Welcome back to Physics 215

1. Welcome back to Physics 215 Today’s agenda: Rotations Relative motion

2. Current homework assignment HW3: • Exam-style problem (print out from course website) • Ch.4 (Knight textbook): 52, 62, 80, 86 • due Wednesday, Sept 19th in recitation Reminder about course website: http://www.phy.syr.edu/courses/PHY215.12Fall/index.html

3. Exam 1: Thursday (9/20/12) • In room 104N (here!) at the usual lecture time • Material covered: • Textbook chapters 1 - 4 • Lectures up through 9/18 (slides online) • Wed/Fri Workshop activities • Homework assignments • Work through practice exam problems (posted on website) • Work on more practice exam problems on Wednesday in recitation workshop

4. Rotations about fixed axis • Linear speed: v = (2pr)/T = w r. Quantity w is called angular velocity •  is a vector! Use right hand rule to find direction of . • Angular accelerationa=Dw/Dt is also a vector! • andparallel angular speed increasing • andantiparallel angular speed decreasing

5. Relating linear and angular kinematics • Linear speed: v = (2pr)/T = w r • Tangential acceleration:atan = r • Radial acceleration: arad = v2/r = 2r

6. Problem – slowing a DVD wI = 27.5 rad/s, a= -10.0 rad/s2 . • 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

7. 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...)

8. Change origin? • Physical laws involve velocities and accelerations which only depend on Dx • Clearly any frame of reference (FOR) with different origin will measure same Dx, v, a, etc.

9. 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?

10. 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.

11. Reference frame(clock, meterstick) carried along by moving object B A

12. B A B A B A

13. B A B A B A

14. B A B A B A

15. Discussion • From point of view of A, car B moves to right. We say the velocity of B relative to A is vBA. Here vBA > 0 • But from point of view of B, car A moves to left. In fact, vAB < 0 • In general, can see that vAB = -vBA

16. Galilean transformation yA yB vBA P vBAt xB xA • xPA = xPB + vBAt -- transformation of coordinates • DxPA/Dt =DxPB/Dt + vBA  vPA = vPB + vBA-- transformation of velocities

17. Discussion • Notice: • It follows that vAB = -vBA • Two objects a and b moving with respect to, say, Earth then find (Pa, Bb, AE) vab = vaE - vbE

18. You are driving East on I-90 at a constant 65 miles per hour. You are passing another car that is going at a constant 60 miles per hour. In your frame of reference (i.e., as measured relative to your car), is the other car 1. going East at constant speed 2. going West at constant speed, 3. going East and slowing down, 4. going West and speeding up.

19. Conclusion • If we want to use (inertial) moving FOR, then velocities are not the same in different frames • However constant velocity motions are always seen as constant velocity • There is a simple way to relate velocities measured by different frames.

20. Why bother? (1) • Why would we want to use moving frames? • Answer: can simplify our analysis of the motion

21. Relative Motion in 2D • Motion may look quite different in different FOR, e.g., ejecting ball from moving cart Earth frame = complicated! Cart frame = simple! Motion of cart

22. Relative Motion in 2D • Consider airplane flying in a crosswind • velocity of plane relative to air, vPA = 240 km/h N • wind velocity, air relative to earth, vAE = 100 km/h E • what is velocity of plane relative to earth, vPE ? vPE = vPA + vAE vAE vPE vPA

23. Why bother? (2) • Have no way in principle of knowing whether any given frame is at rest • Room 104N is NOT at rest (as we have been assuming!)

24. What’s more … • Better hope that the laws of physics don’t depend on the velocity of my FOR (as long as it is inertial …) • Einstein developed Special theory of relativity to cover situations when velocities approach the speed of light

25. The diagram shows the positions of two carts on parallel tracks at successive instants in time. Cart I Cart J Is the average velocity vector of cart J relative to cart I (or, in the reference frame of cart I) in the time interval from 1 to 2…? 1. to the right 2. to the left 3. zero 4. unable to decide

26. Cart I Cart J Is the instantaneous velocity vector of cart J relative to cart I (or, in the reference frame of cart I) at instant 3…? 1. to the right 2. to the left 3. zero 4. unable to decide

27. Cart I Cart J Is the average acceleration vector of cart J relative to cart I (or, in the reference frame of cart I) in the time interval from 1 to 5: 1. to the right 2. to the left 3. zero 4. unable to decide

28. Accelerations? • We have seen that observers in different FORs perceive different velocities • Is there something that they do agree on? • Demo with ball ejected from cart: cart and Earth observer agree on acceleration (time to fall)

29. Acceleration • If car I moves with constant velocity relative to the road, • Then the acceleration of any other object (e.g., car J) measured relative to car I is the same as the acceleration measured relative to the road.

30. Acceleration is same for all inertial FOR! • We have: vPA = vPB + vBA • For velocity of P measured in frame A in terms of velocity measured in B • DvPA/Dt = DvPB/Dt since vBA is constant • Thus acceleration measured in frame A or frame B is same!

31. Reading assignment • Forces, Newton’s Laws of Motion • Ch.5 in textbook • Review for Exam 1 !