00:03:40,676 If particle B is travelling in a circle 98 00:03:40,676 --> 00:03:44,154 with twice the speed of A and twice the radius of A, 99 00:03:44,154 --> 00:03:46,086 what's the ratio of the acceleration 100 00:03:46,086 --> 00:03:48,458 of particle A compared to particle B. 101 00:03:48,458 --> 00:03:51,137 So, particle A is gonna have a centripetal acceleration 102 00:03:51,137 --> 00:03:53,683 of the speed squared over the radius, 103 00:03:53,683 --> 00:03:55,761 and particle B is also gonna have an acceleration 104 00:03:55,761 --> 00:03:57,182 of the speed squared, but this speed 105 00:03:57,182 --> 00:04:00,216 is twice as much of the speed of particle A, 106 00:04:00,216 --> 00:04:01,639 and is travelling in a circle 107 00:04:01,639 --> 00:04:03,716 with twice the radius of particle A. 108 00:04:03,716 --> 00:04:06,012 When we square the two we'll get four over two, 109 00:04:06,012 --> 00:04:08,379 gives us a factor of two times the speed 110 00:04:08,379 --> 00:04:11,042 of A squared over the radius of A. 111 00:04:11,042 --> 00:04:13,212 So, the ratio of the acceleration of particle A 112 00:04:13,212 --> 00:04:15,857 compared to particle B is gonna be one half 113 00:04:15,857 --> 00:04:17,846 since the acceleration of particle A 114 00:04:17,846 --> 00:04:20,485 is half the acceleration of particle B. 115 00:04:20,485 --> 00:04:23,606 Centripetal forces are not a new type of force, 116 00:04:23,606 --> 00:04:25,107 centripetal forces are just one 117 00:04:25,107 --> 00:04:27,387 of the any other forces that we've already met 118 00:04:27,387 --> 00:04:29,618 that happen to be pointing towards the center 119 00:04:29,618 --> 00:04:32,511 of the circle making an object travel in a circle. 120 00:04:32,511 --> 00:04:34,152 So, for a moon going around the Earth, 121 00:04:34,152 --> 00:04:35,894 gravity is the centripetal force. 122 00:04:35,894 --> 00:04:37,932 For a yo-yo going around on a string, 123 00:04:37,932 --> 00:04:39,893 the tension is the centripetal force. 124 00:04:39,893 --> 00:04:42,080 For a skateboarder doing a loopty loop, 125 00:04:42,080 --> 00:04:44,218 the normal force is the centripetal force. 126 00:04:44,218 --> 00:04:46,148 And for a car going around a roundabout, 127 00:04:46,148 --> 00:04:49,226 the static frictional force is the centripetal force. 128 00:04:49,226 --> 00:04:51,852 And these forces still follow Newton's second law, 129 00:04:51,852 --> 00:04:53,729 but using centripetal forces means you're 130 00:04:53,729 --> 00:04:55,450 also going to have to use the expression 131 00:04:55,450 --> 00:04:57,185 for the centripetal acceleration. 132 00:04:57,185 --> 00:04:59,451 Now, if a force is directed radially inward 133 00:04:59,451 --> 00:05:00,847 toward the center of the circle, 134 00:05:00,847 --> 00:05:02,510 you would count that force as positive 135 00:05:02,510 --> 00:05:04,167 since it points in the same direction 136 00:05:04,167 --> 00:05:05,741 as the centripetal acceleration. 137 00:05:05,741 --> 00:05:07,579 And if a force points radially out 138 00:05:07,579 --> 00:05:08,926 from the center of the circle, 139 00:05:08,926 --> 00:05:10,601 you would count that as a negative force. 140 00:05:10,601 --> 00:05:12,867 And if force is directed tangential to the circle, 141 00:05:12,867 --> 00:05:15,433 you wouldn't include it in this calculation at all. 142 00:05:15,433 --> 00:05:17,225 You can include those forces in their own 143 00:05:17,225 --> 00:05:19,032 Newton's second law equation, 144 00:05:19,032 --> 00:05:21,193 but you wouldn't be using V squared over R 145 00:05:21,193 --> 00:05:22,785 for that acceleration. 146 00:05:22,785 --> 00:05:26,016 Those tangential forces change the speed of the object, 147 00:05:26,016 --> 00:05:27,518 but the centripetal force changes 148 00:05:27,518 --> 00:05:29,296 the direction of the object. 149 00:05:29,296 --> 00:05:30,979 So, what's an example problem involving 150 00:05:30,979 --> 00:05:32,671 centripetal forces look like? 151 00:05:32,671 --> 00:05:34,866 Imagine a ball of mass M rolling over 152 00:05:34,866 --> 00:05:37,280 the top of the hill of radius R at a speed S. 153 00:05:37,280 --> 00:05:39,723 And we wanna know, at a top of the hill, 154 00:05:39,723 --> 00:05:41,731 what's the magnitude of the normal force 155 00:05:41,731 --> 00:05:44,044 exerted on the ball by the road. 156 00:05:44,044 --> 00:05:45,558 So, we'll draw our force diagram. 157 00:05:45,558 --> 00:05:47,184 There's gonna be an upward normal force 158 00:05:47,184 --> 00:05:49,020 on the ball from the road, and there's 159 00:05:49,020 --> 00:05:50,630 gonna be a downward force of gravity 160 00:05:50,630 --> 00:05:53,070 on the ball from the Earth, and these two forces 161 00:05:53,070 --> 00:05:55,234 are not going to be equal and opposite. 162 00:05:55,234 --> 00:05:56,383 If they were equal and opposite 163 00:05:56,383 --> 00:05:58,135 they would balance, and if the forces are balanced 164 00:05:58,135 --> 00:06:00,291 the object would maintain its velocity 165 00:06:00,291 --> 00:06:02,104 and keep travelling in a straight line. 166 00:06:02,104 --> 00:06:03,875 But this ball doesn't travel in a straight line, 167 00:06:03,875 --> 00:06:05,617 it starts accelerating downward. 168 00:06:05,617 --> 00:06:07,094 So, this normal force is gonna have to be less 169 00:06:07,094 --> 00:06:08,847 than the force of gravity. 170 00:06:08,847 --> 00:06:10,285 To figure out how much less, we can use 171 00:06:10,285 --> 00:06:12,032 Newton's second law with the formula 172 00:06:12,032 --> 00:06:13,668 for centripetal acceleration. 173 00:06:13,668 --> 00:06:15,848 The speed is S, the radius is R, 174 00:06:15,848 --> 00:06:18,279 the force of gravity is gonna be a positive 175 00:06:18,279 --> 00:06:20,374 centripetal force since it's directed toward 176 00:06:20,374 --> 00:06:21,803 the center of the circle. 177 00:06:21,803 --> 00:06:24,116 The normal force is gonna be a negative centripetal force 178 00:06:24,116 --> 00:06:25,682 since it's directed radially away from 179 00:06:25,682 --> 00:06:27,078 the center of the circle. 180 00:06:27,078 --> 00:06:28,553 Then we divide by the mass, which, 181 00:06:28,553 --> 00:06:30,138 if you solve this for normal force, 182 00:06:30,138 --> 00:06:32,097 gives you the force of gravity 183 00:06:32,097 --> 00:06:35,319 minus M, S squared over R, which makes sense 184 00:06:35,319 --> 00:06:37,637 'cause this normal force has to be less 185 00:06:37,637 --> 00:06:39,253 than the force of gravity. 186 00:06:39,253 --> 00:06:41,637 Newton's universal law of gravity states that 187 00:06:41,637 --> 00:06:44,626 all masses in the universe pull, i.e attract 188 00:06:44,626 --> 00:06:46,435 every other mass in the universe 189 00:06:46,435 --> 00:06:48,207 with gravitational force. 190 00:06:48,207 --> 00:06:49,588 And this force is proportional 191 00:06:49,588 --> 00:06:52,113 to each mass, and inversely proportional 192 00:06:52,113 --> 00:06:54,911 to the square of the center to center distance 193 00:06:54,911 --> 00:06:56,593 between the two masses. 194 00:06:56,593 --> 00:06:58,270 In mathematical form, it just says 195 00:06:58,270 --> 00:07:00,780 that the force of gravity is equal to big G, 196 00:07:00,780 --> 00:07:04,496 a constant, which is 6.67 times 10 to the negative 11th 197 00:07:04,496 --> 00:07:07,168 multiplied by each mass in kilograms, 198 00:07:07,168 --> 00:07:10,049 and then divided by the center to center distance 199 00:07:10,049 --> 00:07:11,830 between the two masses, in other words, 200 00:07:11,830 --> 00:07:14,107 not the surface to surface distance, 201 00:07:14,107 --> 00:07:16,664 but the center to center distance. 202 00:07:16,664 --> 00:07:18,188 And even if these two objects have different 203 00:07:18,188 --> 00:07:19,789 masses, the magnitude of the force 204 00:07:19,789 --> 00:07:22,330 they exert on each other is gonna be the same. 205 00:07:22,330 --> 00:07:24,011 This is illustrated by the formula 206 00:07:24,011 --> 00:07:25,414 since you could swap these two masses 207 00:07:25,414 --> 00:07:26,802 and you get the same number. 208 00:07:26,802 --> 00:07:29,204 And it's also something we know from Newton's third law. 209 00:07:29,204 --> 00:07:31,047 This force of gravity is a vector 210 00:07:31,047 --> 00:07:33,126 and it has a direction, the direction is always such 211 00:07:33,126 --> 00:07:34,890 that it attracts every other mass, 212 00:07:34,890 --> 00:07:37,680 and since this is a force, the unit is in Newtons. 213 00:07:37,680 --> 00:07:39,057 So, what's an example problem involving 214 00:07:39,057 --> 00:07:41,495 Newton's universal law of gravity look like? 215 00:07:41,495 --> 00:07:43,403 Let's say two masses, both of mass M, 216 00:07:43,403 --> 00:07:45,861 exert a gravitational force F on each other. 217 00:07:45,861 --> 00:07:49,144 If one of the masses is exchanged for a mass 3M 218 00:07:49,144 --> 00:07:50,965 and the center to center distance between 219 00:07:50,965 --> 00:07:52,637 the masses is tripled, what would 220 00:07:52,637 --> 00:07:54,616 the new gravitational force be? 221 00:07:54,616 --> 00:07:55,983 We know the gravitational force 222 00:07:55,983 --> 00:07:58,918 is always big G times one of the masses multiplied 223 00:07:58,918 --> 00:08:01,480 by the other mass divided by the center to center 224 00:08:01,480 --> 00:08:02,598 distance squared. 225 00:08:02,598 --> 00:08:04,470 So, the initial force between the two masses 226 00:08:04,470 --> 00:08:07,812 would be big G M times M over R squared, 227 00:08:07,812 --> 00:08:10,637 but the new force with the exchanged values would be 228 00:08:10,637 --> 00:08:13,137 big G times 3M times M divided 229 00:08:14,046 --> 00:08:16,477 by three times the radius squared. 230 00:08:16,477 --> 00:08:18,177 The factor of three squared on the bottom 231 00:08:18,177 --> 00:08:19,879 gives nine, and three divided by nine 232 00:08:19,879 --> 00:08:23,838 is one over three times big G M M over R squared. 233 00:08:23,838 --> 00:08:26,133 So, we can see that the force with the new values 234 00:08:26,133 --> 00:08:29,345 1/3 of the force with the old values. 235 00:08:29,345 --> 00:08:31,268 What's gravitational field mean? 236 00:08:31,268 --> 00:08:33,188 The gravitational field is just another word 237 00:08:33,188 --> 00:08:36,445 for the acceleration due to gravity near an object. 238 00:08:36,446 --> 00:08:38,543 You can visualize a gravitational field 239 00:08:38,543 --> 00:08:42,285 as vectors pointing radially in toward a mass. 240 00:08:42,285 --> 00:08:44,534 All masses create a gravitational field that points 241 00:08:44,534 --> 00:08:47,193 radially in toward them and dies off like 242 00:08:47,193 --> 00:08:50,536 one over R squared the farther you get away from them. 243 00:08:50,536 --> 00:08:53,301 So, the formula for the gravitational field little G 244 00:08:53,301 --> 00:08:56,963 created by a mass M is big G times 245 00:08:56,963 --> 00:09:00,479 the mass creating the field divided by the distance 246 00:09:00,479 --> 00:09:02,779 from the center of the mass to the point where 247 00:09:02,779 --> 00:09:05,656 you're trying to determine the value of the field. 248 00:09:05,656 --> 00:09:07,914 And again, this value for the gravitational field 249 00:09:07,914 --> 00:09:09,805 is going to be equal to the value 250 00:09:09,805 --> 00:09:11,496 for the acceleration due to gravity 251 00:09:11,496 --> 00:09:13,845 of an object placed at that point. 252 00:09:13,845 --> 00:09:15,645 The gravitational field is a vector 253 00:09:15,645 --> 00:09:18,461 since it has a direction, i.e. toward the center 254 00:09:18,461 --> 00:09:19,943 of the object creating it. 255 00:09:19,943 --> 00:09:21,751 And since gravitational field is equivalent 256 00:09:21,751 --> 00:09:23,434 to acceleration due to gravity, 257 00:09:23,434 --> 00:09:25,981 the units are meters per second squared, 258 00:09:25,981 --> 00:09:29,104 but you could also write that as newtons per kilogram, 259 00:09:29,104 --> 00:09:30,786 which is another way of thinking about what 260 00:09:30,786 --> 00:09:32,435 gravitational field means. 261 00:09:32,435 --> 00:09:34,971 Not only is it the acceleration due to gravity 262 00:09:34,971 --> 00:09:36,798 of an object placed at that point, 263 00:09:36,798 --> 00:09:38,832 but it's the amount of the gravitational force 264 00:09:38,832 --> 00:09:42,028 exerted on a mass M placed at that point. 265 00:09:42,028 --> 00:09:44,123 So, you could think of the gravitational field 266 00:09:44,123 --> 00:09:46,429 as measuring the amount of gravitational force 267 00:09:46,429 --> 00:09:49,124 per kilogram at a point in space, 268 00:09:49,124 --> 00:09:51,753 which when rearranged gives you the familiar formula 269 00:09:51,753 --> 00:09:54,535 that the force of gravity is just M times G. 270 00:09:54,535 --> 00:09:55,956 So, what's an example problem involving 271 00:09:55,956 --> 00:09:57,891 gravitational field look like? 272 00:09:57,891 --> 00:10:00,403 Let's say a hypothetical planet X had three times 273 00:10:00,403 --> 00:10:03,274 the mass of Earth and half the radius of Earth. 274 00:10:03,274 --> 00:10:05,432 What would be the acceleration due to gravity 275 00:10:05,432 --> 00:10:08,068 on planet X, i.e. the gravitational field 276 00:10:08,068 --> 00:10:10,649 on planet X, in terms of the acceleration 277 00:10:10,649 --> 00:10:13,195 due to gravity on Earth, which is GE. 278 00:10:13,195 --> 00:10:15,122 So, we know that the gravitational field on Earth 279 00:10:15,122 --> 00:10:17,643 has to be big G times mass of the Earth 280 00:10:17,643 --> 00:10:19,442 over the radius of Earth squared, 281 00:10:19,442 --> 00:10:21,565 which we're calling G sub E, and 282 00:10:21,565 --> 00:10:24,317 the gravitational field on planet X would be 283 00:10:24,317 --> 00:10:27,696 big G times three times the mass of the Earth 284 00:10:27,696 --> 00:10:30,762 divided by half the radius of the Earth squared, 285 00:10:30,762 --> 00:10:32,677 and when we square this factor of a half 286 00:10:32,677 --> 00:10:34,757 we'll get 1/4, which is in the denominator, 287 00:10:34,757 --> 00:10:38,168 so three divided by a 1/4 is 12 times big G 288 00:10:38,168 --> 00:10:40,966 mass of the Earth over radius of the Earth squared, 289 00:10:40,966 --> 00:10:43,425 and since this entire term here is the acceleration 290 00:10:43,425 --> 00:10:45,877 due to gravity on Earth, the acceleration due 291 00:10:45,877 --> 00:10:48,340 to gravity on planet X is gonna be 12 times 292 00:10:48,340 --> 00:10:51,154 the acceleration due to gravity on Earth. 293 00:10:51,154 --> 00:10:53,152 Sometimes when you're solving gravitational problems, 294 00:10:53,152 --> 00:10:55,625 you'll be given the density instead of the mass. 295 00:10:55,625 --> 00:10:58,327 The density is the amount of mass per volume 296 00:10:58,327 --> 00:10:59,606 for a given material. 297 00:10:59,606 --> 00:11:02,171 The symbol for density is the Greek letter rho, 298 00:11:02,171 --> 00:11:03,899 and you can find it by taking the mass 299 00:11:03,899 --> 00:11:05,609 divided by the volume. 300 00:11:05,609 --> 00:11:08,922 So, the units of density are kilograms per meter queued. 301 00:11:08,922 --> 00:11:11,054 And it's not a vector since it has no direction, 302 00:11:11,054 --> 00:11:12,810 but it does let you solve for mass. 303 00:11:12,810 --> 00:11:14,393 If you know the density you could say 304 00:11:14,393 --> 00:11:17,408 that the mass is the density times the volume. 305 00:11:17,408 --> 00:11:20,169 So, what's an example problem involving density look like? 306 00:11:20,169 --> 00:11:22,518 Let's try the hypothetical planet problem again, 307 00:11:22,518 --> 00:11:24,293 but this time instead of being told 308 00:11:24,293 --> 00:11:26,696 that planet X has three times the mass of Earth, 309 00:11:26,696 --> 00:11:28,411 let's say that planet X has three times 310 00:11:28,411 --> 00:11:31,681 the density of Earth, and again, half the radius of Earth. 311 00:11:31,681 --> 00:11:33,373 What would be the acceleration due to gravity 312 00:11:33,373 --> 00:11:35,259 on planet X in terms of the acceleration 313 00:11:35,259 --> 00:11:37,501 due to gravity on Earth GE. 314 00:11:37,501 --> 00:11:38,961 We could write down the formula 315 00:11:38,961 --> 00:11:41,666 for gravitational acceleration or gravitational field, 316 00:11:41,666 --> 00:11:44,111 which is big G M over R squared, 317 00:11:44,111 --> 00:11:45,924 but this time we don't know the mass, 318 00:11:45,924 --> 00:11:47,861 we just know the density, so we want to rewrite 319 00:11:47,861 --> 00:11:49,744 this formula in terms of density, 320 00:11:49,744 --> 00:11:53,789 which we can do by rewriting the M as rho times V, 321 00:11:53,789 --> 00:11:55,895 since density is mass per volume, 322 00:11:55,895 --> 00:11:58,388 and mass is density times volume. 323 00:11:58,388 --> 00:11:59,892 But we don't know the volume of this planet, 324 00:11:59,892 --> 00:12:02,177 we just know the radius, so we need to rewrite volume 325 00:12:02,177 --> 00:12:04,093 in terms of radius, which we could do 326 00:12:04,093 --> 00:12:06,955 since planets are spherical and the volume of a sphere 327 00:12:06,955 --> 00:12:09,566 is four thirds pi R cubed. 328 00:12:09,566 --> 00:12:11,391 We can substitute this expression in 329 00:12:11,391 --> 00:12:13,766 for the volume and finally get an expression 330 00:12:13,766 --> 00:12:16,282 for the acceleration due to gravity of big G 331 00:12:16,282 --> 00:12:20,373 times rho four thirds pi R cubed divided by R squared. 332 00:12:20,373 --> 00:12:23,234 And we can cancel an R squared on the top and the bottom, 333 00:12:23,234 --> 00:12:26,016 which leaves this little G as equaling big G 334 00:12:26,016 --> 00:12:27,766 rho four thirds pi R. 335 00:12:28,808 --> 00:12:30,685 So, the gravitational acceleration on Earth 336 00:12:30,685 --> 00:12:33,494 would be big G rho of Earth four thirds pi 337 00:12:33,494 --> 00:12:34,990 times the radius of Earth. 338 00:12:34,990 --> 00:12:36,862 And the gravitational acceleration 339 00:12:36,862 --> 00:12:38,824 on planet X would be big G times 340 00:12:38,824 --> 00:12:41,374 the density of planet X, which is three times 341 00:12:41,374 --> 00:12:44,314 the density of Earth times four thirds pi 342 00:12:44,314 --> 00:12:47,018 times the radius of planet X, which is one half 343 00:12:47,018 --> 00:12:48,932 the radius of Earth, which when we pull out 344 00:12:48,932 --> 00:12:50,838 the three and factor of a half, 345 00:12:50,838 --> 00:12:53,196 gives us three halves times the expression 346 00:12:53,196 --> 00:12:55,607 for the acceleration due to gravity on Earth. 347 00:12:55,607 --> 00:12:57,166 So, the gravitational acceleration 348 00:12:57,166 --> 00:12:59,378 on planet X is gonna be three halves 349 00:12:59,378 --> 00:13:02,195 the gravitational acceleration on planet Earth. 350 00:13:02,195 --> 00:13:04,319 Gravitational orbits are just a special case 351 00:13:04,319 --> 00:13:06,640 of centripetal acceleration where some object 352 00:13:06,640 --> 00:13:08,225 is orbiting another object due 353 00:13:08,225 --> 00:13:09,862 to the gravitational force. 354 00:13:09,862 --> 00:13:11,284 And if that orbit is a circle, 355 00:13:11,284 --> 00:13:14,072 we can relate the speed, the radius of the orbit, 356 00:13:14,072 --> 00:13:15,721 and the larger mass to each other 357 00:13:15,721 --> 00:13:19,186 using Newton's second law and centripetal acceleration. 358 00:13:19,186 --> 00:13:20,616 You just plugin the acceleration 359 00:13:20,616 --> 00:13:23,243 as the centripetal acceleration, V squared over R, 360 00:13:23,243 --> 00:13:26,294 and since the centripetal force is the force of gravity, 361 00:13:26,294 --> 00:13:27,583 you can plugin the expression for 362 00:13:27,583 --> 00:13:30,208 the force of gravity as the centripetal force, 363 00:13:30,208 --> 00:13:33,637 which is big G M M over the distance between them squared. 364 00:13:33,637 --> 00:13:37,056 And since the mass of the orbiting object cancels 365 00:13:37,056 --> 00:13:39,010 we get an expression that relates to the speed 366 00:13:39,010 --> 00:13:40,644 of the orbiting object, the larger mass 367 00:13:40,644 --> 00:13:42,479 that's pulling that object in, 368 00:13:42,479 --> 00:13:45,653 and the center to center distance between the objects, 369 00:13:45,653 --> 00:13:47,243 which, if we solve this for V, 370 00:13:47,243 --> 00:13:50,595 gives us the square root of big G times the mass 371 00:13:50,595 --> 00:13:52,631 pulling in the object divided by 372 00:13:52,631 --> 00:13:55,123 the center to center distance between the objects. 373 00:13:55,123 --> 00:13:57,318 Note that this formula does not depend on the mass 374 00:13:57,318 --> 00:14:00,037 that's in orbit since that mass canceled out 375 00:14:00,037 --> 00:14:01,223 in the calculation. 376 00:14:01,223 --> 00:14:02,608 So, what's an example problem involving 377 00:14:02,608 --> 00:14:04,489 gravitational orbits look like? 378 00:14:04,489 --> 00:14:07,161 Well, imagine a space station of mass MS 379 00:14:07,161 --> 00:14:09,646 is orbiting at an altitude of 3R 380 00:14:09,646 --> 00:14:13,183 above a planet of mass MP and radius R 381 00:14:13,183 --> 00:14:15,734 as seen in this diagram, and then imagine 382 00:14:15,734 --> 00:14:18,208 a different space station of mass 3MS 383 00:14:18,208 --> 00:14:21,472 is orbiting at an altitude of 2R above a planet 384 00:14:21,472 --> 00:14:26,447 of mass 4MP and a radius of 2R as seen in this diagram, 385 00:14:26,447 --> 00:14:27,879 and we wanna know, if the speed 386 00:14:27,879 --> 00:14:30,695 of the space station of mass MS is V, 387 00:14:30,695 --> 00:14:32,523 then in terms of V, what's the speed 388 00:14:32,523 --> 00:14:34,951 of the space station of mass 3MS? 389 00:14:34,951 --> 00:14:36,636 Well, we just showed that the speed 390 00:14:36,636 --> 00:14:38,473 of an orbiting object is gonna be equal 391 00:14:38,473 --> 00:14:40,822 to the square root of big G times 392 00:14:40,822 --> 00:14:43,149 the mass of the larger object pulling in 393 00:14:43,149 --> 00:14:44,933 the smaller object divided by the 394 00:14:44,933 --> 00:14:48,081 center to center distance between the objects. 395 00:14:48,081 --> 00:14:49,926 And since this formula doesn't involve the mass 396 00:14:49,926 --> 00:14:51,762 of the orbiting object, it doesn't matter 397 00:14:51,762 --> 00:14:53,861 that the objects have different masses, 398 00:14:53,861 --> 00:14:56,082 but the mass of the planet can make a difference. 399 00:14:56,082 --> 00:14:58,438 So, to get the speed of the space station MS, 400 00:14:58,438 --> 00:15:00,851 we could say that it's the square root of Big G 401 00:15:00,851 --> 00:15:04,139 the mass of planet P over the center to center distance, 402 00:15:04,139 --> 00:15:05,764 which is not gonna be the radius 403 00:15:05,764 --> 00:15:07,844 of the planet or the altitude, 404 00:15:07,844 --> 00:15:11,036 it's gonna be the radius of the planet plus the altitude 405 00:15:11,036 --> 00:15:14,241 since this has to be the center to center distance, 406 00:15:14,241 --> 00:15:17,371 which in this case will be 3R plus R, which is 4R. 407 00:15:17,371 --> 00:15:19,157 And now to get the speed of the space station 408 00:15:19,157 --> 00:15:21,656 of mass 3MS, we'll use the same formula, 409 00:15:21,656 --> 00:15:23,690 which is big G mass of the planet, which 410 00:15:23,690 --> 00:15:26,203 in this case is 4MP divided by 411 00:15:26,203 --> 00:15:27,608 the center to center distance, 412 00:15:27,608 --> 00:15:30,701 which in this case would be 2R plus 2R, 413 00:15:30,701 --> 00:15:33,003 and again that's 4R, and if we compare, 414 00:15:33,003 --> 00:15:35,028 the only difference between these expressions 415 00:15:35,028 --> 00:15:37,218 is that there's an extra factor of four within 416 00:15:37,218 --> 00:15:38,502 this square root. 417 00:15:38,502 --> 00:15:40,533 So, if we take that factor out, the square root 418 00:15:40,533 --> 00:15:43,533 of four is two, we'd get two times the expression 419 00:15:43,533 --> 00:15:46,373 for the speed of the space station MS. 420 00:15:46,373 --> 00:15:49,930 So, the space station 3MS is travelling two times 421 00:15:49,930 --> 00:00:00,000 the speed of the space station MS.