1. Physics 218, Lecture XVII1 Physics 218 Lecture 17 Dr. David Toback

2. Physics 218, Lecture XVII2 Checklist for Today Things that were due Monday: –Chapter 8 Quizzes on WebCT Things due Tuesday: –Read Chapters 10 & 11 Things that are due yesterday for Recitation –Chapter 9 problems Things due Monday –Chapter 9 in WebCT

3. Physics 218, Lecture XVII3 The Schedule This Week: (3/17) Chapter 8 quizzes due in WebCT Reading for Chapters 10 & 11 Lecture on Chapter 10 (11 in recitation next week) Chapter 9 and Exam 2 Review in recitation Next Week: (3/24) Chapter 9 due in WebCT (mini-practice exam 2 available) Exam 2 on Tuesday Recitation on Chapters 10 & 11 Reading for Chapters 12 & 13 for Thursday Lecture 12 & 13 on Thursday Following week Chapter 10 & 11 material in WebCT Reading: Chapters 14-16 Lectures on 14-16 (Lectures 1 and 2 of Four) Recitation on Chapters 12 & 13

4. Physics 218, Lecture XVII4 Today’s Lecture: Rest of Chap 10 Center of Mass Center of Mass and Translational Motion Collision & Explosion Problems using: –Conservation of Momentum –Center of Mass

5. Physics 218, Lecture XVII5

6. 6 Center of Mass (CM) What is the “Center of Mass?” More importantly “Why do we care?” This is a special point in space where “it’s as if the object could be replaced by all the mass at that one little point”

7. Physics 218, Lecture XVII7 Center of Mass (CM) Cont… Examples where this is useful: We have a spherical cow that weighs two tons. We can model defenestrating her as if she were a single point We can model the earth moving around the sun as a single point at “the center of the earth” At some level we’ve been assuming these things for doing problems all semester

8. Physics 218, Lecture XVII8 Center of Mass (CM) Cont… Yet another example: there are only a couple of points on a ruler that you can put your finger under and hold it up –Your finger provides the normal force

9. Physics 218, Lecture XVII9 Visual Examples The center of mass has the same trajectory as a point since both have the same acceleration and initial velocity

10. Physics 218, Lecture XVII10 How do you calculate CM? 1.Pick an origin 2.Look at each “piece of mass” and figure out how much mass it has and how far it is (vector displacement) from the origin. Take mass times position 3.Add them all up and divide out by the sum of the masses The center of mass is a displacement vector “relative to some origin”

11. Physics 218, Lecture XVII11 Spelling out the math:

12. Physics 218, Lecture XVII12 2-D Example D h Three balls with masses m 1, m 2 and m 3 are are located at the points given to the right. Where is the center of mass?

13. Physics 218, Lecture XVII13 So what? 2 ways to solve collision/explosion problems: 1.Conservation of Momentum in all directions 2.Watching the Center of Mass Need to be able to do both –Pick easier method –Physics is the same

14. Physics 218, Lecture XVII14 Two balls in outer space Two balls are moving in outer space. They have known masses 2M and 3M and speeds 4V and 2V, respectively, and they collide at the origin. The directions are as shown in the figure. After the collision, the two balls stick together and form a blob. What is the final velocity of the blob? 3M 2M X Y Speed = 4v Speed = 2v (ignore gravity)

15. Physics 218, Lecture XVII15 Toy Rocket Problem Your friend fires a toy rocket into the air with an unknown velocity. You observe that at the peak of its trajectory it has traveled a distance d in the x-direction. It then breaks into two equal mass pieces. Part I falls straight down with no initial velocity. Where does the 2 nd half of the toy end up?

16. Physics 218, Lecture XVII16 Two Balls in Two Dimensions Before a collision, ball 1 moves with speed v 1 in the x direction, while ball 2 is at rest. Both have the same mass. After the collision, the balls go off at angles  and . What are the velocities, v’ 1 and v’ 2, after the collision?  

17. Physics 218, Lecture XVII17 Coming up next week… Homework 9 due Monday in WebCT Make sure you do ALL the quizzes in the learning module (folder) Mini-practice exam 2 and bonus points Exam 2: –Tuesday March 25 th Start Chapters 12-16 in Lecture on Thursday

18. Physics 218, Lecture XVII18

19. Physics 218, Lecture XVII19 Collisions & Explosions Momentum before collision is equal to momentum after collision. True in both X and Y directions separately

20. Physics 218, Lecture XVII20 So what? Now we can show that it’s as if the entire body moves as if it’s a single point Derivation

21. Physics 218, Lecture XVII21 2D Example X1X1 Y1Y1 Three balls with masses m 1, m 2 and m 3 are are located at the points given below. Where is the center of mass? What is the center of mass if all the masses are equal?

22. Physics 218, Lecture XVII22 Exam II Mean without bonus points is 69%. Approximate curve –>100 A+ –>90 A –>80 B –>55? C (but could go higher…) –Below this are the D’s and F’s If you aren’t caught up with the HW and didn’t do well on the exam, either catch up quickly or consider dropping

23. Physics 218, Lecture XVII23 Collisions in Two Dimensions There is nothing new here Simply use the same vector techniques –break things up in the X and Y directions!

24. Physics 218, Lecture XVII24 General Example Two Balls: Ball 1: m 1 with velocity v 1 Ball 2: m 2 with velocity 0. Choose axis so v 1 points in X direction.

25. Physics 218, Lecture XVII25 Simple Example We are given two balls with masses m 1 and m 2 and an origin. The balls are placed at a distance x 1 and x 2 from the origin. Where is the center of mass if: 1. In general 2. m 2 = m 1 ? 3. m 1 = 0

26. Physics 218, Lecture XVII26 CM and Integrals… A rod of length L and mass M has a uniform density. Calculate the center of mass. What if the density varied linearly from 0 to some value at the end? Hint: This requires an integral

27. Physics 218, Lecture XVII27 Three guys on a raft Three guys are hanging out on a raft at the locations given below. The origin is at the left. Where is the center of mass?