VECTORS AND SCALARS

1. VECTORS AND SCALARS  A scalar quantity has only magnitude and is completely specified by a number and a unit. Examples are: mass (2 kg), volume (1.5 L), and frequency (60 Hz). Scalar quantities of the same kind are added by using ordinary arithmetic.

2. A vector quantity has both magnitude and direction. Examples are displacement (an airplane has flown 200 km to the southwest), velocity (a car is moving at 60 km/h to the north), and force (a person applies an upward force of 25 N to a package). When vector quantities are added, their directions must be taken intoaccount.

3. A vectoris represented by an arrowed line whose length is proportional to the vector quantity and whose direction indicates the direction of the vector quantity.

6. The resultant, or sum, of a number of vectors of a particular type (force vectors, for example) is that single vector that would have the same effect as all the original vectors taken together. R

10. VECTOR COMPONENTS A vector in two dimensions may be resolved into two component vectors acting along any two mutually perpendicular directions.

13. 2.1 Draw and calculate the components of the vector F (250 N, 235o) Fx = F cos  = 250 cos (235o) = - 143.4 N Fy = F sin  = 250 sin (235o) = - 204.7 N Fx Fy F

14. VECTOR ADDITION: COMPONENT METHOD To add two or more vectors A, B, C,… by the component method, follow this procedure: 1. Resolve the initial vectors into components x and y. 2. Add the components in the x direction to give Σxand add the components in the y direction to give Σy. That is, the magnitudes of Σxand Σy are given by, respectively: Σx = Ax + Bx + Cx… Σy = Ay + By + Cy…

15.  3. Calculate the magnitudeanddirectionof the resultant R from its components by using the Pythagorean theorem: and

16. 2.2 Three ropes are tied to a stake and the following forces are exerted. Find the resultant force. A (20 N, 0º) B (30 N, 150º) C (40 N, 232º)

17. A (20 N, 0) B (30 N, 150) C (40 N, 232) x-component 20 cos 0 30 cos 150 40 cos 232 Σx = - 30.6 N y-component 20 sin 0 30 sin 150 40 sin 232 Σy = -16.5 N = 34.7 N

19. = 28.3 Since Σx = (-) and Σy = (-) R is in the IIIQuadrant: therefore: 180 + 28.3 = 208.3 R (34.7 N, 208.3)

20. 2.3 Four coplanar forces act on a body at point O as shown in the figure. Find their resultant with the component method. A (80 N, 0) B (100 N, 45) C (110 N, 150) D (160 N, 200)

21. A (80 N, 0)B (100 N, 45) C (110 N, 150)D (160 N, 200) x-component y-component 80 0 100 cos 45100 sin 45 110 cos 150110 sin 150 160 cos 200160 sin 200 Σx = - 95 N Σy = 71 N = 118.6 N

22. = 36.7 Since Σx = (-) and Σy = (+) R is in the IIQuadrant, therefore: 180 - 36.7= 143.3 R (118.6 N, 143.3)

23. = 6.5 N = 29 Since Σx = (+) and Σy = (-) R is in the IV Quadrant, therefore: 360 - 29= 331 R (6.5 N, 331)

24. AP PHYSICS LAB 2.VECTOR ADDITION Objective: The purpose of this experiment is to use the force table to experimentally determine the equilibrant forceof two and three other forces. This result is checked by the component method. A system of forces is in equilibrium when a force called the equilibrant forceis equal and opposite to theirresultant force.

25. Equipment Force Table Set

26. FORCE MASS (kg) FORCE (N) mg = m (9.8 m/s2) DIRECTION F1 F2 Equilibrant FE Resultant FR DATA Table

27. An object that experiences a push or a pull has a force exerted on it. Notice that it is the object that is considered. The object is called the system. The world around the object that exerts forces on it is called the environment. system

28. FORCE Forces can act either through the physical contact of two objects (contact forces: push or pull) or at a distance (field forces: magnetic force, gravitational force).

29. Type of Force and its Symbol Description of Force Direction of Force Applied Force An applied force is a force that is applied to an object by another object or by a person. If a person is pushing a desk across the room, then there is an applied force acting upon the desk. The applied force is the force exerted on the desk by the person. In the direction of the pull or push. FA

30. Type of Force and its Symbol Description of Force Direction of Force Normal Force The normal force is the support force exerted upon an object that is in contact with another stable object. For example, if a book is resting upon a surface, then the surface is exerting an upward force upon the book in order to support the weight of the book. The normal force is always perpendicular to the surface Perpendicular to the surface FN

31. Type of Force and its Symbol Description of Force Direction of Force Friction Force The friction force is the force exerted by a surface as an object moves across it or makes an effort to move across it. The friction force opposes the motion of the object. For example, if a book moves across the surface of a desk, the desk exerts a friction force in the direction opposite to the motion of the book. Opposite to the motion of the object FF

32. Type of Force and its Symbol Description of Force Direction of Force Air Resistance Force Air resistance is a special type of frictional force that acts upon objects as they travel through the air. Like all frictional forces, the force of air resistance always opposes the motion of the object. This force will frequently be ignored due to its negligible magnitude. It is most noticeable for objects that travel at high speeds (e.g., a skydiver or a downhill skier) or for objects with large surface areas. Opposite to the motion of the object FD

33. Type of Force and its Symbol Description of Force Direction of Force Tensional Force Tension is the force that is transmitted through a string, rope, or wire when it is pulled tight by forces acting at each end. The tensional force is directed along the wire and pulls equally on the objects on either end of the wire. In the direction of the pull FT

34. Type of Force and its Symbol Description of Force Direction of Force Gravitational Force (also known as Weight) The force of gravity is the force with which the earth, moon, or other massive body attracts an object towards itself. By definition, this is the weight of the object. All objects upon earth experience a force of gravity that is directed "downward" towards the center of the earth. The force of gravity on an object on earth is always equal to the weight of the object. Straight downward Fg

35. FORCES HAVE AGENTS Each force has a specific identifiable, immediate cause called agent. You should be able to name the agent of each force, for example the force of the desk or your hand on your book. The agent can be animate such as a person, or inanimate such as a desk, floor or a magnet. The agent for the force of gravity is Earth's mass. If you can't name an agent, the force doesn't exist. agent

36. A free-body-diagram (FBD) is a vector diagram that shows all the forces that act on an object whose motion is being studied.  Directions: - Choose a coordinate system defining the positive direction of motion. - Replace the object by a dot and locate it in the center of the coordinate system. - Draw arrows to represent the forces acting on the system.

37. FN FG

38. FN FG

39. FN FF FG

40. FN FGY FGX FG

41. FN FF FG

42. FT FG

43. FT1 FT2 FG

44. FT1 FT2 FG

45. FN FT FG

46. FN1 FN2 FG

47. FN FF FA FG

48. FN FA θ FF FG

49. FT FG

50. FN FF θ FA FG