Lecture 1: Standards and Units, Dimensional Analysis, Scalars, Vectors

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

2. 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. System of units

3. There are seven (7) fundamental units in the SI system: SI system meter (m) - distance or length (d, x, l) (dimension L) second (s) - time (t) (dimension T) kilogram (kg) - mass (m) (dimension M) Ampere (A) - electric current Kelvin (K) - temperature mole (mol) - amount of substance candela (cd) - intensity of light 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. 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 m3 - Area – has the dimension L x L – unit = m2 - Velocity – has the dimension L/T – unit = m/s - Density – has the dimension M/Volume – unit = kg/m3 Other dimensions

5. For time we also use: 1 minute = 60 s 1 hour = 60 minutes = 3600 s Common multipliers hecto- = 102 centi - = 10-2 kilo- = 103 milli- = 10-3 mega- = 106 micro- = 10-6 giga- = 109 nano- = 10-9 tera- = 1012 pico- = 10-12 peta- = 1015 femto- = 10-15

6.  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. Dimensional analysis Ex.: - In a one-dimensional motion with constant acceleration a (dimension = L/T2) , the distance traveled during a time interval t (dimension T) is given by the equation: x = v0*t + ½ a*t2 (v0 is the velocity at the start of the time interval)

7. Dimensional analysis (continued) x = v0*t + ½ a*t2 - 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: v0*t has dimension (L/T)*T = L - correct ½ a*t2 has dimension (L/T2)*T2 = L - correct (the factor of ½ is a dimensionless constant)

8. To convert a velocity from units of km/h to units of • m/s, you must: • multiply by 1000 and divide by 60 • multiply by 1000 and divide by 3600 • multiply by 60 and divide by 600 • multiply by 3600 and divide by 1000 • none of these is correct

9. If x and t represent position and time, respectively, • then the constant A in the equation x = A*cos(B*t) • must • have the dimensions L/T • have the dimensions 1/T • have the dimensions L • have the dimensions L2/T2 • be dimensionless

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

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

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

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

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

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

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

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)? • 1 km • 3.22 km • 1.50 km • 2 km • 1.41 km

18. VERY IMPORTANT: except if

19. Vector decomposition  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. 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. y θ x • The magnitude of the vector is 3 m and the angle • q = 30o. Which statement is correct: • points in the negative x direction; Ay = 1.5 m • points in the positive y direction; Ax = 1.5 m • points in the positive x direction; Ay = 1.5 m • points in the positive x direction; Ax = 1.5 m • none of these is correct

22. Unit Vectors  We have discussed only the algebraic components of the projections ax and ay.  However and are vectors , but with well defined orientations 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. - dimensionless

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

24. Examples of vector decomposition

25. Examples of vector decomposition Translate

26. The magnitude and direction of a vector are given by: • none of the above

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

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

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. Choose a vector of magnitude one: Trigonometric functions y 1 tanq sinq Then: q cosq 1 x i.e. the x and y components of the unitary vector give the values of the sin and cos functions.

31. To convert from degrees to radians (rad) multiply with 2p and divide by 360. Radians and degrees y y q = q = 90o q q = 0 x q q = 0o x q = q = q = 360o q = 180o q = q = 270o Ex: - convert 60o to radians

32. y y y x x x Vector addition using components - Problem - find Step 1 Decompose the vectors on a set of orthogonal axes. 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.

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

34. 2-chances • Vectors and have the following components: • x-component +5 units -6 units • y-component -2 units +2 units • What are the components of vector • Cx = +7 units, Cy = +8 units • Cx = +3 units, Cy = +4 units • Cx = +3 units, Cy = -4 units • Cx = -1 units, Cy = 0 units

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 • = -1 units • = +1 units • = -2.65 units • = +5 units

36. 2-chances • 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? • 2 and 3 • 5 and 10 • 10 and 25 • 3 and 12

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 15o south of east and then walks 8 km due north. How far is the camel then from oasis B? N y D x B A C The displacement left to travel is:

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 -47o (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 f. is in the positive direction of the x axis. What is the magnitude of ? y f x But:

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

40. The choice of a coordinate system is not unique. Coordinate system equivalence 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. Multiplication of a vector with a scalar Operations of multiplication with vectors - the direction of is the same with that of if m > 0 and is opposite if m<0 Multiplication of a vector by a vector - the scalar product - the vector product

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

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

44. The vector product (or “cross” product) of two vectors is defined as a vector: Vector product - 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. The vector product is not commutative. Vector product properties Particular cases:

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

47. Definition: motion is defined as the change of an object’s position with time Motion along a straight line (one – dimensional)  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. 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. Examples of time dependence of position x (m) x (m) x (m) t (s) t (s) t (s)

50. Displacement A change from one position x1 to another position x2: 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