1. Physics 207: Lecture 7, Pg 1 Lecture 7 l Goals:  Solve 1D and 2D problems with forces in equilibrium and non-equilibrium (i.e., acceleration) using Newton’ 1 st and 2 nd laws.  Distinguish static and kinetic coefficients of friction  Differentiate between Newton’s 1 st, 2 nd and 3 rd Laws Assignment: HW4, (Chapters 6 & 7 (1 st part) due 2/17, 9 am, Wednesday) For Thursday, Read Chapter 7 1 st Exam Thursday, Feb. 17 from 7:15-8:45 PM Chapters 1-7

2. Physics 207: Lecture 7, Pg 2 Identifying Forces: Non-contact All objects having mass exhibit a mutually attractive force (i.e., gravity) that is distance dependent At the Earth’s surface this variation is small so little “g” (the associated acceleration) is typically set to 9.80 or 10. m/s 2 F B,G

3. Physics 207: Lecture 7, Pg 3 Contact (i.e., normal) Forces Certain forces act to keep an object in place. These have what ever force needed to balance all others (until a breaking point). F B,T

4. Physics 207: Lecture 7, Pg 4 No net force  No acceleration F B,T Normal force is always  to a surface F B,G (Force vectors are not always drawn at contact points) y

5. Physics 207: Lecture 7, Pg 5 No net force  No acceleration l If zero velocity then “static equilibrium” l If non-zero velocity then “dynamic equilibrium” l This label, static vs. dynamic, is observer dependent l Forces are vectors

6. Physics 207: Lecture 7, Pg 6 A special contact force: Friction l What does it do?  It opposes motion (velocity, actual or that which would occur if friction were absent!) l How do we characterize this in terms we have learned?  Friction results in a force in a direction opposite to the direction of motion (actual or, if static, then “inferred”)! mama F F APPLIED f f FRICTION gmggmg N i j

7. Physics 207: Lecture 7, Pg 7 Friction... l Friction is caused by the “microscopic” interactions between the two surfaces:

8. Physics 207: Lecture 7, Pg 8 Friction... l Force of friction acts to oppose motion:  Parallel to a surface N  Perpendicular to a Normal force. mama F ffFffF gmggmg N i j

9. Physics 207: Lecture 7, Pg 9 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?

10. Physics 207: Lecture 7, Pg 10 Important notes l Many contact forces are conditional and, more importantly, they are not necessarily constant l We have a general notion of forces is from everyday life. l In physics the definition must be precise.  A force is an action which causes a body to accelerate. (Newton’s Second Law) l On a microscopic level, all forces are non-contact

11. Physics 207: Lecture 7, Pg 11 Pushing and Pulling Forces l A rope is an example of something that can pull l You arm is an example of an object that can push or push

12. Physics 207: Lecture 7, Pg 12 Examples of Contact Forces: A spring can push

13. Physics 207: Lecture 7, Pg 13 A spring can pull

14. Physics 207: Lecture 7, Pg 14 Ropes provide tension (a pull) In physics we often use a “massless” rope with opposing tensions of equal magnitude

15. Physics 207: Lecture 7, Pg 15 Forces at different angles Case 1 Case 2 F mg N Case1: Downward angled force with friction Case 2: Upwards angled force with friction Cases 3,4: Up against the wall Questions: Does it slide? What happens to the normal force? What happens to the frictional force? mg Cases 3, 4 mg N N F F f f f

16. Physics 207: Lecture 7, Pg 16 Free Body Diagram A heavy sign is hung between two poles by a rope at each corner extending to the poles. Eat at Bucky’s A hanging sign is an example of static equilibrium (depends on observer) What are the forces on the sign and how are they related if the sign is stationary (or moving with constant velocity) in an inertial reference frame ?

17. Physics 207: Lecture 7, Pg 17 Free Body Diagram Step two: Sketch in force vectors Step three: Apply Newton’s 2 nd Law (Resolve vectors into appropriate components)  mgmg  T1T1 T2T2 Eat at Bucky’s   T1T1 T2T2 mgmg Step one: Define the system

18. Physics 207: Lecture 7, Pg 18 Free Body Diagram Eat at Bucky’s   T1T1 T2T2 mgmg Vertical : y-direction 0 = -mg + T 1 sin  1 + T 2 sin  2 Horizontal : x-direction 0 = -T 1 cos  1 + T 2 cos  2

19. Physics 207: Lecture 7, Pg 19 Exercise, Newton’s 2 nd Law A. P + C < W B. P + C > W C. P = C D. P + C = W A woman is straining to lift a large crate, without success. It is too heavy. We denote the forces on the crate as follows: P is the upward force being exerted on the crate by the person C is the contact or normal force on the crate by the floor, and W is the weight (force of the earth on the crate). Which of following relationships between these forces is true, while the person is trying unsuccessfully to lift the crate? (Note: force up is positive & down is negative)

20. Physics 207: Lecture 7, Pg 20 Mass l We have an idea of what mass is from everyday life. l In physics:  Mass (in Phys 207) is a quantity that specifies how much inertia an object has (i.e. a scalar that relates force to acceleration) (Newton’s Second Law) l Mass is an inherent property of an object. l Mass and weight are different quantities; weight is usually the magnitude of a gravitational (non-contact) force. “Pound” (lb) is a definition of weight (i.e., a force), not a mass!

21. Physics 207: Lecture 7, Pg 21 Inertia and Mass l The tendency of an object to resist any attempt to change its velocity is called Inertia l Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity (acceleration) If mass is constant then If force constant  l Mass is an inherent property of an object l Mass is independent of the object’s surroundings l Mass is independent of the method used to measure it l Mass is a scalar quantity l The SI unit of mass is kg |a| m

22. Physics 207: Lecture 7, Pg 22 Exercise Newton’s 2 nd Law A. increasing B. decreasing C. constant in time D. Not enough information to decide l An object is moving to the right, and experiencing a net force that is directed to the right. The magnitude of the force is decreasing with time (read this text carefully). l The speed of the object is

23. Physics 207: Lecture 7, Pg 23 Exercise Newton’s 2 nd Law A. B B. C C. D D. F E. G A 10 kg mass undergoes motion along a line with a velocities as given in the figure below. In regards to the stated letters for each region, in which is the magnitude of the force on the mass at its greatest?

24. Physics 207: Lecture 7, Pg 24 Home Exercise Newton’s 2 nd Law A. 4 x as long B. 2 x as long C. 1/2 as long D. 1/4 as long A constant force is exerted on a cart that is initially at rest on an air table. The force acts for a short period of time and gives the cart a certain final speed s. Air Track Cart Force In a second trial, we apply a force only half as large. To reach the same final speed, how long must the same force be applied (recall acceleration is proportional to force if mass fixed)?

25. Physics 207: Lecture 7, Pg 25 Home Exercise Newton’s 2 nd Law Solution Air Track Cart Force (B) 2 x as long F = ma Since F 2 = 1/2 F 1 a 2 = 1/2 a 1 We know that under constant acceleration, v = a  t So, a 2  t 2 = a 1  t 1 we want equal final velocities 1/2 a 1 /  t 2 = a 1 /  t 1  t 2 = 2  t 1

26. Physics 207: Lecture 7, Pg 26 Home Exercise Newton’s 2 nd Law A. 8 x as far B. 4 x as far C. 2 x as far D. 1/4 x as far A force of 2 Newtons acts on a cart that is initially at rest on an air track with no air and pushed for 1 second. Because there is friction (no air), the cart stops immediately after I finish pushing. It has traveled a distance, D. Air Track Cart Force Next, the force of 2 Newtons acts again but is applied for 2 seconds. The new distance the cart moves relative to D is:

27. Physics 207: Lecture 7, Pg 27 Home Exercise Solution Air Track Cart Force (B) 4 x as long We know that under constant acceleration,  x = a (  t) 2 /2 (when v 0 =0) Here  t 2 =2  t 1, F 2 = F 1  a 2 = a 1

28. Physics 207: Lecture 7, Pg 28 Home Exercise: Physics in an Elevator Gravity and Normal Forces A woman in an elevator is accelerating upwards The normal force exerted by the elevator on the woman is, (A) greater than (B) the same as (C) less than the force due to gravity acting on the woman (REMEMBER: Draw a FREE BODY DIAGRAM)

29. Physics 207: Lecture 7, Pg 29 Moving forces around l Massless strings: Translate forces and reverse their direction but do not change their magnitude (we really need Newton’s 3 rd of action/reaction to justify) l Massless, frictionless pulleys: Reorient force direction but do not change their magnitude string T1T1 -T 1 T1T1 T2T2 -T 2 | T 1 | = | -T 1 | = | T 2 | = | T 2 |

30. Physics 207: Lecture 7, Pg 30 Scale Problem l You are given a 1.0 kg mass and you hang it directly on a fish scale and it reads 10 N (g is 10 m/s 2 ). l Now you use this mass in a second experiment in which the 1.0 kg mass hangs from a massless string passing over a massless, frictionless pulley and is anchored to the floor. The pulley is attached to the fish scale. l What force does the fish scale now read? 1.0 kg 10 N ? 1.0 kg

31. Physics 207: Lecture 7, Pg 31 Scale Problem l Step 1: Identify the system(s). In this case it is probably best to treat each object as a distinct element and draw three force body diagrams.  One around the scale  One around the massless pulley (even though massless we can treat is as an “object”)  One around the hanging mass l Step 2: Draw the three FBGs. (Because this is a now a one-dimensional problem we need only consider forces in the y-direction.) ? 1.0 kg

32. Physics 207: Lecture 7, Pg 32 Static and Kinetic Friction l Friction exists between objects and its behavior has been modeled. At Static Equilibrium: A block, mass m, with a horizontal force F applied, N Direction: A force vector  to the normal force vector N and the vector is opposite to the direction of acceleration if  were 0. Magnitude: f is proportional to the applied forces such that f s ≤  s N   s called the “coefficient of static friction”

33. Physics 207: Lecture 7, Pg 33 Case study... big F l Dynamics: x-axis i :ma x = F  K N y-axis j : ma y = 0 = N – mg or N = mg soF  K mg = m a x axmaxaxmax F gmggmg N i j  K mg v fkfk fkfk

34. Physics 207: Lecture 7, Pg 34 Case study... little F l Dynamics: x-axis i :ma x = F  K N y-axis j : ma y = 0 = N – mg or N = mg soF  K mg = m a x axmaxaxmax F gmggmg N i j  K mg v fkfk fkfk

35. Physics 207: Lecture 7, Pg 35 Friction: Static friction Static equilibrium: A block with a horizontal force F applied, As F increases so does f s F m1m1 FBD fsfs N mg  F x = 0 = -F + f s  f s = F  F y = 0 = - N + mg  N = mg

36. Physics 207: Lecture 7, Pg 36 Static friction, at maximum (just before slipping) Equilibrium: A block, mass m, with a horizontal force F applied, N Direction: A force vector  to the normal force vector N and the vector is opposite to the direction of acceleration if  were 0. Magnitude: f S is proportional to the magnitude of N f s =  s N F m fsfs N mg

37. Physics 207: Lecture 7, Pg 37 Kinetic or Sliding friction (f k < f s ) Dynamic equilibrium, moving but acceleration is still zero As F increases f k remains nearly constant (but now there acceleration is acceleration) F m1m1 FBD fkfk N mg  F x = 0 = -F + f k  f k = F  F y = 0 = - N + mg  N = mg v f k =  k N

38. Physics 207: Lecture 7, Pg 38 Sliding Friction: Quantitatively N l Direction: A force vector  to the normal force vector N and the vector is opposite to the velocity. fN l Magnitude: f k is proportional to the magnitude of N fN g  f k =  k N ( =  K  mg in the previous example) The constant  k is called the “coefficient of kinetic friction” Logic dictates that  S >  K for any system

39. Physics 207: Lecture 7, Pg 39 Coefficients of Friction Material on Material  s = static friction  k = kinetic friction steel / steel0.60.4 add grease to steel0.10.05 metal / ice0.0220.02 brake lining / iron0.40.3 tire / dry pavement0.90.8 tire / wet pavement0.80.7

40. Physics 207: Lecture 7, Pg 40 An experiment Two blocks are connected on the table as shown. The table has unknown static and kinetic friction coefficients. Design an experiment to find  S Static equilibrium: Set m 2 and add mass to m 1 to reach the breaking point. Requires two FBDs m1m1 m2m2 m2gm2g N m1gm1g T T Mass 2  F x = 0 = -T + f s = -T +  S N  F y = 0 = N – m 2 g fSfS Mass 1  F y = 0 = T – m 1 g T = m 1 g =  S m 2 g   S = m 1 /m 2

41. Physics 207: Lecture 7, Pg 41 A 2 nd experiment Two blocks are connected on the table as shown. The table has unknown static and kinetic friction coefficients. Design an experiment to find  K. Dynamic equilibrium: Set m 2 and adjust m 1 to find place when a = 0 and v ≠ 0 Requires two FBDs m1m1 m2m2 m2gm2g N m1gm1g T T Mass 2  F x = 0 = -T + f f = -T +  k N  F y = 0 = N – m 2 g fkfk Mass 1  F y = 0 = T – m 1 g T = m 1 g =  k m 2 g   k = m 1 /m 2

42. Physics 207: Lecture 7, Pg 42 An experiment (with a ≠ 0) Two blocks are connected on the table as shown. The table has unknown static and kinetic friction coefficients. Design an experiment to find  K. Non-equilibrium: Set m 2 and adjust m 1 to find regime where a ≠ 0 Requires two FBDs T Mass 2  F x = m 2 a = -T + f k = -T +  k N  F y = 0 = N – m 2 g m1m1 m2m2 m2gm2g N m1gm1g T fkfk Mass 1  F y = m 1 a = T – m 1 g T = m 1 g + m 1 a =  k m 2 g – m 2 a   k = (m 1 (g+a)+m 2 a)/m 2 g

43. Physics 207: Lecture 7, Pg 43 Sample Problem l You have been hired to measure the coefficients of friction for the newly discovered substance jelloium. Today you will measure the coefficient of kinetic friction for jelloium sliding on steel. To do so, you pull a 200 g chunk of jelloium across a horizontal steel table with a constant string tension of 1.00 N. A motion detector records the motion and displays the graph shown. l What is the value of μ k for jelloium on steel?

44. Physics 207: Lecture 7, Pg 44 Sample Problem  F x =ma = F - f f = F -  k N = F -  k mg  F y = 0 = N – mg  k = (F - ma) / mg & x = ½ a t 2  0.80 m = ½ a 4 s 2 a = 0.40 m/s 2  k = (1.00 - 0.20 · 0.40 ) / (0.20 ·10.) = 0.46

45. Physics 207: Lecture 7, Pg 45 Recap l Assignment: HW4, (due 2/17, 9 am, Wednesday) l Read Chapter 7 l 1st Exam Wednesday, Feb. 17 from 7:15-8:45 PM