1. Physics 207: Lecture 9, Pg 1 Lecture 9 l Today:  Review session Assignment: For Thursday, Read Chapter 8, first four sections Exam Wed., Feb. 18 th from 7:15-8:45 PM Chapters 1-7 One 8½ X 11 note sheet and a calculator (for trig.) Place: Room 2103: All Sections

2. Physics 207: Lecture 9, Pg 2 Textbook Chapters l Chapter 1 Concept of Motion l Chapter 2 1D Kinematics l Chapter 3 Vector and Coordinate Systems l Chapter 4 Dynamics I, Two-dimensional motion l Chapter 5 Forces and Free Body Diagrams l Chapter 6 Force and Newton’s 1 st and 2 nd Laws l Chapter 7 Newton’s 3 rd Law Exam will reflect most key points (but not all) ~30% of the exam will be more conceptual ~70% of the exam is problem solving

3. Physics 207: Lecture 9, Pg 3 The flying bird in the cage l You have a bird in a cage that is resting on your upward turned palm. The cage is completely sealed to the outside (at least while we run the experiment!). The bird is initially sitting at rest on the perch. It decides it needs a bit of exercise and starts to fly. Question: How does the weight of the cage plus bird vary when the bird is flying up, when the bird is flying sideways, when the bird is flying down? l So, what is holding the airplane up in the sky?

4. Physics 207: Lecture 9, Pg 4 Example with pulley l A mass M is held in place by a force F. Find the tension in each segment of the massless ropes and the magnitude of F.  Assume the pulleys are massless and frictionless. The action of a massless frictionless pulley is to change the direction of a tension. This is an example of static equilibrium.  M T5T5 T4T4 T3T3 T2T2 T1T1 F

5. Physics 207: Lecture 9, Pg 5 Example with pulley l A mass M is held in place by a force F. Find the tension in each segment of the rope and the magnitude of F.  Assume the pulleys are massless and frictionless.  Assume the rope is massless. The action of a massless frictionless pulley is to change the direction of a tension. Here F = T 1 = T 2 = T 3 = T Equilibrium means  F = 0 for x, y & z For example: y-dir ma = 0 = T 2 + T 3 – T 5 and ma = 0 = T 5 – Mg So T 5 = Mg = T 2 + T 3 = 2 F  T = Mg/2  M T5T5 T4T4 T3T3 T2T2 T1T1 F

6. Physics 207: Lecture 9, Pg 6 Example l The velocity of an object as a function of time is shown in the graph at right. Which graph below best represents the net force vs time relationship for this object? (E)

7. Physics 207: Lecture 9, Pg 7 Another Example A 200 kg truck accelerates eastwards on a horizontal road in response to a gradually increasing frictional force from the ground. There is an unsecured 50 kg block sitting on the truck bed liner. There is friction between the block and the bed liner. An accelerometer is mounted in the truck. The block accelerates with the truck until the acceleration reaches 10 m/s 2. At that instant the block begins to slide and the truck’s accelerometer now reports a value of 11 m/s 2. What are the coefficients of static and kinetic friction?  S =1.0  k =0.6 0 acceleration time 11 10

8. Physics 207: Lecture 9, Pg 8 Example Wedge with friction A mass m slides with friction down a wedge of angle  at constant velocity. The wedge sits at rest on a frictionless surface and abuts a wall. What is the magnitude of the force of the wall on the block? m v  mg N fkfk FBD block

9. Physics 207: Lecture 9, Pg 9 Example Wedge with friction A mass m slides with friction down a wedge of mass M & angle  at constant velocity. The wedge sits at rest on a frictionless surface and abuts a wall. What is the magnitude of the force of the wall on the block? m v  mg N fkfk FBD block FBD wedge -N -f k Mg FwFw F 3 rd Law

10. Physics 207: Lecture 9, Pg 10 Example Wedge with friction A mass m slides with friction down a wedge of mass M & angle  at constant velocity. The wedge sits at rest on a frictionless surface and abuts a wall. What is the magnitude of the force of the wall on the block? mg N fkfk FBD block x-dir:  F x = 0 = -f k + mg sin  f k = mg sin  y-dir:  F y = 0 = N - mg cos   N = mg cos  y x 

11. Physics 207: Lecture 9, Pg 11 Example Wedge with friction A mass m slides with friction down a wedge of mass M & angle  at constant velocity. The wedge sits at rest on a frictionless surface and abuts a wall. What is the magnitude of the force of the wall on the block? Notice that mg cos  sin  mg cos  sin  Force wall = 0 But there are faster ways. FBD wedge mg cos  Mg FwFw F mg sin    mg cos  sin  

12. Physics 207: Lecture 9, Pg 12 Example Another setting Three blocks are connected on the table as shown. The table has a coefficient of kinetic friction of  K =0.40, the masses are m 1 = 4.0 kg, m 2 = 1.0 kg and m 3 = 2.0 kg. (A) What is the magnitude and direction of acceleration on the three blocks ? (B) What is the tension on the two cords ? m1m1 T1T1 m2m2 m3m3

13. Physics 207: Lecture 9, Pg 13 Another example with a pulley Three blocks are connected on the table as shown. The table has a coefficient of kinetic friction of  K =0.40, the masses are m 1 = 4.0 kg, m 2 = 1.0 kg and m 3 = 2.0 kg. (A) FBD (except for friction) (B) So what about friction ? m1m1 T1T1 m2m2 m3m3 m2gm2g N m3gm3g m1gm1g T3T3 T1T1

14. Physics 207: Lecture 9, Pg 14 Problem recast as 1D motion Three blocks are connected on the table as shown. The center table has a coefficient of kinetic friction of  K =0.40, the masses are m 1 = 4.0 kg, m 2 = 1.0 kg and m 3 = 2.0 kg. m1m1 m2m2 m3m3 m2gm2g N m3gm3g m1gm1g T3T3 T1T1 frictionless m 1 g > m 3 g and m 1 g > (  k m 2 g + m 3 g) and friction opposes motion (starting with v = 0) so f f is to the right and a is to the left (negative) f

15. Physics 207: Lecture 9, Pg 15 Problem recast as 1D motion Three blocks are connected on the table as shown. The center table has a coefficient of kinetic friction of  K =0.40, the masses are m 1 = 4.0 kg, m 2 = 1.0 kg and m 3 = 2.0 kg. m1m1 m2m2 m3m3 m2gm2g N m3gm3g m1gm1g T3T3 T1T1 frictionless x-dir: 1.  F x = m 2 a =  k m 2 g - T 1 + T 3 m 3 a = m 3 g - T 3 m 1 a =  m 1 g + T 1 Add all three: ( m 1 + m 2 + m 3 ) a =  k m 2 g  m 3 g – m 1 g f T3T3 T1T1

16. Physics 207: Lecture 9, Pg 16 Another example with friction and pulley l Three 1 kg masses are connected by two strings as shown below. There is friction,, between the stacked masses but the table top is frictionless. l Assume the pulleys are massless and frictionless. l What is T 1 ? M M M T1T1 friction coefficients  s =0.4 and  k =0.2

17. Physics 207: Lecture 9, Pg 17 Chapter 2

18. Physics 207: Lecture 9, Pg 18 Chapter 2 Also average speed and average velocity

19. Physics 207: Lecture 9, Pg 19 Chapter 3

20. Physics 207: Lecture 9, Pg 20 Chapter 3

21. Physics 207: Lecture 9, Pg 21 Chapter 4

22. Physics 207: Lecture 9, Pg 22 Chapter 4

23. Physics 207: Lecture 9, Pg 23 Chapter 5

24. Physics 207: Lecture 9, Pg 24 Chapter 5 & 6

25. Physics 207: Lecture 9, Pg 25 Chapter 6

26. Physics 207: Lecture 9, Pg 26 Chapter 7

27. Physics 207: Lecture 9, Pg 27 Chapter 7

28. Physics 207: Lecture 9, Pg 28 Short word problems l After breakfast, I weighed myself and the scale read 588 N. On my way out, I decide to take my bathroom scale in the elevator with me. What does the scale read as the elevator accelerates downwards with an acceleration of 1.5 m/s 2 ? W= (1.0-1.5/9.8) 588 N l A bear starts out and walks 1 st with a velocity of 0.60 j m/s for 10 seconds and then walks at 0.40 i m/s for 20 seconds. What was the bear’s average velocity on the walk? What was the bear’s average speed on the walk (with respect to the total distance travelled) ?

29. Physics 207: Lecture 9, Pg 29 Conceptual Problem The pictures below depict cannonballs of identical mass which are launched upwards and forward. The cannonballs are launched at various angles above the horizontal, and with various velocities, but all have the same vertical component of velocity. (d)

30. Physics 207: Lecture 9, Pg 30 Conceptual Problem Let W B and W F be the weight of the bird and the feeder respectively. Let T be the tension in the wire and N be the normal force of the feeder on the bird. Which of the following free-body diagrams best represents the birdfeeder? (The force vectors are not drawn to scale and are only meant to show the direction, not the magnitude, of each force.) A bird sits in a birdfeeder suspended from a tree by a wire, as shown in the diagram at left. (f)

31. Physics 207: Lecture 9, Pg 31 Graphing problem The figure shows a plot of velocity vs. time for an object moving along the x-axis. Which of the following statements is true? (C) (A) The average acceleration over the 11.0 second interval is -0.36 m/s 2 (B) The instantaneous acceleration at t = 5.0 s is -4.0 m/s 2 (C) Both A and B are correct. (D) Neither A nor B are correct.

32. Physics 207: Lecture 9, Pg 32 Conceptual Problem A block is pushed up a 20º ramp by a 15 N force which may be applied either horizontally (P1) or parallel to the ramp (P2). How does the magnitude of the normal force N depend on the direction of P? (B) (A) N will be smaller if P is horizontal than if it is parallel the ramp. (B) N will be larger if P is horizontal than if it is parallel to the ramp. (C) N will be the same in both cases. (D) The answer will depend on the coefficient of friction. 20 °

33. Physics 207: Lecture 9, Pg 33 Conceptual Problem A cart on a roller-coaster rolls down the track shown below. As the cart rolls beyond the point shown, what happens to its speed and acceleration in the direction of motion (D)? A. Both decrease. B. The speed decreases, but the acceleration increases. C. Both remain constant. D. The speed increases, but acceleration decreases. E. Both increase. F. Other

34. Physics 207: Lecture 9, Pg 34 Conceptual Problem l A person initially at point P in the illustration stays there a moment and then moves along the axis to Q and stays there a moment. She then runs quickly to R, stays there a moment, and then strolls slowly back to P. Which of the position vs. time graphs below correctly represents this motion? (2)

35. Physics 207: Lecture 9, Pg 35 The inclined plane coming and going (not static): the component of mg along the surface < kinetic friction l Exercise left for home but you should find that the block will always come to rest. Another type of problem: l A 8.0 kg rocket provides 80 N of thrust. A strong 10 m long rope is attached from a pivot to the rocket. If everything is horizontal and there is no friction describe the motion of the rocket from rest when the rocket has the following angles (90, 45 and 0 degrees).

36. Physics 207: Lecture 9, Pg 36 Sample Problem l A 200 kg wood crate sits in the back of a truck. The coefficients of friction between the crate and the truck are μ s = 0.9 and μ k = 0.5. The truck starts moving up a 20° slope. What is the maximum acceleration the truck can have without the crate slipping out the back? l Solving:  Visualize the problem, Draw a picture if necessary  Identify the system and make a Free Body Diagram  Choose an appropriate coordinate system  Apply Newton’s Laws with conditional constraints (friction)  Solve

37. Physics 207: Lecture 9, Pg 37 Sample Problem l A physics student on Planet Exidor throws a ball that follows the parabolic trajectory shown. The ball’s position is shown at one-second intervals until t = 3 s. At t = 1 s, the ball’s velocity is v = (2 i + 2 j) m/s. a. Determine the ball’s velocity at t = 0 s, 2 s, and 3 s. b. What is the value of g on Planet Exidor? -2 m/s 2

38. Physics 207: Lecture 9, Pg 38 Another question to ponder How high will it go? l One day you are sitting somewhat pensively in an airplane seat and notice, looking out the window, one of the jet engines running at full throttle. From the pitch of the engine you estimate that the turbine is rotating at 3000 rpm and, give or take, the turbine blade has a radius of 1.00 m. If the tip of the blade were to suddenly break off (it occasionally does happen with negative consequences) and fly directly upwards, then how high would it go (assuming no air resistance and ignoring the fact that it would have to penetrate the metal cowling of the engine.)

39. Physics 207: Lecture 9, Pg 39 Lecture 9 Assignment: For Thursday, Read Chapter 8, first four sections Exam Wed., Feb. 18 th from 7:15-8:45 PM Chapters 1-7 One 8½ X 11 note sheet and a calculator (for trig.) Place: Room 2103: All Sections