Physics 218 Lecture 17

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

2. 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 Physics 218, Lecture XVII

3. 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 Physics 218, Lecture XVII

4. 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 Physics 218, Lecture XVII

5. Physics 218, Lecture XVII

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” Physics 218, Lecture XVII

7. 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 Physics 218, Lecture XVII

8. 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 Physics 218, Lecture XVII

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

10. How do you calculate CM? • Pick an origin • 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 • 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” Physics 218, Lecture XVII

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

12. 2-D Example Three balls with masses m1, m2 and m3 are are located at the points given to the right. Where is the center of mass? h D Physics 218, Lecture XVII

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

14. 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? Y Speed = 4v (ignore gravity) 2M X Speed = 2v 3M Physics 218, Lecture XVII

15. 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 2nd half of the toy end up? Physics 218, Lecture XVII

16. Two Balls in Two Dimensions Before a collision, ball 1 moves with speed v1 in the x direction, while ball 2 is at rest. Both have the same mass. After the collision, the balls go off at angles Q and -Q. What are the velocities, v’1 and v’2, after the collision? Q -Q Physics 218, Lecture XVII

17. 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 25th • Start Chapters 12-16 in Lecture on Thursday Physics 218, Lecture XVII

18. End of Lecture Notes Physics 218, Lecture XVII

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

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

21. 2D Example Three balls with masses m1, m2 and m3 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? Y1 X1 Physics 218, Lecture XVII

22. 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 Physics 218, Lecture XVII

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

24. General Example Two Balls: Ball 1: m1 with velocity v1 Ball 2: m2 with velocity 0. Choose axis so v1 points in X direction. Physics 218, Lecture XVII

25. Simple Example We are given two balls with masses m1 and m2 and an origin. The balls are placed at a distance x1 and x2 from the origin. • Where is the center of mass if: • In general • m2 =m1 ? • m1 = 0 Physics 218, Lecture XVII

26. 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 Physics 218, Lecture XVII

27. 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? Physics 218, Lecture XVII