Chapter 2 - Scalars and Vectors PDF

Outcome 1: Identify the use of S.I. system of measurement and various properties of vectors. Legal Disclaimer: Copied & Modified from College Physics by Raymond A. Serway & Chris Vuille (Book Chapter 25: Geometric Optics, Chapter 27: Wave optics)  Learnin...

Outcome 1: Identify the use of S.I. system of measurement and various properties of vectors. Legal Disclaimer: Copied & Modified from College Physics by Raymond A. Serway & Chris Vuille (Book Chapter 25: Geometric Optics, Chapter 27: Wave optics)  Learning objectives Scalar and vector quantities Vector Representation Type of vectors Cartesian or Rectangular coordinates of a vector Addition of vectors Components of a vector Unit vectors Vector addition  Scalars and Vectors Scalars Vectors A physical quantity that can be A physical quantity that can be explained completely by its magnitude completely described by magnitude is called a scalar quantity. and direction is called a vector Ex: Time, mass, distance, length, quantity. volume, temperature, and energy Ex: Displacement, velocity, and force  Vector Representation Graphical Method Symbolic Method Vectors can be represented by a Vector diagram, Symbolically, a vector is represented using an arrow in the Cartesian system. by a Bold letter or by using an arrow The length of the line represents its magnitude. on the symbol of a vector quantity. For example: The arrow head shows the direction of vector. Velocity = V = 𝑣Ԧ Acceleration = a = 𝑎Ԧ Force = F = 𝐹Ԧ  Cartesian or Rectangular coordinate system Two reference lines are drawn at right angles to each other. The reference lines are called X axis & Y axes: X-axis is drawn horizontal with positive direction to the right. Y-axis is drawn vertical with positive direction upward They are called coordinate axes and their point of intersection is the ORIGIN (O). Different types of vectors Type Definition example How find resultant vector Magnitude Direction Like Have same directions but Add the magnitude vectors different magnitudes of the two vectors (parallel to each other) (A+B) Direction of both Equal Have same directions and vectors Double the vectors equal magnitudes (parallel to magnitude (2A) each other) Opposite Have opposite directions but Vectors equal magnitudes (antiparallel to each other or zero angle between them is 180O) Unlike Have opposite directions and Subtract the Direction vectors have different magnitudes magnitude the two of bigger (antiparallel) vectors vector  Vector Addition – 4 cases  CASE 1: When two vectors point in the SAME direction, simply add them together.  CASE 2: When two vectors point in the OPPOSITE direction, simply subtract them.  CASE 3: When two vectors are PERPENDICULAR to each other, use the PYTHAGOREAN THEOREM.  CASE 4: If there are 2 or more vectors in RANDOM DIRECTIONS, use the component method. Components of a Vector A vector 𝐴 in a plane is described by a pair of its vector coordinates. The x-coordinate of vector 𝑨 is called its x- component and the y-coordinate of vector 𝑨 is called its y-component. The vector x-component is a vector denoted by 𝐴𝑥 The vector y-component is a vector denoted by 𝐴𝑦 Components of a Vector The X-component of a vector is the projection along the x-axis Ax  A cos The y-component of a vector is the projection along the y-axis. Ay  A sin This assumes the angle θ is measured with respect to the positive x-axis, anticlockwise. Properties of vector components Quadrant Ax Ay Ist + + IInd - + IIIrd - - IVth + _ Finding Magnitude and direction of a Vector from its components If Ax and Ay are the components of a vector, The magnitude of 𝑨 is Direction of 𝑨 is given as Polar Coordinates (r, θ) The points are labeled (r, θ) as shown next, r represents the distance from the origin to the point having Cartesian coordinates (x, y), and θ is the angle between r and a fixed axis. The quantities r and θ are known as the Polar Coordinates of the point. To analyze r and θ, keep in mind that this fixed axis is usually the positive x axis, and θ is usually measured counterclockwise from it Naming vector directions Class Work Problems Q1. Calculate the X and Y components of the four force vectors shown in the figure. (Make sure to use the values of the angle correctly in the equations). Q2. A person walked a distance 65 m, in a direction 25 East of North. Calculate the horizontal (X) and vertical (Y) components of his displacement. Unit vectors If 𝐴𝑥 𝑎𝑛𝑑𝐴𝑦 are the components A Vector with magnitude 1 in any of a vector 𝐴Ԧ , it can be direction. represented as, 𝐴Ԧ = 𝐴𝑥 𝑖Ԧ + 𝐴𝑦 𝑗Ԧ Unit vectors along the directions X, Y and Z are represented by 𝑖Ԧ , 𝑗Ԧ , 𝑘 A 3 dimensional vector has x, y respectively. and z components. Then, 𝐴Ԧ = 𝐴𝑥 𝑖Ԧ + 𝐴𝑦 𝑗Ԧ +𝐴𝑧 𝑘 X 𝒊Ԧ Then, the magnitude of the vector is, Y 𝒋Ԧ X𝒌 𝐴 = (𝐴𝑥 2 + 𝐴𝑦 2 + 𝐴𝑧 2 )  Vector Addition Let the sum of vectors 𝐴Ԧ and 𝐵 is represented by 𝑅 𝑹=𝑨+𝑩 Then, 𝑹𝒙 = 𝑨𝒙 + 𝑩𝒙 , 𝑹𝒚 = 𝑨𝒚 + 𝑩𝒚 and 𝑹𝒛 = 𝑨𝒛 + 𝑩𝒛 So, 𝑹 = 𝑹𝒙 𝒊Ԧ + 𝑹𝒚 𝒋Ԧ + 𝑹𝒛 𝒌 That means, 𝑹 = 𝑨𝒙 + 𝑩𝒙 𝒊Ԧ + 𝑨𝒚 + 𝑩𝒚 𝒋Ԧ + (𝑨𝒛 + 𝑩𝒛 )𝒌 Class Work Problems Q3. Represent the four vectors in Q1 using the unit vectors. Class Work Problems Q4. Find the sum of two vectors A and B lying in the 𝑥𝑦 plane and given by 𝐴Ԧ = (2.0𝑖 + 2.0𝑗) 𝑚 and 𝐵 = (2.0𝑖 + 4.0𝑗) 𝑚 Q5. A particle undergoes three consecutive displacements: 𝑑1 = (15𝑖 + 30𝑗 + 12𝑘) 𝑐𝑚, 𝑑2 = (23𝑖 − 14𝑗 − 5.0𝑘) 𝑐𝑚, and 𝑑3 = (−13𝑖 + 15𝑗) 𝑐𝑚. Find the components of the resultant displacement and its magnitude. Q6. A man travels 75km in a direction 40 west of north. Find the components and express the displacement using unit vectors. (−48.20𝑖Ƹ + 57.45𝑗)Ƹ Class Work Problems Q6. An airplane heads due north at 100 m/s through a 30 m/s cross wind blowing from east to the west. Determine the magnitude of the resultant velocity of the airplane. Ans: 𝒗𝒑𝒍𝒂𝒏𝒆 = 100𝑗Ƹ 𝑚/𝑠 𝒗𝒘𝒊𝒏𝒅 = −30𝑖Ƹ 𝑚/𝑠 𝒗𝑹 = 100𝑖Ƹ − 30𝑗Ƹ 𝑚/𝑠 𝑣𝑅 = 104.4 𝑚/𝑠  Practice Problems 1) A vector has an 𝑥 component of 25 units and a 𝑦 component of 40 units. Find the magnitude of this vector. (Ans: 47.17 units) 2) A person walks 25m due south and 35m due west. a) Express the resultant displacement using unit vectors. b) Calculate the magnitude of the resultant displacement. (Ans: −𝟑𝟓𝒊Ƹ − 𝟐𝟓𝒋,Ƹ 43.01 m) 3) A delivery truck travels 18 blocks north, 10 blocks east and 16 blocks south. What is the final displacement (magnitude) from the origin? Assume the blocks are equal length. (Ans: 10𝒊Ƹ + 𝟐𝒋,Ƹ 10.198 m)  Multiple Choice Questions 1) Which of the following quantities have magnitude and direction? a) Mass b) Time c) Temperature d) Force 2) Magnitude of the vector 3𝑖Ƹ + 2𝑗Ƹ − 𝑘෠ is ______________. a) 3.74 units b) 4 units c) 3.60 units d) 3.46 units 3) The vector 𝐴Ԧ = −7𝑖Ƹ − 2𝑗Ƹ , it lies in ____________ quadrant. a) 1st quadrant b) 2nd quadrant c) 3rd quadrant d) 4th quadrant 4) An object is travelling in a direction, 60 south of east. Identify the correct statement. a) Both X and Y components are positive b) Both X and Y components are negative c) X component is positive and Y component is negative d) X component is negative and Y component is positive