1. 1 Lecture 1  Standards and units  Dimensional analysis  Scalars  Vectors  Operations with vectors and scalars

2. 2 System of units In order to communicate the result of a measurement, one must give units. The units given to mass, length, and time, etc., form the basis of different systems of units; other units are derived from them Two common systems of units you will encounter are International System SI (aka metric) system and the British System.

3. 3 SI system There are seven (7) fundamental units in the SI system: Ampere (A) - electric current Kelvin (K) - temperature mole (mol) - amount of substance candela (cd) - intensity of light meter(m) - distance or length (d, x, l) (dimension L) second (s) - time (t) (dimension T) kilogram (kg) - mass (m) (dimension M) We will use the SI system in this class. You will have to know the conversion factors from the British System. Ex.: 1 inch = 2.54 cm; 1 pound ~ 0.454 kg; 1 mile ~ 1.61 km

4. 4 Other dimensions The dimensions for all other physical quantities are derived from the fundamental ones: Ex.:- Volume – has the dimension L x L x L – so the unit is m 3 - Area – has the dimension L x L – unit = m 2 - Velocity – has the dimension L/T – unit = m/s - Density – has the dimension M/Volume – unit = kg/m 3

5. 5 Common multipliers For time we also use: 1 minute = 60 s 1 hour = 60 minutes = 3600 s hecto- = 10 2 centi - = 10 -2 kilo- = 10 3 milli- = 10 -3 mega- = 10 6 micro- = 10 -6 giga- = 10 9 nano- = 10 -9 tera- = 10 12 pico- = 10 -12 peta- = 10 15 femto- = 10 -15

6. 6 Dimensional analysis Whenever you solve quantitatively a physics problems make sure you check that the equations yield the correct dimensions for the quantity. Good way to catch errors gross errors, but will not tell you if the quantitative result is correct. Ex.: - In a one-dimensional motion with constant acceleration a (dimension = L/T 2 ), the distance traveled during a time interval t (dimension T) is given by the equation: x = v 0 *t + ½ a*t 2 (v 0 is the velocity at the start of the time interval)

7. 7 Dimensional analysis (continued) x = v 0 *t + ½ a*t 2 - Since x is a distance and has the dimension L each term on the right side of the equation must have the dimension L. Check: v 0 *t has dimension (L/T)*T = L - correct ½ a*t 2 has dimension (L/T 2 )*T 2 = L - correct (the factor of ½ is a dimensionless constant)

8. 8 To convert a velocity from units of km/h to units of m/s, you must: (A) multiply by 1000 and divide by 60 (B) multiply by 1000 and divide by 3600 (C) multiply by 60 and divide by 600 (D) multiply by 3600 and divide by 1000 (E) none of these is correct

9. 9 If x and t represent position and time, respectively, then the constant A in the equation x = A*cos(B*t) must (A) have the dimensions L/T (B) have the dimensions 1/T (C) have the dimensions L (D) have the dimensions L 2 /T 2 (E) be dimensionless

10. 10 The dimensions of two quantities MUST be identical if you are either ___________ or ____________ the quantities. (A) adding; multiplying (B) subtracting; dividing (C) multiplying; dividing (D) adding; subtracting (E) all of these are correct Hint: - do not mix oranges and apples

11. 11 Scalars Quantities described by a single number (magnitude or absolute value) + unit Ex.: temperature (T) time (t)

12. 12 Vectors Quantities described not only by magnitude but also by direction. Ex.: velocity ( ) displacement ( ) The arrows indicate that we deal with a vector

13. 13 Vector properties magnitude Negative of a vector (reciprocal) Negative of a vector (reciprocal) but Multiplication with a scalar – m has the same direction as Multiplication with a scalar – m has the same direction as and magnitude = and magnitude =

14. 14 Geometrical addition of vectors Mathematically the sum of two vectors Mathematically the sum of two vectors Geometrically using the graphical representation: Geometrically using the graphical representation: - triangle method (head-to tail) - parallelogram method Note: it is OK to translate a vector parallel to itself

15. 15 Properties of addition Vector addition is commutative: Vector addition is commutative: Vector addition is associative: Vector addition is associative:

16. 16 Vector subtraction Vector subtraction – subtracting from is equivalent to adding to. Vector subtraction – subtracting from is equivalent to adding to.

17. 17 If you move east in a straight line 1 km and then north the same distance, how far will you find yourself from the starting point (in a straight line to the origin)? (A) 1 km (B) 3.22 km (C) 1.50 km (D) 2 km (E) 1.41 km

18. 18 VERY IMPORTANT: except if

19. 19 Vector decomposition Less cumbersome technique for vector addition than the geometrical method. Less cumbersome technique for vector addition than the geometrical method. Step 1 Choose a system of rectangular coordinates (cartesian coordinates) Step 2 Resolve the vector by projecting it on the x, y (2-dimensional case) axes, by drawing Perpendicular lines from the two ends of the vector to the axes.

20. 20 Vector decomposition (continued) Step 3 Geometrically the components of the vector will then be: If we know the components of a vector and want to find its magnitude and direction then:

21. 21 x y θ The magnitude of the vector is 3 m and the angle  = 30 o. Which statement is correct: (A) points in the negative x direction; A y = 1.5 m (B) points in the positive y direction; A x = 1.5 m (C) points in the positive x direction; A y = 1.5 m (D) points in the positive x direction; A x = 1.5 m (E) none of these is correct

22. 22 Unit Vectors We have discussed only the algebraic components of the projections a x and a y. We have discussed only the algebraic components of the projections a x and a y. However and are vectors, but with well defined orientations However and are vectors, but with well defined orientations along the directions chosen for the system of coordinates. along the directions chosen for the system of coordinates. Define a pair of unit vectors parallel with the x and y axis and oriented in their positive direction. Define a pair of unit vectors parallel with the x and y axis and oriented in their positive direction. - dimensionless

23. 23 Vector decomposition - Summary Choose a set of orthogonal coordinates and project the vector in its components. Introduce a set of dimensionless unit vectors oriented in the positive direction of the axes.

24. 24 Examples of vector decomposition

25. 25 Examples of vector decomposition Translate

26. 26 The magnitude and direction of a vector are given by: (A) (B) (C) (D) (E) none of the above

27. 27 Which diagram best describes vector in a cartesian system of coordinates (x, y)? x y xxx x y y y y (A)(B) (C)(D)(E) 2-chances

28. 28  = 30 o  = 150 o  = 210 o  =330 o Ex.:

29. 29 Important – remember this convention  Always measure the angles from the positive direction of the x axis in the counter –clockwise direction.  If you measure the angle clockwise you will have to add a negative sign in order not to lose the information regarding the direction of your vectors in the analysis.

30. 30 Trigonometric functions  x y Choose a vector of magnitude one : Then: i.e. the x and y components of the unitary vector give the values of the sin and cos functions. sin  cos  tan   

31. 31 Radians and degrees  x y            x y   To convert from degrees to radians (rad) multiply with 2  and divide by 360. Ex: - convert 60 o to radians

32. 32 Vector addition using components Step 1 Decompose the vectors on a set of orthogonal axes. - Problem - find Step 2 Add algebraically the components on each axis to obtain the components of the sum vector. Step 3 Construct the sum vector using its components. x y x y x y

33. 33 Example: Calculate the sum of the following two: and First express with components: Then add the components on each axis:

34. 34 Vectors and have the following components: x-component +5 units -6 units y-component -2 units +2 units What are the components of vector (A)C x = +7 units, C y = +8 units (B)C x = +3 units, C y = +4 units (C)C x = +3 units, C y = -4 units (D)C x = -1 units, C y = 0 units 2-chances

35. 35 Vectors and have the following components: x-component +5 units -6 units y-component -2 units +2 units What is the magnitude of vector (A) = -1 units (B) = +1 units (C) = -2.65 units (D) = +5 units

36. 36 Two vector quantities, whose directions can be altered at will, can have a resultant whose length is between the limits 5 and 15. What could the magnitudes of these two vector quantities be? (A) 2 and 3 (B) 5 and 10 (C) 10 and 25 (D) 3 and 12 2-chances

37. 37 Problem 23 (page 54) Oasis B is 25 km due east of oasis A. Starting from oasis A, a camel walks 24 km in a direction 15 o south of east and then walks 8 km due north. How far is the camel then from oasis B? A  B x y The displacement left to travel is: C D

38. 38 Sample Problem 3-6 (page 47) Three vectors satisfy the relation. has a magnitude of 22.0 units and is directed at an angle of -47 o (clockwise) from the positive direction on an x axis. has a magnitude of 17 units and is directed counterclockwise from the positive direction of the x axis by an angle . is in the positive direction of the x axis. What is the magnitude of ? x y  But:

39. 39 Sample Problem 3-6 (page 47) (continued) x y

40. 40 Coordinate system equivalence The choice of a coordinate system is not unique. The system that we have been using so far is convenient because it looks “proper” (its axes are parallel with the paper or blackboard edges) However the coordinate systems are equivalent since the magnitude and orientation of a vector is not affected by the system in which it is analyzed.

41. 41 Operations of multiplication with vectors Multiplication of a vector with a scalar Multiplication of a vector by a vector - the scalar product - the vector product - the direction of is the same with that of if m > 0 and is opposite if m<0

42. 42 Scalar product The scalar product (or “dot” product) of two vectors is defined as:  If we work with components We used: Can be generalized to 3 – dimensions (see also the textbook)

43. 43 Scalar product properties The scalar product is commutative Particular cases: The scalar product is distributive

44. 44 Vector product The vector product (or “cross” product) of two vectors is defined as a vector: - with magnitude: The vector is perpendicular on the plane formed by the vectors and and its direction is determined by the right hand rule.

45. 45 Vector product properties The vector product is not commutative. Particular cases:

46. 46 Vector product components - using the unit vector properties:  determinant notation We will review the vector product later. Use this for reference.

47. 47 Motion along a straight line (one – dimensional) Definition: motion is defined as the change of an object’s position with time In this chapter we will impose a series of restrictions: - motion is constrained in one dimension, i.e. along a straight line (typically along the x or y axis). - the motion can be either in the positive or in the negative direction of the axis used. - for now, we will neglect the forces (pushes or pulls) that determine an object to move). - the moving object is either a particle or an object that moves like a particle (all its points move in the same direction at the same rate (speed).

48. 48 Position of an object For a one-dimensional motion the position of the object is specified by a single coordinate x. In order to describe the position of an object: - take a snapshot of the object at different times and record its position. - plot this position as a function of time x = x(t) - you might need to fit the plot to obtain the missing points.

49. 49 Examples of time dependence of position t (s) x (m) t (s) x (m) t (s) x (m)

50. 50 Displacement A change from one position x 1 to another position x 2 : Note: displacement is a vector even if in this chapter we will not specify it all the time. Consequently, make sure that the sign (i.e. direction) is not ignored. but

51. 51 There are four pairs of initial and final positions. Which pair(s) gives a negative displacement: (a) –3m, 5m (b) –3m, -7m (c) 3m, -3.4m (d) –3m, 0m (A) (a) and (b) (B) (b) and (d) (C) (c) and (d) (D) only (b) (E) (b) and (c)

52. 52 For a certain interval of time, is the magnitude of the displacement always equal to the distance traveled? (A) Yes (B) No

53. 53 Average velocity One characteristic of the motion is the rate of change of the object (particle) position. The initial and final position of the two objects are the same. Average velocity v avg is the ratio of the displacement and the time interval over which it has occurred. -Dimension is: [v avg ]= L/T -Unit: m/s

54. 54 Graphic interpretation of the velocity The velocity is the slope of the straight line that connects the two particular points over which the displacement is calculated. x t Slope of this line t1t1 t2t2 x1x1 x2x2

55. 55 According to the following graph, when the two bodies (1) and (2) have the same velocity: (A) t = 0 s (B) t = 5 s (C) t = 10 s (D) Never (E) None of the above x (m) t (s)    

56. 56 Average speed Average speed is the ratio of the total distance traveled (Ex. number of meters moved over the time interval Because the time interval is always positive the average speed is always positive

57. 57 Example: A car goes form city A to city B and then returns to city C (see figure). If it takes 10 minutes to drive from A to B, and 14 minutes from B to C, and the car was stationed in city B for half an hour, what are the average velocity and speed? AB C Note: - typically the value of the average velocity and speed are different

58. 58 Instantaneous velocity To obtain the velocity at any instance, the time interval  t over which the average velocity is calculated is reduced - v is the derivative of x with respect to t. - v is the slope of the tangent to the position versus time curve at the point of interest. Slope of this line = average velocity Speed = the magnitude of instantaneous velocity. It does not contain any indication about the direction of motion. x(t)

59. 59 The time dependence of the position of an object is shown in the figure. At which point is the object at rest (zero velocity)? (A) D and F (B) A and C (C) B and E (D) B, D, E and F (E) A, C, E and F x (m) t (s) A B C D E F

60. 60 Acceleration Acceleration – characterizes the rate of change in the velocity of an object (particle) -Dimension is: [a avg ]=L/T 2 -Unit: m/s 2 v t v1v1 v2v2 t1t1 t2t2 -a avg – is the slope of this line

61. 61 Instantaneous Acceleration Instantaneous acceleration – derivative of velocity with respect to time

62. 62 The acceleration’s sign A negative sign for the acceleration does not necessarily means that the speed of an object is decreasing. Ex.: - an object starts from rest and increases its speed to (– 10 m/s) in 5s. x v - the acceleration is negative even if the objects accelerates, because the motion is in the negative direction of the x axis Note: - if the sign of the velocity and acceleration of an object are the same, the speed of the object increases. If the signs are opposite the speed decreases.

63. 63 Problem: The position of a car versus time is described by the following graph. a) Find the displacement and total distance traveled in 60 s. b) Plot the time dependence of the velocity c) Calculate the average velocity and average speed d) Calculate the average acceleration in the time interval 0 s to 60 s. e) Plot the time dependence of the acceleration (comment on the result)

64. 64 a) b) c)

65. 65 d) e) Note: - the measurement of the time dependence of the position was poor and it does not completely reflect the reality.

66. 66 More real measurement:

67. 67 What is v x at t = 1 s? (A) +3 m/s (B) +2 m/s (C) -2 m/s (D) -3 m/s (E) None of the above

68. 68 What is v x at t = 3 s? (A) +3 m/s (B) +2 m/s (C) -2 m/s (D) -3 m/s (E) None of the above

69. 69 What is the average acceleration between t = 1s and t = 8 s? (A) ~ 0.43 m/s 2 (B) ~ 0.21 m/s 2 (C) ~ - 0.43 m/s 2 (D) ~ 0 m/s 2 (E) None of the above 2 chances

70. 70 Constant acceleration case When the acceleration is constant the average acceleration and instantaneous acceleration are equal: Eq. (1)