1. Lecture 5 Chapter 5 Using Newton’s Laws Friction Circular motion Drag Forces Numerical integration Misconceptions

2. Assignment 2 Chp 4 4-52, 4-67, 4-75 Chp 5 5-30, 5-44 Due Friday 12:00 Mar 11

3. Friction You are standing still, then begin to walk. What was the external forced that caused you to accelerate? Hint: It is very hard to start walking if you are standing on ice. What force causes a car to accelerate when a traffic light turns green?

4. Frictional Forces Models of friction See Chabay and Sherwood Matter and Ineractions Volume 1 ISBN 0-471-35491-0 Model of dry friction 3 or 4 asperites support top block. Temporarily weld together The Friction and Lubrication of Solids F. P. Bowden and D. Tabor, Oxford University Press 1964 Friction is an attractive force between two surfaces that is a result of the vector sum of many electrical forces between the surface atoms of the two different bodies. Only about 10 -4 of the surface atoms actually contribute.

5. No motion and no horizontal forces What is the free body diagram without friction?

6. No Motion What is the free body diagram with friction but no motion?

7. Still No Motion with larger Force F

8. Increase F more and now you get acceleration Static friction force decreases to kinetic friction and its value remans constant and now you get acceleration

9. Constant velocity

10. How is the frictional force related to the normal? The maximum value equals is where N is the normal force. Above we would have The coefficient of static friction ranges from 0 to 1.2 Kinetic Friction: If we increase F until the block starts to move, the friction force decreases to mg F fsfs Fixed block N

11. Problem Solving with Newton’s 2nd Law involving friction Vector sum of external forces in x direction = ma x Vector sum of external forces in y direction = ma y If no acceleration, then set sum equal to 0

12. 12/15/2015Physics 631.   N mg +x +y  Mass on incline plane at rest with impending motion. Find the coefficient of static friction What is the free body diagram? h d

13. 12/15/2015Physics 631. Free Body Diagram Apply Newton's 2nd law in the x directions and using f s =  s N x direction  N mg +x +y 

14. 12/15/2015Physics 631. Free Body Diagram Apply Newton's 2nd law in the y direction  N mg +x +y  Apply Newtons 2nd Law

15. 12/15/2015Physics 631.  N mg +x +y 

16. Find a and T for the Atwood’s machine with friction between M and surface. T T T fkfk

17. T mg -y Mg N T f T T T fkfk +y +x Solve for a and then T

18. Find a and T for the Atwood’s machine with friction between M and surface. T T T fkfk

19. Question: What is the minimum magnitude force required to start the crate moving? T  N fsfs mg +x +y Tcosφ x components y components

20. Question: What is the initial acceleration if  k  Newton's 2nd law T  N fsfs mg +x +y  cos  x components y components

21. Inertial Drag Force and Terminal Velocity Drag force: Whenever you have a body like a ball moving through a medium that behaves like a fluid, there will be a drag force opposing the motion. Imagine a falling ball slowed down due to elastic collisions with air molecules. Simply pushing the air out of the way. air molecules ball v A dy. Inertial drag

22. Inertial Drag Force and Terminal Velocity air molecules ball v A dy. Inertial drag

23. Terminal speeds in air where m is the mass of the falling ball Solve for v 0 Using Newton’s 2nd law, Stokes-Napier Law

24. TERMINAL SPEEDS IN AIR Object` Speed (m/s) Speed (mph) Feather0.40.9 Snowflake12.2 BB920 Mouse1329 Tennis ball3166 Baseball4286 Sky diver60 -120134 -268 Cannonball250560 Show demo of falling feather in vacuum

25. How to solve this equation? Two ways One way is to use a spread sheet in Exel. Ball Air Gravity

26. Use Newtons 2nd Law Using 2nd Law Initial component of momentum: Initial force on ball: Find new p

27. Newtons 2nd Law

28. Go to Excel Spread Sheet 631 Website: Lecture 3 Materials

29. =C16+(g-(b_1/m_1)*C16*C16)*delta_t =D16+1/2*(C16+C17)*delta_t

32. We can also solve the equation to get the velocity as a function of time before it reaches terminal velocity. Let b = 1/2C 

33. Solving equation continued

35. Now show comparison of this solution with numerical integration with Excel.

36. Comparison The curve modeled by velocity squared for terminal velocity Differs from the true equation due to a large delta t. When delta t becomes small enough the two curves are Indistinguishable.

37. ma = - kv Whenever you have a body moving through a liquid there will be a drag force opposing the motion. Here the drag force is proportional to - kv. Viscous drag. Water Resistance and Drag Forces A 1000 kg boat in the water shuts off its engine at 90 km/hr. Find the time required to slow down to 45 km/hr due to a water drag force equal to -70v Newtons, where v is the speed of the boat. Let k = 70 N/s/m.

38. v/v 0 = 45/90 =1/2 t = m/k ln 2=1000/70 ln 2 = 9.9 s

39. UNIFORM CIRCULAR MOTION Centripetal Acceleration: accelerates a body by changing the direction of the body’s velocity without changing the speed. There must be a force also pointing radially inward to make this motion. Examples: –Ball on a string : show demo: Force is produced by the weight of the mass and transmitted by the tension in the string. – Moon in orbit around the earth: gravitational force –A car making a sharp turn: friction –A carousel; friction and contact forces Demo: pushing bowling ball with broom in a circle 39

40. Uniform circular motion Centripetal force is really not a new force like gravity, tension, friction. Motion of earth around sun – centripetal force is a result of gravity Rock whirled around on a string – centripetal force is a result of tension Sometimes it is a result of friction or the normal force

41. 41 CENTRIPETAL ACCELERATION:. Δv points radially inward

42. 42 CENTRIPETAL ACCELERATION. Triangles are similar

43. 43 Centripetal Acceleration. And,  so Magnitude of Period of the motion Magnitudes are related by due to similar triangles

44. What is the magnitude of a c and its direction for a radius of r = 0.5 m and a period of T= 2 s, Need to find v What is the direction of a c ? 44 INWARD

45. A ball is being whirled around on a string. The string breaks. Which path does the ball take? v a c e db 45 QUALITATIVE QUIZ

46. VERTICAL CIRCULAR MOTION Down is negative, Up is positive N N At the top: Minimum v for N = 0: (apparent weightlessness) At the bottom: Apparent weight = N = mv 2 /r + mg Weigh more r

47. What do we mean by Fictitious Forces F f = - ma (the fictitious force always acts in the opposite direction of acceleration) T ma mg  TT T FfFf 

48. In a vertically accelerated reference frame, eg. an elevator, what is your apparent weight? Apparent weight = N Upwards is positive Downwards is negative Example of fictitious force (F f = - ma) Upward acceleration Downward acceleration Free fall a = g N = 0 Weightless condition

49. Figure 5.34 Find the T in the cord between A and B and find the acceleration a

50. Wednesday Agenda Any questions about homework Assignment 2 See Section 2-8 page 38 and problem 5-102 on numerical integration Also see 5-87, 5-92, 5-100 More about centripetal acceleration Conceptual questions from Chapter 5 Go on to Chapter 7- Skip Chapter 6 for now

51. Figure 5.18 What happens in between?

52. Figure 5.56 What is the tension T in the cord and the tangential a t and centripetal a c acceleration as a function of m,g, r, v and theta?

53. Figure 5.20 Find v velocity and T period in terms of g, l, and angle theta.

54. Figure 5.24 f

55. Conceptual Questions Chapter 5