The Math Behind Circular Motion - By MITK12Videos
Transcript
| 00:11 | circular motion ? How does it work ? Why did | |
| 00:19 | that ball start to move in a circle ? Turns | |
| 00:23 | out all you need to get something moving in a | |
| 00:25 | circle is a force on that object towards the center | |
| 00:28 | of the circle . In this case the string pulls | |
| 00:32 | the ball into the center , forcing it in a | |
| 00:35 | circle . What would happen to the ball ? Is | |
| 00:38 | it no longer felt the strings ? Force ? My | |
| 00:41 | friends , Anna and I tested this by slicing the | |
| 00:44 | string with a knife . He set the ball in | |
| 00:47 | circular motion and I sliced this string with the knife | |
| 00:51 | that was fast with no string , pulling it to | |
| 00:53 | the center , the ball went straight up into the | |
| 00:55 | air , so without an inward force objects in circular | |
| 00:58 | motion would move tangent to the circle . Let's see | |
| 01:02 | that again . Without the force pulling it to the | |
| 01:05 | center , it goes straight up tangent to the circle | |
| 01:07 | it was moving . Let's write down some equations to | |
| 01:11 | see how this motion works . If you don't know | |
| 01:13 | all of the math I'm about to describe . That's | |
| 01:15 | okay . Just pay attention to which direction each thing | |
| 01:17 | faces when the spell is moving in a circle . | |
| 01:20 | Let's call its position peak . It's convention to define | |
| 01:24 | omega as its angular velocity angles per second . Let's | |
| 01:29 | say it took the ball t . Seconds to get | |
| 01:31 | to point P . So P is at the angle | |
| 01:34 | omega times T . We can break pee into an | |
| 01:38 | X component and a Y component since the angles omega | |
| 01:41 | T . If we assume the circle's radius is one | |
| 01:45 | , the X component is cosine omega T and the | |
| 01:48 | Y component is sign of omega T . Remember that | |
| 01:53 | the velocity with no strings attached faces , tangent to | |
| 01:56 | the circle velocity metrics . Help position changes with time | |
| 02:01 | . All right . The velocity V . Based on | |
| 02:03 | how the position P changes in time , the X | |
| 02:11 | component of V has derivative minus omega times sine of | |
| 02:15 | omega T . The Y component of V has derivative | |
| 02:20 | Omega times . Cosine of omega T . See how | |
| 02:25 | V and P are always perpendicular . Let's calculate the | |
| 02:30 | acceleration on the ball from the strings . Force acceleration | |
| 02:34 | measures how the velocity changes in time and it's pointed | |
| 02:37 | in the direction of the strings force so the ball | |
| 02:40 | accelerates into the middle of the circle . The X | |
| 02:45 | component is minus omega squared co sign of omega T | |
| 02:49 | . The derivative of these X component , the Y | |
| 02:53 | component is minus omega squared sine of omega T . | |
| 02:58 | There are two key points . The only force on | |
| 03:01 | an object in circular motion is towards the center of | |
| 03:03 | the circle and if you cut that fourth , the | |
| 03:07 | object continues moving in a line tangent to the circle | |
| 03:11 | . I hope you have fun experimenting with circular motion | |
| 03:15 | at home . You don't need knives . |
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