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

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

Physics 218, Lecture XV

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

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

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

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

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

Spelling out the math: Physics 218, Lecture XV

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

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

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

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

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

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

End of Lecture Notes Physics 218, Lecture XV

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

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

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

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

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

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

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

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

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