1. Physics 207: Lecture 8, Pg 1 Lecture 9 l Goals  Describe Friction in Air (Ch. 6)  Differentiate between Newton’s 1 st, 2 nd and 3 rd Laws  Use Newton’s 3 rd Law in problem solving Assignment: HW4, (Chap. 6 & 7, due 10/5) 1 st Exam Thurs., Oct. 7 th from 7:15-8:45 PM Chapters 1-7 in rooms 2103, 2141, & 2223 Chamberlin Hall

2. Physics 207: Lecture 8, Pg 2 Friction in a viscous medium Drag Force Quantified With a cross sectional area, A (in m 2 ), coefficient of drag of 1.0 (most objects),  sea-level density of air, and velocity, v (m/s), the drag force is: D = ½ C  A v 2  c A v 2 in Newtons c = ¼ kg/m 3 In falling, when D = mg, then at terminal velocity l Example: Bicycling at 10 m/s (22 m.p.h.), with projected area of 0.5 m 2 exerts a force of ~30 Newtons  At low speeds air drag is proportional to v but at high speeds it is v 2  Minimizing drag is often important

3. Physics 207: Lecture 8, Pg 3 Newton’s Third Law: If object 1 exerts a force on object 2 (F 2,1 ) then object 2 exerts an equal and opposite force on object 1 (F 1,2 ) F 1,2 = -F 2,1 IMPORTANT: Newton’s 3 rd law concerns force pairs which act on two different objects (not on the same object) ! For every “action” there is an equal and opposite “reaction”

4. Physics 207: Lecture 8, Pg 4 Force Pairs vs. Free Body Diagrams Consider the following two cases (a falling ball and ball on table), Compare and contrast Free Body Diagram and Action-Reaction Force Pair sketch

5. Physics 207: Lecture 8, Pg 5 Free body diagrams mgmgmgmg F B,T = N Ball Falls For Static Situation N = mg

6. Physics 207: Lecture 8, Pg 6 Force Pairs 1 st and 2 nd Laws  Free-body diagram Relates force to acceleration 3 rd Law  Action/reaction pairs Shows how forces act between objects F B,E = -mg F B,T = N F E,B = mg F B,E = -mg F E,B = mg F T,B = -N

7. Physics 207: Lecture 8, Pg 7 Example (non-contact) Consider the forces on an object undergoing projectile motion F B,E = - m B g EARTH F E,B = m B g F B,E = - m B g F E,B = m B g Question: By how much does g change at an altitude of 40 miles? (Radius of the Earth ~4000 mi)

8. Physics 207: Lecture 8, Pg 8 Note on Gravitational Forces Newton also recognized that gravity is an attractive, long-range force between any two objects. When two objects with masses m 1 and m 2 are separated by distance r, each object “pulls” on the other with a force given by Newton’s law of gravity, as follows:

9. Physics 207: Lecture 8, Pg 9 Cavendish’s Experiment F = m 1 g = G m 1 m 2 / r 2 g = G m 2 / r 2 If we know big G, little g and r then will can find m 2 the mass of the Earth!!!

10. Physics 207: Lecture 8, Pg 10 Example (non-contact) Consider the force on a satellite undergoing projectile motion 40 km above the surface of the earth: F B,E = - m B g EARTH F E,B = m B g F B,E = - m B g F E,B = m B g Compare: g = G m 2 / 4000 2 g ’ = G m 2 / (4000+40) 2 g / g ’ = / (4000+40) 2 / 4000 2 = 0.98

11. Physics 207: Lecture 8, Pg 11 A conceptual question: A flying bird in a 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 Follow up question: So, what is holding the airplane up in the sky?

12. Physics 207: Lecture 8, Pg 12 Static Friction with a bicycle wheel l You are pedaling hard and the bicycle is speeding up. What is the direction of the frictional force? l You are breaking and the bicycle is slowing down What is the direction of the frictional force?

13. Physics 207: Lecture 8, Pg 13 Exercise Newton’s Third Law A. greater than B. equal to C. less than A fly is deformed by hitting the windshield of a speeding bus.  v The force exerted by the bus on the fly is, that exerted by the fly on the bus.

14. Physics 207: Lecture 8, Pg 14 Exercise 2 Newton’s Third Law A. greater than B. equal to C. less than A fly is deformed by hitting the windshield of a speeding bus.  v The magnitude of the acceleration, due to this collision, of the bus is that of the fly. Same scenario but now we examine the accelerations

15. Physics 207: Lecture 8, Pg 15 Exercise 3 Newton’s 3rd Law A. 2 B. 4 C. 6 D. Something else a b l Two blocks are being pushed by a finger on a horizontal frictionless floor. l How many action-reaction force pairs are present in this exercise?

16. Physics 207: Lecture 8, Pg 16 Force pairs on an Inclined plane Forces on the block Block weight is mg Normal Force Friction Force “Normal” means perpendicular   mg cos  f  x y mg sin 

17. Physics 207: Lecture 8, Pg 17 Force pairs on an Inclined plane Forces on the block (equilibrium case) Normal Force Friction Force  f=  N x y Forces on the plane

18. Physics 207: Lecture 8, Pg 18 Force pairs on an Inclined plane Forces on the block (non equilibrium case) Normal Force Friction Force  f x y Forces on the plane

19. Physics 207: Lecture 8, Pg 19 Example: Friction and Motion l A box of mass m 1 = 1 kg is being pulled by a horizontal string having tension T = 40 N. It slides with friction (  k = 0.5) on top of a second box having mass m 2 = 2 kg, which in turn slides on a smooth (frictionless) surface.  What is the acceleration of the second box ? 1 st Question: What is the force on mass 2 from mass 1? (A) a = 0 N (B) a = 5 N (C) a = 20 N (D) can’t tell m2m2m2m2 T m1m1m1m1  slides with friction (  k =0.5  ) slides without friction a = ? v

20. Physics 207: Lecture 8, Pg 20 Example Solution l First draw FBD of the top box: m1m1 N1N1 m1gm1g T f k =  K N 1 =  K m 1 g v

21. Physics 207: Lecture 8, Pg 21 l Newtons 3 rd law says the force box 2 exerts on box 1 is equal and opposite to the force box 1 exerts on box 2. m1m1 f f 1,2 =  K m 1 g = 5 N m2m2 f= f f 2,1 = -f 1,2 l As we just saw, this force is due to friction: Example Solution Action Reaction (A) a = 0 N (B) a = 5 N (C) a = 20 N (D) can’t tell

22. Physics 207: Lecture 8, Pg 22 l Now consider the FBD of box 2: m2m2 f f 2,1 =  k m 1 g m2gm2g N2N2 m1gm1g Example Solution

23. Physics 207: Lecture 8, Pg 23 l Finally, solve F x = ma in the horizontal direction: m2m2 f f 2,1 =  K m 1 g  K m 1 g = m 2 a Example Solution = 2.5 m/s 2

24. Physics 207: Lecture 8, Pg 24 Home Exercise Friction and Motion, Replay A box of mass m 1 = 1 kg, initially at rest, is now pulled by a horizontal string having tension T = 10 N. This box (1) is on top of a second box of mass m 2 = 2 kg. The static and kinetic coefficients of friction between the 2 boxes are  s =1.5 and  k = 0.5. The second box can slide freely (frictionless) on an smooth surface. Compare the acceleration of box 1 to the acceleration of box 2 ? m2m2m2m2 T m1m1m1m1  friction coefficients  s =1.5 and  k =0.5 slides without friction a2a2 a1a1

25. Physics 207: Lecture 8, Pg 25 Home Exercise Friction and Motion, Replay in the static case A box of mass m 1 = 1 kg, initially at rest, is now pulled by a horizontal string having tension T = 10 N. This box (1) is on top of a second box of mass m 2 = 2 kg. The static and kinetic coefficients of friction between the 2 boxes are  s =1.5 and  k = 0.5. The second box can slide freely on an smooth surface (frictionless). If no slippage then the maximum frictional force between 1 & 2 is (A) 20 N (B) 15 N (C) 5 N (D) depends on T m2m2m2m2 T m1m1m1m1  friction coefficients  s =1.5 and  k =0.5 slides without friction a2a2 a1a1

26. Physics 207: Lecture 8, Pg 26 Home Exercise Friction and Motion f s = 10 N and the acceleration of box 1 is Acceleration of box 2 equals that of box 1, with |a| = |T| / (m 1 +m 2 ) and the frictional force f is m 2 a (Notice that if T were in excess of 15 N then it would break free) m2m2m2m2 T m1m1m1m1  friction coefficients  s =1.5 and  k =0.5 slides without friction a2a2 a1a1 T g m 1 g N fSfS f S   S N =  S m 1 g = 1.5 x 1 kg x 10 m/s 2 which is 15 N (so m 2 can’t break free)

27. Physics 207: Lecture 8, Pg 27 Recap l Wednesday: Review for exam