1. Goals for Chapter 4 To understand the meaning of force in physics To view force as a vector and learn how to combine forces To understand the behavior of a body on which the forces “balance”: Newton’s First Law of Motion

2. Goals for Chapter 4 To learn the relationship between mass, acceleration, and force: Newton’s Second Law of Motion To relate mass and weight To see the effect of action-reaction pairs: Newton’s Third Law of Motion To learn to make free-body diagrams

3. 5-1 Newton's Laws We will focus on Newton's three laws of motion… but o Newtonian mechanics is valid for “everyday” situations o It is not valid for speeds which are an appreciable fraction of the speed of light o Viewed as an approximation of general relativity o It is not valid for objects on the scale of atomic structure o Viewed as an approximation of quantum mechanics o And not valid for objects in accelerating frames of reference!

4. 5-1 Newton's First and Second Laws Before Newtonian mechanics: o Some influence (force) was thought necessary to keep a body moving o The “natural state” of objects was at rest This seems intuitively reasonable (due to friction) But envision a frictionless surface o Does not slow an object o The object would keep moving forever at a constant speed o Friction is a force!

5. What are some properties of a force?

6. Forces Characteristics of forces: o Unit: N, the newton; 1 N = 1 kg m/s 2 o Forces are vectors Net force is the vector sum of all forces on an object Principle of superposition for forces: o A net force has the same impact as a single force with identical magnitude and direction

7. There are four common types of forces #1: The normal force: When an object pushes on a surface, the surface pushes back on the object perpendicular to the surface. This is a contact force.

8. 5-2 Some Particular Forces The normal force: o If you are standing on a surface, the push back on you from the surface (due to deformation) is the normal force o Normal means perpendicular

9. There are four common types of forces #2: Friction force: This force occurs when a surface resists sliding of an object and is parallel to the surface. Friction is a contact force.

10. There are four common types of forces II #3: Tension force: A pulling force exerted on an object by a rope or cord. This is a contact force.

11. There are four common types of forces II #4: Weight: The pull of gravity on an object. This is a long-range force, not a contact force, and is also a “field” force.

12. 5-2 Some Particular Forces The gravitational force: o A pull that acts on a body, directed toward a second body o Generally we consider situations where the second body is Earth In free fall (+y direction chosen as UP, with no drag from the air):

13. 5-2 Some Particular Forces In free fall (+y direction chosen as UP, with no drag from the air): We can write it as a vector:

14. 5-2 Some Particular Forces Weight : The name given to the gravitational force that one body (like the Earth) exerts on an object o It is a force measured in newtons (N) o It is directed downward towards the center

15. Example To relate weight to mass, consider an apple in free fall. The only force on the apple is the gravitational force which results in an acceleration of g. Applying Newton’s 2 nd Law F net = ma where F net = F g = W and a =g

16. Example To relate weight to mass, consider an apple in free fall. The only force on the apple is the gravitational force which results in an acceleration of g. Applying Newton’s 2 nd Law F net = ma where F net = F g = W and a =g F g = W = mg Thus, W = mg (mass – weight relationship)

17. Don’t confuse mass and weight! Keep in mind that the Newton is a unit of force, not mass. A 2.49 x 10 4 N Rolls-Royce Phantom traveling in +x direction makes an emergency stop; the x component of the net force acting on its -1.83 x 10 4 N. What is its acceleration?

18. 5-2 Some Particular Forces Measuring weight: o Use a balance to compare a body to known masses, find its mass, and compute its weight o Use a spring scale that measures weight on a calibrated scale o Weight is not the same as mass: a pan balance will read the same for different values of g, a scale will read differently for different values of g

19. 5-2 Some Particular Forces Weight must be measured when the body is not accelerating vertically o E.g., in your bathroom, or on a train o But not in an elevator o Or on DROP ZONE o Or in orbit….

20. 5-2 Some Particular Forces

21. Example Normal force for a block resting on a horizontal surface that is: o Accelerating vertically at a y : o Vertically at rest:

22. 5-2 Some Particular Forces Example Normal force for a block resting on a horizontal surface that is: o Accelerating vertically at a y : o Vertically at rest: Answer: (a) equal to mg (no acceleration) (b) greater than mg (with positive acceleration)

23. 5-2 Some Particular Forces Frictional force or friction: o Occurs when one object slides or attempts to slide over another o Directed along the surface, opposite to the direction of intended motion

24. 5-2 Some Particular Forces Tension force: o A cord (or rope, etc.) is attached to a body and pulled taut o Cord pulls on the body with force T directed along the cord, AWAY from the body! o The cord is said to be under tension o The tension in the cord is T and is assumed to be the same EVERYWHERE along the cord!

25. What are the magnitudes of common forces?

26. Forces! A force: o Is a “push or pull” acting on an object o A net force causes acceleration o Newton’s First Law of Motion: If there is no NET force on an object it will remain in the same state of motion

27. Newton’s First Law Simply stated: “An object at rest tends to stay at rest, an object in motion tends to stay in uniform motion.”

28. Newton’s First Law More properly, “A body acted on by zero net force moves with constant velocity and zero acceleration.”

29. Newton’s First Law In part (a) a net force acts, causing acceleration. In part (b) the net force is zero, resulting in no acceleration.

30. 5-1 Newton's First and Second Laws Newton's first law is true in a non-accelerating frame Inertial frames:

31. 5-1 Newton's First Newton's first law is not true in all frames You go around a corner in a car, and seem to slide SIDEWAYS!? Air moving south from the north pole towards the equator seems to move to the WEST!!!?? You are on a bus moving away from a stop sign, and your physics books slides backwards with no one touching it!!!!???

32. 5-1 Newton's First and Second Laws Newton's first law is not true in accelerating frames (a): a frictionless puck, pushed from the north pole, viewed from space (b): the same situation, viewed from the ground Over long distances, the ground is a non-inertial frame In (b), a fictitious force would be needed to explain deflection

33. When is Newton’s first law valid? You are on roller skates in a stopped BART car… The car starts to accelerate forwards. What happens to you? If no net force acts on you, you maintain a constant velocity (0!)

34. When is Newton’s first law valid? You are on roller skates in a moving BART car… The car starts to slow - accelerate backwards. What happens to you? If no net force acts on you, you maintain a constant velocity

35. When is Newton’s first law valid? If no net force acts on you, you maintains a constant velocity (a vector!) But as seen in the non-inertial frame of the accelerating vehicle, it appears that you are being pushed to the outside! Newton’s first law is valid only in non-accelerating inertial frames.

36. When is Newton’s first law valid?

39. If NO net force acts on rider, rider maintains a constant velocity. But as seen in non-inertial frame of the accelerating vehicle, it appears that rider is being pushed. Newton’s first law is valid only in non-accelerating inertial frames.

40. Newton’s First Law “An object at rest tends to stay at rest, an object in motion tends to stay in uniform motion.”

41. Newton’s First Law Mathematically, “A body acted on by zero net force moves with constant velocity and zero acceleration.”

42. Newton’s First Law II In part (a) net force acts, causing acceleration. In part (b) net force = 0 resulting in no acceleration.

43. Drawing force vectors Use a vector arrow to indicate the magnitude and direction of the force.

44. Superposition of forces Several forces acting at a point on an object have the same effect as their vector sum acting at the same point.

45. Decomposing a force into its component vectors Choose coordinate system with perpendicular x and y axes. F x and F y are components of force along axes. Use trigonometry to find force components.

46. 5-1 Newton's First and Second Laws Principle of superposition for forces: o A net force has the same impact as a single force with identical magnitude and direction o Restate 1 st law even more correctly:

47. Notation for the vector sum Vector sum of all forces on an object is resultant of forces The net force.

48. Superposition of forces Force vectors are added using components: R x = F 1x + F 2x + F 3x + … R y = F 1y + F 2y + F 3y + … F1 F2 F3

49. Superposition of forces Force vectors are added using components: R x = F 1x + F 2x + F 3x + … R y = F 1y + F 2y + F 3y + … F1 F2 F3 F3 x F1 x Rx

50. Superposition of forces Force vectors are added using components: R x = F 1x + F 2x + F 3x + … R y = F 1y + F 2y + F 3y + … F1 F2 y F3 F1 y F3 y Ry

51. Superposition of forces

52. 5-1 Newton's First and Second Laws Superposition Examples

53. 5-1 Newton's First and Second Laws Superposition Examples Answer: (c), (d), (e)

54. Decomposing a force into its component vectors Choose perpendicular x and y axes. F x and F y are the components of a force along these axes. Use trigonometry to find these force components.

55. Notation for the vector sum—Figure 4.7 The vector sum of all the forces on an object is called the resultant of the forces or the net forces.

56. Superposition of forces Force vectors are most easily added using components: R x = F 1x + F 2x + F 3x + …, R y = F 1y + F 2y + F 3y + …. See Example 4.1 (which has three forces).

57. Newton’s Second Law If the net force on an object is not zero, it causes the object to accelerate.

58. An object undergoing uniform circular motion An object in uniform circular motion is accelerated toward center of circle. THEREFORE Net force on object must point toward the center of the circle.

59. Force and acceleration The acceleration of an object is directly proportional to the net force  on the object.

60. Mass and acceleration The acceleration of an object is inversely proportional to the object’s mass if the net force remains fixed.

61. Newton’s second law of motion The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to the mass of the object. The SI unit for force is the newton (N). 1 N = 1 kg·m/s 2

62. Using Newton’s Second Law—Example

63. 5-1 Newton's First and Second Laws What is mass? o “the mass of a body is the characteristic that relates a force on the body to the resulting acceleration” o Mass is a measure of a body’s resistance to change in motion o Literally, # of protons, neutrons, & electrons (regardless of how they are grouped into kinds of elements!) o It is not the same as weight, density, size etc. o Same force on LARGER mass produces smaller acceleration!

64. 5-1 Newton's First and Second Laws Example Apply identical 8.0 N force to various bodies: o Mass: 1kg → acceleration: 8 m/s 2 o Mass: 2kg → acceleration: 4 m/s 2 o Mass: 0.5kg → acceleration: 16 m/s 2 And if you saw an acceleration from the application of this force… o Acceleration: 2 m/s 2 Then you can deduce… o Mass = 4 kg

65. 5-1 Newton's First and Second Laws Summarize these behaviors as: As an equation, we write: Identify the body in question, and only include forces that act on that body!

66. 5-1 Newton's First and Second Laws As an equation, we write: Separate applied forces into components along coordinate system axes

67. 5-1 Newton's First and Second Laws If the net force on a body is zero: o Its acceleration is zero o The forces and the body are in equilibrium o But there may still be forces! They just must cancel!

68. Newton’s Second Law If the net force on an object is zero, the object will not accelerate.

69. Newton’s Second Law If the net force on an object is not zero, it causes the object to accelerate.

70. Newton’s Second Law If the net force on an object is not zero, it causes the object to accelerate.

71. An object undergoing uniform circular motion An object in uniform circular motion is accelerated toward the center of the circle. So net force on object must point toward the center of the circle.

72. Force and acceleration Magnitude of acceleration of an object is directly proportional to the net force  on the object. | | = F/m

73. Mass and acceleration Magnitude of acceleration of an object is inversely proportional to object’s mass if net force remains fixed. | | = F/m

74. Newton’s second law of motion The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to the mass of the object. The SI unit for force is the newton (N). 1 N = 1 kg·m/s 2

75. Using Newton’s Second Law Worker pushes box of mass 40 kg, with constant force of 20N. What is acceleration?

76. Using Newton’s Second Law

77. Using Newton’s Second Law II—Example Shove bottle of mass 0.45 kg at initial speed of 2.8 m/s a distance of 1 m before it stops. What is the magnitude and direction of force on bottle?

78. Using Newton’s Second Law II—Example Shove bottle of mass 0.45 kg at initial speed of 2.8 m/s a distance of 1 m before it stops. What is the magnitude and direction of force on bottle?

79. Using Newton’s Second Law II—Example

80. We know: Displacement in x = +1.0 m Initial x velocity = +2.8 m/s Final x velocity = 0 m/s THREE THINGS!

81. Using Newton’s Second Law II—Example v f 2 = v i 2 + 2a  x So… a = (v f 2 - v i 2 )/2  x a = - 3.9 m/s 2 NOTE a points in the direction of f! +x

82. Systems of units SI: force in Newtons, distance in meters, mass in kg. British: force in pounds, distance in feet, & mass in slugs. CGS: force in dynes, distance in centimeters, & mass in grams

83. Mass and weight The weight of an object (on the earth) is the gravitational force that the earth exerts on it. The weight W of an object of mass m is W = mg The value of g depends on altitude. On other planets, g will have an entirely different value than on the earth.

84. A euro in free fall

85. Don’t confuse mass and weight! Keep in mind that the newton is a unit of force, not mass.

86. Newton’s Third Law If you exert a force on a body, the body always exerts a force (the “reaction”) back upon you. A force and its reaction force have the same magnitude but opposite directions. These forces act on different bodies.

87. Objects interact when they push or pull on each other: We can write this law as a scalar or vector relation: We call these two forces a third-law force pair Any time any two objects interact, there is a third-law force pair

88. We call these two forces a third-law force pair Any time any two objects interact, there is a third-law force pair

89. Applying Newton’s Third Law: Objects at rest An apple rests on a table. Identify the forces that act on it and the action-reaction pairs.

90. Applying Newton’s Third Law: Objects at rest An apple rests on a table. Identify the forces that act on it and the action-reaction pairs.

91. Applying Newton’s Third Law: Objects at rest An apple rests on a table. Identify the forces that act on it and the action-reaction pairs.

92. Applying Newton’s Third Law: Objects at rest An apple rests on a table. Identify the forces that act on it and the action-reaction pairs.

93. Applying Newton’s Third Law: Objects in motion A person pulls on a block across the floor. Identify the action-reaction pairs.

94. Applying Newton’s Third Law: Objects in motion A person pulls on a block across the floor. Identify the action-reaction pairs.

95. Applying Newton’s Third Law: Objects in motion A person pulls on a block across the floor. Identify the action-reaction pairs.

96. Applying Newton’s Third Law: Objects in motion A person pulls on a block across the floor. Identify the action-reaction pairs.

97. A paradox? If an object pulls back on you just as hard as you pull on it, how can it ever accelerate?

98. 5-3 Applying Newton's Laws

99. Third-law force pairs:

100. 5-3 Applying Newton's Laws

101. Answer: (a) they increase (b) yes (c) they begin to decrease slowly (the gravitational force of Earth decreases with height—negligible in an actual elevator) (d) yes

102. A paradox? If an object pulls back on you just as hard as you pull on it, how can it ever accelerate?

103. Free-body diagrams A free-body diagram is a sketch showing all the forces acting on an object.

104. Free Body Diagrams To solve problems with forces, we often draw a free body diagram The only body shown is the one we are solving for Forces are drawn as vector arrows with their tails on the body Coordinate system shown Acceleration is NEVER part of a free body diagram – only forces on a body are present.

105. Free Body Diagrams

109. Answer: F 3 = 2 N to the left in both cases

110. 5-2 Some Particular Forces

112. Real world – Tension will distort a cord/cable/string! They stretch! Ideal world - A massless and unstretchable cord exists only as a connection between two bodies o It pulls on both with the same force, T o True even if the bodies and cord are accelerating, and even if the cord runs around a massless, frictionless pulley o These are useful simplifying assumptions

113. 5-2 Some Particular Forces

114. Answer: (a) equal to 75 N (b) greater than 75 N (c) less than 75 N

115. 5-3 Applying Newton's Laws Sample Problem A block of mass M = 3.3 kg, connected by a cord and pulley to a hanging block of mass m = 2.1 kg, slides across a frictionless surface

116. Draw the forces involved Treat the string as unstretchable, the pulley as massless and frictionless, and each block as a particle 5-3 Applying Newton's Laws Draw a free-body diagram for each mass Apply Newton's 2nd law (F = ma) to each block → 2 simultaneous eqs. Eliminate unknowns (T) that are the same, and solve for the acceleration

117. For the sliding block: For the hanging block: Note the direction of “a” for the hanging block!!! Combining we get: 5-3 Applying Newton's Laws

118. For the sliding block: For the hanging block: Combining we get: Plugging in we find a = 3.8 m/s 2 and T = 13 N Does this make sense? Check dimensions are correct, check that a < g, check that T < mg (otherwise acceleration would be upward), check limiting cases (e.g., g = 0, M = 0, m = ∞) 5-3 Applying Newton's Laws