1. Physics 218 Lecture 15
Dr. David Toback
Physics 218, Lecture XV

2. Today’s Lecture: Rest of Chap 8
Center of Mass
Center of Mass and Translational Motion
Collision & Explosion Problems using:
Conservation of Momentum
Physics 218, Lecture XV

3. Physics 218, Lecture XV

4. What is the “Center of Mass?”
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 XV

5. 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 XV

6. 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 XV

7. 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 XV

8. 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 XV

9. Spelling out the math:
Physics 218, Lecture XV

10. 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 XV

11. So what? Conservation of Momentum in all directions
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 XV

12. 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
(ignore gravity)
Speed = 4v 2M X
Speed = 2v 3M
Physics 218, Lecture XV

13. 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 and that it 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 XV

14. 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 XV

15. Next Week Homework 7: Due Monday Reading for Tuesday:
Chapter 9: Rotational Motion
Reading Questions: Q9.1 & Q9.7
Exam 2: Thursday, October 26th
Mini-practice exam for Exam 2
Bonus points available!!!
Reading & questions for Chap 10
Tuesday, after exam
Physics 218, Lecture XV

16. End of Lecture Notes
Physics 218, Lecture XV

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

18. 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 XV

19. 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 XV

20. Exam II Mean without bonus points is 69%. Approximate curve
>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 XV

21. 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 XV

22. General Example Two Balls: Ball 1: m1 with velocity v1
Choose axis so v1 points in X direction.
Physics 218, Lecture XV

23. 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 XV

24. 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 XV

25. 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 XV