1. Units and Prefixes SI units Units must accompany quantities at all times, otherwise the quantities are meaningless. If a person writes mass = 1, do they mean 1 gram, 1 kilogram or 1 tonne? The Système International d Unitè (SI) system of units is used by modern scientists worldwide and all equations and constants are defined in terms of this system of units. When substituting numbers into equations, you must make sure that the quantities are in SI units. Your answers will also be in SI units, and the units are part of the answer, otherwise the answer is meaningless. The SI units system uses fundamental units, which include the: metre (m) to measure length kilogram (kg) to measure mass second (s) to measure time These quantities are fundamental because they cannot be expressed in terms of other quantities. Note that even though a centimetre, millimetre and gram are related to metres and kilograms, they are NOT SI units. You must always use metres for length and kilograms for mass. Copyright MATRIX EDUCATION 2014 Page 10 of 199 Our Students Come First!
The SI units also include derived units used to describe other quantities that are used in science. These units can be expressed in terms of the fundamental units. Some units you will encounter in this course are: Quantity Units SI Units Equivalent SI Units Speed Metres per second (ms -1 ) Acceleration Metres per second squared (ms -2 ) Force Newton (N) (kg ms -2 ) Energy Joule (J) (kg m 2 s -2 ) Momentum Kilograms metres per second (kg ms -1 ) Impulse Newton Seconds (Ns) (kg ms -1 ) Prefixes are commonly used to indicate the size of a quantity, e.g. 1 kilometre or 1 megabyte. A kilometre is 1000 metres; the prefix kilo means one thousand and a megabyte is one million bytes; the prefix mega means one million. If a quantity is given using a prefix, you must multiply it by the appropriate factor to convert it to SI units before using it in calculations. Prefix Symbol Factor Giga G x 10 9 (billion) Mega M x 10 6 (million) Kilo K x 10 3 (thousand) Centi C x 10-2 (one hundredth) Milli m x 10-3 (one thousandth) Micro μ x 10-6 (one millionth) Nano n x 10-9 (one billionth) Copyright MATRIX EDUCATION 2014 Page 11 of 199 Our Students Come First!
2. Scalars and Vectors Students learn to: distinguish between scalar and vector quantities in equations Definitions Physics is a science based on measurement. Some things are measurable, like physical quantities, and some aren t. Physical quantities, things we can measure, can be separated into two main categories scalars and vectors. A scalar is any measurement that has a magnitude but doesn t have a direction. A scalar requires only a number and a unit to be complete. Mathematically, all scalars can be considered the same. Mass and area are examples of scalar quantities. Australia has an area of 7,617,930m 2. A vector is any quantity that has a magnitude and a direction associated with it. We say a city is 1000 km south or a car travels at 60 km/h east. Mathematically, all vectors are the same, but vectors are different to scalars. Classify the following quantities as either scalars or vectors: Distance (d), displacement (r ), acceleration (a ), speed (v), mass (m), impulse (I ) time (t), force (F ), energy (E), power (P), momentum (p ), velocity (v ) SCALAR VECTOR Copyright MATRIX EDUCATION 2014 Page 12 of 199 Our Students Come First!
An arrow above a letter or boldface type is often used to indicate a vector quantity. When working with scalars, the mathematics we know will suffice. However, because directions are involved in vectors, they need their own version of mathematics for operations such as addition, subtraction, decomposition, multiplication, division and so forth. We will now look into a few types of vector operations. Vector representation Vector quantities can be added, but first we must learn how to represent them. N θ x Vectors are normally represented as arrows, as shown above. In a vector: The length of the line, x, represents the magnitude of the vector in some units. The arrow gives the direction and has a head and a tail. The head points in the specified direction. The magnitude (x), units and direction (θ) are clearly labelled. In the space below, draw a scaled diagram representing the vector 5 cm N40 W, clearly labelling all important details. Copyright MATRIX EDUCATION 2014 Page 13 of 199 Our Students Come First!
Vector addition The resultant vector is the vector that results from the joining of two or more vectors, head-to-tail. v 1 + v 2 The steps for the vector addition of two vectors v 1 and v 2 is outlined below: 1) Choose an origin and sketch one of the vectors, v 1. N v 1 2) Join the tail of the second vector, v 2, to the head of the first vector. N v 1 v 2 3) Connect the tail of the first vector to the head of the second vector. This is the resultant vector, v 1 + v 2. N v 1 v 2 v 1 + v 2 Copyright MATRIX EDUCATION 2014 Page 14 of 199 Our Students Come First!
4) Determine the magnitude and direction of the resultant vector using accurate scale drawings and trigonometry methods. Pythagoras Theorem Cosine rule (a 2 = b 2 + c 2 2bc cos A) N θ v 1 v 2 v 1 + v 2 DEMONSTRATION: Vector addition simulator. Which of the following show the vector addition diagrams matched correctly to the respective vector addition equations? I II III IV A C C A B A A B B B C C I II III IV A+C=B A+B=C A+B=C A+B=C C+B=A C+A=B C+B=A A+C=B A+C=B A+C=B A+B=C A+B=C A+C=B B+A=C B+A=B C+A=B Copyright MATRIX EDUCATION 2014 Page 15 of 199 Our Students Come First!
Find the resultant vector of the addition of the following vectors. In each case show the vector addition. a) v 1 = 10 m/s East and v 2 = 30 m/s East. 3 b) There are two forces on a toy wagon, F 1 = 15 N East and F 2 = 25 N North. 4 c) Hazel takes thirty steps West, twenty-two steps North, forty-four steps East, and twenty-nine steps South. Assume all her steps are the same size. How many steps would she have to take from her starting point to end in the same spot if she had gone directly? In what direction would she have to walk? 5 Copyright MATRIX EDUCATION 2014 Page 16 of 199 Our Students Come First!
Vector decomposition It is useful to think of a vector as the sum of two other vectors at right angles to each other. These two vectors are called components of the original vector. Consider the figure shown below. v vy vx = + vy vx v y is a component of a vector v in the vertical direction. Express v y in terms of and v. 6 v x is a component of a vector v in the horizontal direction. Express v x in terms of and v. 7 The vectors w and y below can be decomposed into vertical components (w y, y y) and horizontal components (w x, y x). Draw and label these components. θ y w θ Copyright MATRIX EDUCATION 2014 Page 17 of 199 Our Students Come First!
The figure shown below is a diagram of the addition of two vectors, v 1 and v 2. The dotted vector v R is the result vector. vr The vectors v 1 and v 2 can be decomposed. The components are shown. We can use the components of the two vectors to find the resultant vector, v R. 8sin10 vr N 10sin30 10cos30 8cos10 Copyright MATRIX EDUCATION 2014 Page 18 of 199 Our Students Come First!
To determine the magnitude of the resultant vector, we use Pythagoras Theorem: v R = (10cos30 + 8cos10) 2 + (10sin30 + 8sin10) 2 = 8 To determine the direction of the resultant vector: tanθ = Opposite 10sin30 + 8sin10 = Adjacent 10cos30 + 8cos10 θ = 9 Therefore, v R = N E Remember, when giving the direction of the resultant vector, you must give the angle between the vector and the vertical line joining North and South. NOTE TO STUDENTS There are two common ways used to state a direction: as a true bearing or a compass bearing. True bearing is measured in a clockwise direction from the North and given in a standard three-digit notation. Compass bearings are directions given from the North or the South. For example, a true bearing of 135 is equivalent to the compass bearing S45 E. DID YOU KNOW? The function Pol(x,y) on your Casio calculator can quickly determine the length of the hypotenuse and the adjacent angle in a right angled triangle. Copyright MATRIX EDUCATION 2014 Page 19 of 199 Our Students Come First!
Find the resultant of the two vectors using the method of components. In each case, make a rough sketch of the vector addition and the vector components. v 1 = 15 m/s N30 E and v 2 = 25 m/s N45 E 10 45 30 v 1 = 15 m/s N45 E and v 2 = 25 m/s S30 E 11 45 30 Copyright MATRIX EDUCATION 2014 Page 20 of 199 Our Students Come First!
Vector subtraction Just as we can add vectors, we can also subtract them. Vector subtraction is an operation that is important in relative motion, which we will discuss later. When we subtract two vectors, v 2 v 1, it is equivalent to adding two vectors, v 2 + ( v 1). What is the meaning of the vector v 1? v 1 is a vector equal in magnitude to v 1 but opposite in direction to it. See the figure below. v1 v1 Vectors v 2 and v 2 are shown below. v2 v2 An example of vector subtraction is shown below. v1 v2 v1 = + v2 = v1 v2 Copyright MATRIX EDUCATION 2014 Page 21 of 199 Our Students Come First!
Subtract the following vectors (v 2 v 1). In each case make a sketch of the vector subtraction. v 1 = 10 ms 1 East and v 2 = 30 ms 1 East 12 v 1 = 10 ms 1 East and v 2 = 30 ms 1 West 13 F 1 = 15 N East and F 2 = 25 N South [F 2 F 1] 14 Copyright MATRIX EDUCATION 2014 Page 22 of 199 Our Students Come First!
3. Distance & Displacement Distance (r) [m] Distance is a scalar quantity. It refers to the length of the entire path travelled by a body. The SI unit for distance is metres (m). Displacement (r ) [m + direction] Displacement is a vector quantity. It refers to the change in position of a body, or the difference between where you started and where you end. Displacement is specified by both magnitude (m) and direction. The difference between distance and displacement is illustrated below. Source: http://web2.uwindsor.ca/courses/physics/high_schools/2005/brownian_motion/distance.jpg A displacement and the distance travelled can be of the same magnitude. Give an example of when this could be the case. 15 However, since displacement gives information about the overall result, it must include the direction of the finishing point from the starting point. Displacement can be zero even if the distance travelled is very large. Copyright MATRIX EDUCATION 2014 Page 23 of 199 Our Students Come First!
Consider the following example. A bee leaves its hive to gather nectar. Its journey takes it 200 m due north from the hive, then 100 m due east, then 300 m due south and then finally on a beeline straight back to the hive. Draw a scale diagram of the bee s journey. Determine the distance travelled by the bee. 17 Determine the bee s displacement at the end of its journey. 18 Copyright MATRIX EDUCATION 2014 Page 24 of 199 Our Students Come First!
4. Lesson Review Questions A car travels 40 km south and then 30 km west in 1 hour. What is the total distance travelled? What is the car s straight-line distance from its starting point at the end of the hour? What is the car s total displacement? While on holidays, Sabiha drives 140 km north-west to visit the Hunter Valley and then 260 km S30 o E to go surfing in Wollongong. Draw a diagram to represent this journey. What is the total distance she travelled that day? What is her final displacement? Copyright MATRIX EDUCATION 2014 Page 25 of 199 Our Students Come First!
Jackson takes 2 minutes to jog 500 m east, then 3 minutes to jog 800 m west, then 5 minutes to return home. Draw a diagram to represent his journey. What is the total distance travelled? What is his displacement after 2 minutes? What is his displacement after 5 minutes? What is his displacement after 10 minutes? Copyright MATRIX EDUCATION 2014 Page 26 of 199 Our Students Come First!
Subtract the following vectors (a 2 a 1), making a sketch of the vector subtraction. a 1= 10 ms 2 N45 E and a 2 = 8 ms 2 West Copyright MATRIX EDUCATION 2014 Page 27 of 199 Our Students Come First!
ANSWERS 3 40 m/s East 4 29.15 N N31 E 5 15.65 (16 steps) S63 E 6 vy = vsinθ 7 vx = vcosθ 8 17.73 9 21 10 39.7 m/s N39 E 11 25.6 m/s S64 E 12 20 ms -1 East 13 40 ms -1 West 14 29.1 N N31 W 15 A ball rolled on a horizontal surface in a straight line. 17 Distance travelled = 200 + 100 + 300 + 141.4 = 741.4 m 18 0 m. It starts and ends its journey at the same point (its hive). Copyright MATRIX EDUCATION 2014 Page 28 of 199 Our Students Come First!