1. Centripetal Acceleration and Centripetal Force

2. Acceleration Speeding up Slowing down Changing direction

3. Think of spinning an object on the end of a string. How would you describe it to someone? The easiest thing to say is that it moves in the shape of a circle. ◦ This means you should be able to measure the radius of the circle, and from that calculate things like circumference. C =2 π r A = π r 2

4. You might also notice that since it is going in a circle, the motion of the bob as it moves around repeats itself over and over again. ◦ We refer to this as periodic motion, because periodically ("every so often") it gets back to where it started and keeps on going. ◦ Other objects, like pendulums, can also be described as periodic, as long as they repeat the same motion over and over again. The bob is also an example of two dimensional motion, just like a projectile. ◦ It is moving in the x and y directions at the same time, not in a perfect straight line.

5. You might remember that when we originally talked about acceleration we defined it as a change in the speed and/or direction of the movement of an object. While the bob is going around in a circle, its speed might be constant (and usually will be in the problems you do), but what is happening to the direction it is traveling?

6. At any particular instant in time, the velocity of the bob spinning in a circle is pointing in the same direction as the tangent line to the circle. In Figure 1 you can see that the velocity vectors are pointing in different directions as the object moves clockwise. That means that the direction the bob is traveling is constantly changing! By definition, this means that the bob is accelerating. If it wasn’t accelerating, it would be traveling in a straight line. So how do we calculate the acceleration of the bob?

7. Centripetal Acceleration

8. What is the centripetal acceleration of a person in a car driving at 60km/h in a traffic circle that is 120m across? Change the velocity into metres per second, and since the measurement of the circle is a diameter, divide it by two. v = 17 m/s r = 60.0 m a c = ? a c = v 2 / r = 17 2 / 60.0 a c = 4.6 m/s 2

9. Centripetal comes from Greek meaning “centre seeking” The bob is basically being pulled in towards the centre of the circle, which is why it follows a curved path around the centre. It’s as though it were seeking the centre of the circle, even though it never gets there. Do not confuse this with the word “centrifugal”, which doesn’t even exist! ◦ A person in a car going around in a circle might say that a “centrifugal force” was pushing them across the seat out of the circle. ◦ In fact, it is just your body obeying Newton’s first law… it is trying to move in a straight line. ◦ The car is trying to keep you going in a circle, which is centripetal.

10. Centripetal Force By now you should be saying to yourself that if an object going in a circle is accelerating, there must be a force acting on it. This agrees with Newton’s Laws that a force results in acceleration. If there were no forces involved, the object should be moving at a constant speed in a straight line, which it isn’t. We should be able to come up with some formulas for forces that relate directly to circular motion.

11. If you have already calculated the centripetal acceleration of an object moving in a circle from one of the formulas we have studied so far, then just go ahead and use F = ma to calculate the force. Because this is a force causing an object to move in a circle, we refer to the force as a centripetal force. This is written in formulas as F c A lot of the time you won’t calculate the acceleration first, so we should have a couple of formulas to be able to calculate the centripetal force based on the information you do have.

13. Centripetal Force

15. You might have seen movies with an astronaut in training spinning around and around in this big machine to get ready for their flight. The device is called a centrifuge, like the one pictured below (click on it to go to the Nasa website for the Center for Gravitational Biology Research).Center for Gravitational Biology Research https://www.youtube.com/watch?v=EXQKl-28zn0 http://videos.howstuffworks.com/discovery/29332-extreme-engineering-acceleration-and-g-force-video.htm http://videos.howstuffworks.com/discovery/44485-bad-universe-jet-jockey-video.htm

16. Let’s look at the forces on a 100 kg person’s body, and try to relate it back to regular Earth gravity (one “gee” = 9.81m/s 2 ). Keep in mind that just standing on the ground the person in this example would normally weigh (F g = mg) 9.81e2N. What is the centripetal force acting on this 100kg person if he is spun around in a 8.80m circle at… 10.0m/s? F c = mv 2 / r = (100)(10.0) 2 / 8.80 F c = 1.14e3 N 15.0m/s? F c = mv 2 / r = (100)(15.0) 2 / 8.80 F c = 2.56e3 N