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

Physics 218, Lecture XVII2 Today’s Lecture Conservation of Momentum Example Collisions in two or three dimensions Center of Mass Center of Mass and Translational Motion

Physics 218, Lecture XVII3

4 Solving Cons of Momentum Problems Is this a system where I can easily use conservation of momentum? If so, draw two diagrams: Before & After Sum up all the momenta before and after → set them equal Use those equations to solve for what you’re looking for

Physics 218, Lecture XVII5 Before…

Physics 218, Lecture XVII6 After…

Physics 218, Lecture XVII7 Collisions in Two Dimensions There is nothing new here Simply using the same vector techniques to break things up in the X and Y directions like we’ve been doing over and over again

Physics 218, Lecture XVII8 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 v’ 1 and v’ 2 after the collision?  

Physics 218, Lecture XVII9 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 new blob. a) What is the velocity of the blob? b) Is mechanical energy conserved in this collision? If not, how much heat is lost. 3M 2M X Y Speed = 4v Speed = 2v (ignore gravity and the gravitational attraction between them)

Physics 218, Lecture XVII10 Center of Mass (CM) What is the “Center of Mass?”: More importantly “why do we care?” This is a special point within an object It’s the point where “it’s as if the object could be replaced by all the mass at that one little point” For example, if we have a spherical cow which weighs a ton, it’s as if we could model the cow as a point at her center

Physics 218, Lecture XVII11 Center of Mass (CM) Cont… Another example is treating the earth (or the moon) as a point at “the center of its mass” Yet another example: there are only a couple of points on a pencil that you can put your finger under and hold it up

Physics 218, Lecture XVII12 Bottom Line in CM: This concept is what allows us to do the problems pretending all the mass is in one place… We’ve actually been assuming this since the beginning

Physics 218, Lecture XVII13 Examples

Physics 218, Lecture XVII14 How do you calculate CM? 1.Pick an origin 2.Look at each “piece of mass” and figure out how much it weighs and how far it is from the origin 3.Add them all up and divide out by the mass The center of mass is always “relative to some origin”

Physics 218, Lecture XVII15 Spelling it out:

Physics 218, Lecture XVII16 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

Physics 218, Lecture XVII17 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?

Physics 218, Lecture XVII18 Rocket Problem A two stage rocket is on a trajectory. At the peak it has traveled a distance d and it breaks into two equal mass pieces. Part I falls straight down with no initial velocity Where does the 2 nd half of the rocket end up?

Physics 218, Lecture XVII19 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 XVII20 Next Time Class is cancelled for Thursday Next week we’ll begin Rotational Motion

Physics 218, Lecture XVII21 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.

Physics 218, Lecture XVII22 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 XVII23 Translation of Center of Mass This is why this stuff is cool… It’s as if the entire body moves as if it’s a single point Derivation