Physical quantities units and standard What do we measure? - Physical quantities Units - a unit is a measure of the quantity that is defined to be exactly 1 (one) Examples: meter, mile, gram, kilogram Standard - a reference to which all the other examples of the quantity are compared Base quantities, their units and standards The International System of Units QuantityUnit nameUnit symbolStandard LengthMeterm Distance traveled by light in 1/299,792,458 second TimeSeconds Time required for 9,192,631,770 periods of radiation emitted by cesium atoms MassKilogramkg Platinum-iridium cylinder in International Bureau of Weights and Measures at Sevres, near Paris 1

Other systems of units CGSE: centimeter, gram, second 1m = 100cm 1kg = 1000g British engineering system This system has force instead of mass as one of its basic quantities, which are feet, pounds, and seconds. 1 m = ft; 1 inch = 2.54 cm 1 kg = slug (Not the same as weight!) on Earth 1 kg weighs lb, on the Moon 1 kg weighs lb 2

Multiples of Units yocto-y zepto-z atto-a femto-f pico-p nano-n micro-  milli-m centi-c 10 3 kilo-k 10 6 mega-M 10 9 giga-G tera-T peta-P exa-E zetta-Z yotta-Y Conversion of units: Multiply by the appropriate representation of 1 to cancel the unwanted units away Converting between metric units, is easy, as all it involves is powers of 10. Example 1: Convert 3kg into gram Example 2: Convert 10 mph into m/s 3

A vector has magnitude as well as direction Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude Some scalar quantities: mass, time, temperature VECTORS Geometric presentation : Notations: - letter with arrow; a – bold font Magnitude (length of the vector): Some properties: 4

Vector addition (geometric) Two vectors: Several vectors Subtraction 5

Question 2: A person walks 3.0 mi north and then 4.0 mi west. The length and direction of the net displacement of the person are: 1) 25 mi and 45˚ north of east 2) 5 mi and 37˚ north of west 3) 5 mi and 37˚ west of north 4) 7 mi and 77˚ south of west Question 3: Consider the following three vectors: What is the correct relationship between the three vectors? Question 1: Which of the following arrangements will produce the largest resultant when the two vectors of the same magnitude are added? AB C 6

x y axax ayay 2 - Dimensional Vectors (2D) 7

Example 1: a x = 3m, a y = - 4m. Find a and . Example 2: a = 4,  = 30 ◦. Find a x, a y. Example 3:  = 30 ◦, a y = 3. Find a x, and a. Question: What is the unit vector in the direction of vector a ? 8

z y x 3 - Dimensional Vectors (3D) 9

X Y Z Right-hand rule X Y Z 10

Vector addition Geometric:Algebraic: Multiplication by number Triangle inequality:Properties: 11

Scalar product (dot product)  Properties of the dot product Angle between vectors: Projections 12

B projected onto A: Component of A perpendicular to B: A projected onto B: Component of B perpendicular to A: Projections     13

Question 3: Which pair of vectors will have the largest value for A·B? 1) A B 3) 2) A B 60° A B 30° A projected onto B Question 1: What is equal to? Question 2: What is angle (in degrees) between the following two vectors? 1) 0 2) 45 3) 90 4)135 More visual: A·B = BA projected onto B 14

Vector product (cross product) A B A×B Right-hand rule 15

Properties of vector product z y x Right-handed coordinate systems 16

Question 1: Vectors A, B and C are on the plane of the screen. They are drawn to scale. Compare the magnitude of these two cross products: A) |A×B| > |A×C| B) |A×B| = |A×C| C) |A×B| < |A×C| And they both point out of the screen. The cross product selects the part of B that is perpendicular to the direction of A. A B C Bsinθ Bcosθ Question 2: 17

Question 3: Consider two nonzero vectors A and B with an angle Φ between them. Question 4: What is angle (in degrees) between vectors A and B? A)45 B) 90 C)135 D)180 1.Into page 2.Out of page Questions 3 & 4: Right-handed Cartesian coordinate system. What is the direction of the +z axis? What is the direction of the +x axis? 18

Question 5: Question 6: Question 7: What is angle (in degrees) between vectors A and B? 19