Chapter 4. Force and Motion

1. Chapter 4. Force and Motion Chapter Goal: To establish a connection between force and motion.

2. Student Learning Objectives • To recognize what does and does not constitute a force. • To identify the specific forces acting on an object. • To draw an accurate free-body diagram of an object. • To begin the process of understanding the connection between force and motion.

3. A force is an interaction between two objects • A force is a push or a pull on an object. • If I push a book across a table, the book pushes me back (inanimate objects can exert force!) • A force is a vector. It has both a magnitude and a direction. • The force (interaction) has the same magnitude for both me and the book. However the direction of the force on me is opposite to the direction of the force on the book.

4. A force is an interaction between two objects • A force on an object requires an agent. The agent is another object • This is another way of confirming that a force is an interaction between objects. It takes two to tango! • A force is either a contact force or a long- range force. • Gravity is the only long-range force we will study this semester. It is an interaction between two objects, but they are not necessarily in contact. • All other forces (this semester) only exist when the two objects are in contact.

5. Drawing force vectors

6. Combining Forces

7. What is the net force on the object? Your answer should include both magnitude and direction (in the form of an angle from the x-axis) 3 N 4 N

8. What is the net force on the object? Individual Forces Net Force 5 N 3 N 4 N

9. A Short Catalog of Forces - Gravity • Gravity is a long-range attractive force between two objects. • In this class, our emphasis is on the interaction between the Earth and objects on or near its surface (the weight force). • Your text uses W for the weight force.

10. A Short Catalog of Forces: Tension Ropes, strings, cables and hooks exert a tensional force. These agents always pull, they never push. Although it seems strange, the sled exerts a tension force on the rope.

11. A Short Catalog of Forces - Normal The normal force is a contact force between 2 surfaces that is perpendicular to the surfaces

12. A Short Catalog of Forces - friction The frictional force is a contact force between 2 surfaces that is parallel to the surfaces

13. A Short Catalog of Forces Other forces are usually shown as an with an identifying subscript.

14. You’ve just kicked a rock, and it is now sliding across the ground about 2 meters in front of you. Which of these forces act on the rock? • Gravity, acting downward • The normal force, acting upward • The force of the kick, acting in the  direction of motion • Friction, acting opposite the direction of  motion • All of the above

15. You’ve just kicked a rock, and it is now sliding across the ground about 2 meters in front of you. Which of these forces act on the ball? You may choose more than one answer. • Gravity, acting downward • The normal force, acting upward • The force of the kick, acting in the  direction of motion • Friction, acting opposite the direction of  motion • All of the above • Ans: A, B, and D

16. Free-body diagrams • Identify all forces acting on the object. • Draw a coordinate system. If motion is along an incline, the coordinates system should be tilted relative to true horizontal and true vertical. • Represent the object as a dot at the origin. • Draw vectors representing each of the identified forces and label with the appropriate symbol.

17. Extension 1 – Freebody diagrams

18. Mass An object’s mass (m) is a measure of the amount of matter that it contains. The SI unit of mass is the kilogram (kg). The English unit of mass is the slug, which is virtually never used. The mass of an object does not change unless you add or subtract matter from it. The mass of an object is the same on any planet and in deep space. Scales do not measure mass, except for a balance scale, which compares one mass to another.

19. Weight • The weight of an object on or above the earth is the gravitational force that the earth exerts on the object (FG or W). • Since weight is a force, the SI unit of weight is the Newton (N); the English unit is the pound. • The weight of an object changes on other planets since the gravitational force of other planets are different than that of the earth. • Scales do not measure weight, at least not directly.

20. Apparent weight or “what the scale says” • Scales measure a normal force or a tension force (i.e.hanging scales). • In most situations, the force measured by the scale is numerically equal to the weight force. • In accelerating reference frames (e.g. the ever-popular elevator with a scale), the scale reading will differ from the true weight. This is called apparent weight. • Metric scales assume measurement on earth and are calibrated to read in mass units (kg). American scales read in English force units. It’s a culture thing.

21. 4.7 The Gravitational Force Newton’s Law of Universal Gravitation Every particle in the universe exerts an attractive force on every other particle. A particle is a piece of matter, small enough in size to be regarded as a mathematical point. The force that each exerts on the other is directed along the line joining the particles.

22. 4.7 The Gravitational Force For two particles that have masses m1 and m2and are separated by a distance r, the force has a magnitude given by:

23. 4.7 The Gravitational Force If m1 has a mass of 12 kg and m2 has a mass of 25 kg and the two objects are 1.2 meters apart, what is the gravitational force between them?

24. 4.7 The Gravitational Force The gravitational force between 2 objects is usually pretty small, unless one of the object is a planet!

25. If m1 has a mass of 12 kg and m2 has a mass of 25 kg and the two objects are twice as far apart as before (i.e. 2.4 m), what is the gravitational force between them? Twice as much? Half as much? Or something else

26. The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. Compared to , the magnitude of the force is • one quarter as big. • half as big. • the same size. • twice as big. • four times as big.

27. The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. Compared to , the magnitude of the force is • one quarter as big. • half as big. • the same size. • twice as big. • four times as big.

28. 4.7 The Gravitational Force Definition of Weight The weight of an object on or above the earth is the gravitational force that the earth exerts on the object. The weight always acts downwards, toward the center of the earth. On or above another astronomical body, the weight is the gravitational force exerted on the object by that body. SI Unit of Weight: newton (N)

29. 4.7 The Gravitational Force and W = mg Calculate g ME = 5.98 x 1024 kg RE = 6.38 x 106 m (don’t forget to square!)

30. 4.7 The Gravitational Force Relation Between Mass and Weight g = 9.80 m/s2 on Earth

31. 4.7 The Gravitational Force Relation Between Mass and Weight

32. 4.3 Newton’s Second Law of Motion Newton’s Second Law An object of mass m, subject to forces F1 , F2 , etc. will undergo an acceleration with a magnitude directly proportional to the net force and inversely proportional to the mass: The direction of the acceleration is the same as the direction of the net force.

33. 4.3 Newton’s Second Law of Motion Newton’s First Law This is a special case of the 2nd Law. If the net force equals 0, there is no acceleration and the velocity of the object will not change. If it is at rest, it will stay at rest. • In this case the object is in equilibrium. • Static equilibrium – object is at rest for a finite period • Dynamic equilibrium –object is moving at constant velocity.

34. Problem-Solving Strategy for Newton’s Law Problems

35. Problem-Solving Strategy: Equilibrium Problems If the net force is equal to zero, must the object be at rest? The next example shows the answer to this:

36. Newton’s 1st Law: Towing a car up a hill A car with a weight (not mass) of 15,000 N is being towed up a 20° frictionless slope at constant velocity. The tow rope is parallel to the slope surface. What is the tension on the tow rope? Is this a Newton’s 2nd Law problem (ΣF = ma) or a Newton’s 1st Law problem (ΣF = 0)? Draw a free body diagram for this problem.

37. Newton’s 1st Law: Towing a car up a hill A car with a weight (not mass) of 15,000 N is being towed up a 20° frictionless slope at constant velocity. The tow rope is parallel to the slope surface. What is the tension on the tow rope? What part of the story indicates that ΣF = 0?

38. Newton’s 1st Law: Towing a car up a hill Once the pictorial representation is completed, use Newton’s 1st Law in component form: T and n have only one non-zero component, but FGhas non-zero components in both x and y directions. How do we write the components of FG in terms of θ ?

39. Newton’s 1st Law: Towing a car up a hill Solving the 2nd equation tells us that n = FG cos θ, which is not necessary information for this problem.

40. Problem-Solving Strategy for Newton’s 2nd Law Problems Use the problem-solving strategy outlined for Newton’s 1st Law problems to draw the free body diagram and determine known quantities. Use Newton’s Law in component form to find the values for any individual forces and/or the acceleration. If necessary, the object’s trajectory (time, velocity, position, acceleration) can be determined by using the equations of kinematics. Reverse # 2 and 3 if necessary.

41. Newton’s 2nd Law – Speed of a Towed Car • Find the acceleration using Newton’s second law in component form. • Find the velocity using one of the kinematic equations. • Note that in this problem, the car’s mass, not the weight, is given.

42. Speed of a towed car – pictorial representation of motion, motion diagram and free body diagram

43. Speed of a towed car

44. EXAMPLE 6.3 Speed of a towed car Our pictorial analysis (and common sense) indicate that there is no acceleration along the y-axis.

45. EXAMPLE 6.3 Speed of a towed car

46. Apparent Weight – What the scale says (even if there is no scale!) • The value the scale reads when the object is accelerating or being acted upon by some other force (e.g the buoyant force in water) • Not equal to the true weight (FG = mg) • Apparent weight is either a normal force (step on scale) or tension force (hanging scale) • An object inside an accelerating elevator has the same acceleration as the elevator. • Acceleration is not a force so don’t include directly on free body diagram.

47. Typical FBD for an apparent weight problem n FG = mg, but n does not!

48. Apparent Weight – What the scale would say, even if there is no scale It takes the elevator in a skyscraper 4.0 s to reach a constant speed of 10.0 m/s. A 60.0 kg passenger gets on at the ground floor. What is her apparent weight during those 4 seconds? Use the problem-solving strategy (repeated in next slide).

49. Problem-Solving Strategy for Newton’s Law Problems

50. Problem-Solving Strategy for Newton’s 2nd Law Problems Use Newton’s Law in component form to find the values for any individual forces and/or the acceleration. If necessary, the object’s trajectory (time, velocity, position, acceleration) can be determined by using the equations of kinematics. Reverse # 1 and 2 if necessary (Hint: it is!)