1. Physics 215 – Fall 2014Lecture 03-21 Welcome back to Physics 215 Today’s agenda Motion along curved paths, circles Tangential and radial components of acceleration Rotations Introduction to relative motion

2. Physics 215 – Fall 2014Lecture 03-22 Current homework assignment HW3: –Exam-style problem (print out from course website) –Ch.4 (Knight textbook): 52, 62, 80, 84 –due Wednesday, Sept 17 th in recitation

3. Physics 215 – Fall 2014Lecture 03-23 Exam 1: next Thursday (9/18/14) In room 208 (here!) at the usual lecture time Material covered: –Textbook chapters 1 - 4 –Lectures up through 9/16 (slides online) –Wed/Fri Workshop activities –Homework assignments Exam is closed book, but you may bring calculator and one handwritten 8.5” x 11” sheet of notes. Work through practice exam problems (posted on website) Work on more practice exam problems next Wednesday in recitation workshop

4. Physics 215 – Fall 2014Lecture 03-24 Acceleration vector for object speeding up from rest at point A ?

5. Physics 215 – Fall 2014Lecture 03-25 What if the speed is changing? Consider acceleration for object on curved path starting from rest Initially, v 2 /r = 0, so no radial acceleration But a is not zero! It must be parallel to velocity

6. Physics 215 – Fall 2014Lecture 03-26 Acceleration vectors for object speeding up: Tangential and radial components (or parallel and perpendicular)

7. Physics 215 – Fall 2014Lecture 03-27 Sample problem A Ferris wheel with diameter 14.0 m, which rotates counter-clockwise, is just starting up. At a given instant, a passenger on the rim of the wheel and passing through the lowest point of his circular motion is moving at 3.00 m/s and is gaining speed at a rate of 0.500 m/s 2. (a) Find the magnitude and the direction of the passenger’s acceleration at this instant. (b) Sketch the Ferris wheel and passenger showing his velocity and acceleration vectors.

8. Physics 215 – Fall 2014Lecture 03-28 Summary Components of acceleration vector: Parallel to direction of velocity: (Tangential acceleration) – “How much does speed of the object increase?” Perpendicular to direction of velocity: (Radial acceleration) – “How quickly does the object turn?”

9. Physics 215 – Fall 2014Lecture 03-29 Ball going through loop-the-loop

10. Physics 215 – Fall 2014Lecture 03-210 Rotations about fixed axis Linear speed: v = (2  r)/T =  r. Quantity  is called angular velocity  is a vector! Use right hand rule to find direction of . Angular acceleration  t is also a vector! –  and  parallel  angular speed increasing –  and  antiparallel  angular speed decreasing

11. Physics 215 – Fall 2014Lecture 03-211 A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center as point P. The angular velocity of Q is 1.twice as big as P 2.the same as P 3.half as big as P 4.none of the above

12. Physics 215 – Fall 2014Lecture 03-212 A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center as point P. The linear velocity of Q is 1.twice as big as P 2.the same as P 3.half as big as P 4.none of the above

13. Physics 215 – Fall 2014Lecture 03-213 Relating linear and angular kinematics Linear speed: v = (2  r)/T =  r Tangential acceleration: a tan = r  Radial acceleration: a rad = v 2 /r =  2 r

14. Physics 215 – Fall 2014Lecture 03-214 Problem – slowing a DVD  I = 27.5 rad/s,  = -10.0 rad/s 2 how many revolutions per second? linear speed of point on rim? angular velocity at t = 0.30 s ? when will it stop? 10.0 cm.

15. Physics 215 – Fall 2014Lecture 03-215 Kinematics Consider 1D motion of some object Observer at origin of coordinate system measures pair of numbers (x, t) – (observer) + coordinate system + clock called frame of reference (x, t) not unique – different choice of origin changes x (no unique clock...)

16. Physics 215 – Fall 2014Lecture 03-216 Change origin? Physical laws involve velocities and accelerations which only depend on  x Clearly any frame of reference (FOR) with different origin will measure same  x, v, a, etc.

17. Physics 215 – Fall 2014Lecture 03-217 Inertial Frames of Reference Actually can widen definition of FOR to include coordinate systems moving at constant velocity Now different frames will perceive velocities differently... Accelerations?

18. Physics 215 – Fall 2014Lecture 03-218 Moving Observer Often convenient to associate a frame of reference with a moving object. Can then talk about how some physical event would be viewed by an observer associated with the moving object.

19. Physics 215 – Fall 2014Lecture 03-219 Reference frame (clock, meterstick) carried along by moving object A B

20. Physics 215 – Fall 2014Lecture 03-220 A B A B A B

21. Physics 215 – Fall 2014Lecture 03-221 A B A B A B

22. Physics 215 – Fall 2014Lecture 03-222 A B A B A B

23. Physics 215 – Fall 2014Lecture 03-223 Discussion From point of view of A, car B moves to right. We say the velocity of B relative to A is v BA. Here v BA > 0 But from point of view of B, car A moves to left. In fact, v AB < 0 In general, can see that v AB = -v BA

24. Physics 215 – Fall 2014Lecture 03-224 Galilean transformation xAxA xBxB v BA P v BA t x PA = x PB + v BA t -- transformation of coordinates   x PA  t  x PB /  t + v BA  v PA = v PB + v BA -- transformation of velocities yByB yAyA

25. Physics 215 – Fall 2014Lecture 03-225 Discussion Notice: –It follows that v AB = -v BA –Two objects a and b moving with respect to, say, Earth then find (P  a, B  b, A  E) v ab = v aE - v bE

26. Physics 215 – Fall 2014Lecture 03-226 Reading assignment Relative motion 4.4 in textbook Review for Exam 1 !