1. Physics 207: Lecture 6, Pg 1 Lecture 6 Chapter 4 l Discuss circular motion  Discern differing reference frames and understand how they relate to particle motion Chapters 5 & 6  Recognize different types of forces and know how they act on an object in a particle representation  Identify forces and draw a Free Body Diagram  Solve problems with forces in equilibrium (a=0) and non- equilibrium (a≠0) using Newton’s 1 st & 2 nd laws. Assignments: HW3 (Chapters 4 & 5), finish reading Chapter 6 Exam 1: Thurs. Oct. 7 from 7:15-8:45 PM Chapters 1-7

2. Physics 207: Lecture 6, Pg 2 Concept Check Q1. You drop a ball from rest, how much of the acceleration from gravity goes to changing its speed? A. All of it B. Most of it C. Some of it D. None of it Q2. A hockey puck slides off the edge of the table, at the instant it leaves the table, how much of the acceleration from gravity goes to changing its speed? A. All of it B. Most of it C. Some of it D. None of it

3. Physics 207: Lecture 6, Pg 3 Circular Motion (quick “review”) Angular position  Arc traversed s = r  Tangential speed | v T | =  s /  t & if  t  0 ds / dt = r d  /dt Angular velocity  = d  /dt =  v T | / r Radial acceleration  a r | =  v T 2 / r =  2 r r  vTvT s

4. Physics 207: Lecture 6, Pg 4 Uniform Circular Motion (UCM,  =0) Period (T): The time required to do one full revolution, 360 ° or 2  radians Frequency (f): 1/T, number of “cycles” per unit time Angular velocity or angular speed  = 2  f = 2  /T Positive  counter clockwise Negative  clockwise r  vtvt s

5. Physics 207: Lecture 6, Pg 5 Mass-based separation with a centrifuge How many g’s (1 g is ~10 m/s 2 )? a r = v T 2 / r =  2 r f = 6000 rpm = 100 rev. per second is typical with r = 0.10 m a r = (2  10 2 ) 2 x 0.10 m/s 2 BeforeAfter bb5 a r = 4 x 10 4 m/s 2 or ca. 4000 g’s !!!

6. Physics 207: Lecture 6, Pg 6 g’s with respect to humans l 1 gStanding l 1.2 gNormal elevator acceleration (up). l 1.5-2g Walking down stairs. l 2-3 gHopping down stairs. l 3.5 gMaximum acceleration in amusement park rides (design guidelines). l 4 gIndy cars in the second turn at Disney World (side and down force). l 4+ gCarrier-based aircraft launch. l 10 gThreshold for blackout during violent maneuvers in high performance aircraft. l 11 gAlan Shepard in his historic sub orbital Mercury flight experience a maximum force of 11 g. l 20 gColonel Stapp’s experiments on acceleration in rocket sleds indicated that in the 10-20 g range there was the possibility of injury because of organs moving inside the body. Beyond 20 g they concluded that there was the potential for death due to internal injuries. Their experiments were limited to 20 g. l 30 gThe design maximum for sleds used to test dummies with commercial restraint and air bag systems is 30 g. Comment: In automobile accidents involving rotation severe injury or death can occur even at modest speeds

7. Physics 207: Lecture 6, Pg 7 Circular Motion Radial acceleration  a r | =  v T 2 / r =  2 r Angular acceleration  = d 2  /dt =  a T | / r s = s 0 + v T0  t + ½ a T  t 2 r  vTvT s

8. Physics 207: Lecture 6, Pg 8 A bad day at the lab…. l In 1998, a Cornell campus laboratory was seriously damaged when the rotor of an ultracentrifuge failed while in use. l Description of the Cornell Accident -- On December 16, 1998, milk samples were running in a Beckman. L2-65B ultracentrifuge using a large aluminum rotor. The rotor had been used for this procedure many times before. Approximately one hour into the operation, the rotor failed due to excessive mechanical stress caused by the g-forces of the high rotation speed. The subsequent explosion completely destroyed the centrifuge. The safety shielding in the unit did not contain all the metal fragments. The half inch thick sliding steel door on top of the unit buckled allowing fragments, including the steel rotor top, to escape. Fragments ruined a nearby refrigerator and an ultra-cold freezer in addition to making holes in the walls and ceiling. The unit itself was propelled sideways and damaged cabinets and shelving that contained over a hundred containers of chemicals. Sliding cabinet doors prevented the containers from falling to the floor and breaking. A shock wave from the accident shattered all four windows in the room. The shock wave also destroyed the control system for an incubator and shook an interior wall.

9. Physics 207: Lecture 6, Pg 9 Example Question l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity? l 3 What is the tangential speed of a point on the rim during this initial period? 4 Sketch the  (angular displacement) versus time plot. l 5 What is the average angular velocity over the 1 st 10 seconds? l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the angular acceleration be? l 7 What is the magnitude and direction of the acceleration after 10 seconds?

10. Physics 207: Lecture 6, Pg 10 Example l A horizontally mounted disk 2.0 meters in diameter (1.0 m in radius) spins at constant angular speed such that it first undergoes (1) 10 counter clockwise revolutions in 5.0 seconds and then, again at constant angular speed, (2) 2 counter clockwise revolutions in 5.0 seconds. l 1 What is T the period of the initial rotation? T = time for 1 revolution = 5 sec / 10 rev = 0.5 s

11. Physics 207: Lecture 6, Pg 11 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. 2 What is  the initial angular velocity?  = d  /dt =  /  t  = 10 2π radians / 5 seconds = 12.6 rad / s ( also 2  f = 2  / T )

12. Physics 207: Lecture 6, Pg 12 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 3 What is the tangential speed of a point on the rim during this initial period? | v T | = ds/dt = (r d  /dt = r  | v T | = r  = 1 m 12.6 rad/ s = 12.6 m/s

13. Physics 207: Lecture 6, Pg 13 Example l A horizontally mounted disk 1 meter in radius spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. 4 Sketch the  (angular displacement) versus time plot.  =   +    t r  vtvt s

14. Physics 207: Lecture 6, Pg 14 Sketch of  vs. time time (seconds)        (radians)  =   +   t  =  +  5  rad  =   +   t  =   rad + (  5) 5 rad  = 24 rad

15. Physics 207: Lecture 6, Pg 15 Example l 5 What is the average angular velocity over the 1 st 10 seconds?

16. Physics 207: Lecture 6, Pg 16 Sketch of  vs. time time (seconds)        (radians)  =   +   t  =  +  5  rad  =   +   t  =   rad + (  5) 5 rad  = 24 rad 5 Avg. angular velocity =  /  t = 24  /10 rad/s

17. Physics 207: Lecture 6, Pg 17 Example 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must be the angular acceleration  ? Key point …..  is associated with tangential acceleration (a T ).

18. Physics 207: Lecture 6, Pg 18 Tangential acceleration?  =  o +  o  t +  t 2 (from plot, after 10 seconds) 24  rad = 0 rad + 0 rad/s  t + ½ (a T /r)  t 2 48  rad 1m / 100 s 2 = a T r  vtvt s 1 2 aTaT r l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the “tangential acceleration” be? l 7 What is the magnitude and direction of the acceleration after 10 seconds?

19. Physics 207: Lecture 6, Pg 19 Non-uniform Circular Motion For an object moving along a curved trajectory, with varying speed Vector addition: a = a r + a T (radial and tangential) arar aTaT a

20. Physics 207: Lecture 6, Pg 20 Tangential acceleration? a T = 0.48  m / s 2 and v T = 0 + a T  t= 4.8  m/s = v T a r = v T 2 / r = 23  2 m/s 2 r  vtvt s l 7 What is the magnitude and direction of the acceleration after 10 seconds? Tangential acceleration is too small to plot!

21. Physics 207: Lecture 6, Pg 21 Do different observers see the same physics Relative motion and frames of reference l Reference frame S is stationary Reference frame S ’ is moving at v o This also means that S moves at – v o relative to S ’ l Define time t = 0 as that time when the origins coincide

22. Physics 207: Lecture 6, Pg 22 Relative Velocity The positions, r and r ’, as seen from the two reference frames are related through the velocity, v o, where v o is velocity of the r ’ reference frame relative to r  r ’ = r – v o  t l The derivative of the position equation will give the velocity equation  v ’ = v – v o l These are called the Galilean transformation equations l Reference frames that move with “constant velocity” (i.e., at constant speed in a straight line) are defined to be inertial reference frames (IRF); anyone in an IRF sees the same acceleration of a particle moving along a trajectory.  a ’ = a (dv o / dt = 0)

23. Physics 207: Lecture 6, Pg 23 Central concept for problem solving: “x” and “y” components of motion treated independently. l Example: Man on cart tosses a ball straight up in the air. l You can view the trajectory from two reference frames: Reference frame on the ground. Reference frame on the moving cart. y(t) motion governed by 1) a = -g y 2) v y = v 0y – g  t 3) y = y 0 + v 0y – g  t 2 /2 x motion: x = v x t Net motion : R = x(t) i + y(t) j (vector)

24. Physics 207: Lecture 6, Pg 24 Relative Velocity l Two observers moving relative to each other generally do not agree on the outcome of an experiment (path) l For example, observers A and B below see different paths for the ball

25. Physics 207: Lecture 6, Pg 25 Acceleration in Different Frames of Reference l Again: The derivative of the velocity equation will give the acceleration equation  v ’ = v – v 0  a ’ = a l The acceleration of the particle measured by an observer in one frame of reference is the same as that measured by any other observer moving at a constant velocity relative to the first frame.

26. Physics 207: Lecture 6, Pg 26 Home Exercise, Relative Motion l You are swimming across a 50 m wide river in which the current moves at 1 m/s with respect to the shore. Your swimming speed is 2 m/s with respect to the water. You swim across in such a way that your path is a straight perpendicular line across the river. l How many seconds does it take you to get across? 2m/s 1m/s50m a) b) c) d)

27. Physics 207: Lecture 6, Pg 27 Home Exercise, l The time taken to swim straight across is (distance across) / (v y ) Choose x axis along riverbank and y axis across river y x l Since you swim straight across, you must be tilted in the water so that your x component of velocity with respect to the water exactly cancels the velocity of the water in the x direction: 1m/s m/s y x 2m/s

28. Physics 207: Lecture 6, Pg 28 No Net Force, No acceleration…a demo exercise l In this demonstration we have a ball tied to a string undergoing horizontal UCM (i.e. the ball has only radial acceleration) 1 Assuming you are looking from above, draw the orbit with the tangential velocity and the radial acceleration vectors sketched out. 2 Suddenly the string brakes. 3 Now sketch the trajectory with the velocity and acceleration vectors drawn again.

29. Physics 207: Lecture 6, Pg 29 Chaps. 5, 6 & 7 What causes motion? (What is special about acceleration?) What are forces ? What kinds of forces are there ? How are forces and changes in motion related ?

30. Physics 207: Lecture 6, Pg 30 Newton’s First Law and IRFs An object subject to no external forces moves with constant velocity if viewed from an inertial reference frame (IRF). If no net force acting on an object, there is no acceleration. l The above statement can be used to define inertial reference frames.

31. Physics 207: Lecture 6, Pg 31 IRFs  An IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”.  If one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector!  In many cases (i.e., Chapters 5, 6 & 7) the surface of the Earth may be viewed as an IRF

32. Physics 207: Lecture 6, Pg 32 Newton’s Second Law The acceleration of an object is directly proportional to the net force acting upon it. The constant of proportionality is the mass. l This is a vector expression: F x, F y, F z l Units The metric unit of force is kg m/s 2 = Newtons (N) The English unit of force is Pounds (lb)

33. Physics 207: Lecture 6, Pg 33 Lecture 6 Assignment: Read rest of chapter 6