Chapter 1 PHYS-110 PDF

Overview Any Picture you think it can represent this chapter Chapter 1 1  Learning Outcomes After studying this chapter, you will be able to: 1. Define vector quantity. 2. Differ...

Overview Any Picture you think it can represent this chapter Chapter 1 1  Learning Outcomes After studying this chapter, you will be able to: 1. Define vector quantity. 2. Differentiate between vector and scalar quantities. 3. Understand the Cartesian coordinate system. 4. Resolve any vector and find its components. 5. Calculate the magnitude and direction of vectors. 6. Identify the unit vectors ( magnitude and direction) on three axes. 7. Write a vector in a unit vector notation. 8. Add vectors by components. 9. Multiply vectors by a scalar (either +ve or - ve no.). 10. Calculate the scalar product of two vectors in terms of their magnitudes and angle between them. Chapter 1 2  1.6 ( VECTORS ) What is a Vector ? (Difference between vector and scalar quantity) Cartesian Coordinate System (2D &3D) Chapter (1) Resolve vectors and find their components Vector Length and Direction Unit Vector (write a vector in a unit vector notation) Vector addition using components Multiplication of Vectors with a Scalar Vectors multiplication (Scalar product) Chapter 1 3  What is a Vector? Vectors are mathematical description of quantities which have magnitude and direction. The magnitude of a vector is a non-negative number often associated with a physical unit. Vectors have a starting point (tail) and an ending point (arrow) which points to a specific direction. Vectors are denoted by a letter with a small horizontal arrow pointing to the right above it (𝒙 ). Vector quantities are important in physics. Chapter 1 4  What is a Vector ? (Difference between vector and scalar quantities) Vector quantities PHYSICAL QUANTITIES magnitude direction scalar Quantites Example magnitude Displacement Velocity Example Speed Temperature time Chapter 1 5  Cartesian Coordinate System A Cartesian coordinate system is defined as a set of two or more axes with angles of 90° between each them. Hence they are perpendicular or orthogonal to one another. In a one-dimensional coordinate system we have one axis (e.g 𝒙): The position of a point can be written as P=(P𝒙 ) On the Y- axis Positive direction P𝒚=+3 cm origin 0 The point P has the x-coordinate Negative P𝒙 = - 2.5 cm direction Chapter 1 6  Cartesian Coordinate System We can define a two-dimensional coordinate system: ▪ By typically labelling the horizontal axis 𝒙 and the vertical axis 𝒚. ▪ We can then specify any point P (position) in 2- dimensional space by specifying its coordinates P = (P𝒙, P𝒚 ) P = (3, 4) We can define a three-dimensional coordinate system: ▪ By typically labelling the horizontal axis 𝒙, the vertical axis y and the third orthogonal axis z. ▪ We can then specify any point P (position) in 3- dimensional space by specifying its coordinates P = (Px, Py, Pz) P = (3, 4, 3) Chapter 1 7  Resolve a vector and find its components ▪ Resolving a vector is the process of finding its components. ▪ A component is the projection of the vector on an axis. ▪ We know from mathematics that: ▪ We know from mathematics that: 𝑎 𝑎𝑦 𝑐𝑜𝑠ϴ = 𝑥 𝑐𝑜𝑠α = 𝑎 𝑎 𝑎𝑦 𝑎𝑥 𝑠𝑖𝑛ϴ = 𝑠𝑖𝑛α = 𝑎 𝑎 Chapter 1 8  Resolve a vector and find its components A more general way of finding vector components the angle is measured with respect to x-axis ՜ ՜ 𝒂𝒙 : +𝒗𝒆, 𝒂𝒚 : +𝒗𝒆 𝒃𝒙 : −𝒗𝒆, 𝒃𝒚 : +𝒗𝒆 𝑏 𝑎 𝒂𝒙 = 𝑎 cos 𝜃1 𝒃𝒙 = −𝑏 cos 𝜃2 𝒃 𝒚 𝒂𝒚 𝒂𝒚 = 𝑎 sin 𝜃1 𝒃𝒚 = 𝑏 sin 𝜃2 𝒃𝒙 𝜃2 𝜃1 𝒂𝒙 𝒄𝒙 𝜃3 𝜃4 𝒅𝒙 𝒄𝒚 𝒅𝒚 ՜ ՜ 𝒄𝒙 : −𝒗𝒆, 𝒄𝒚 : −𝒗𝒆 𝑐 𝑑 𝒅𝒙 : +𝒗𝒆, 𝒅𝒚 : −𝒗𝒆 𝒄𝒙 = −𝑐 cos 𝜃3 𝒅𝒙 = 𝑑 cos 𝜃4 𝒄𝒚 = −𝑐 sin 𝜃3 𝒅𝒚 = −𝑑 sin 𝜃4 Chapter 1 9  Resolve a vector and find its components More general ways of finding vector component Angle is measured with respect to y-axis 𝒃𝒙 : −𝒗𝒆, 𝒃𝒚 : +𝒗𝒆 ՜ ՜ 𝒂𝒙 : +𝒗𝒆, 𝒂𝒚 : +𝒗𝒆 𝑏 𝑎 𝒂𝒙 = 𝑎 sin 𝜃1 𝒃𝒙 = −𝑏 sin 𝜃2 𝒃𝒚 𝒂𝒚 𝒂𝒚 = 𝑎 cos 𝜃1 𝒃𝒚 = 𝑏 cos 𝜃2 𝜃2 𝜃1 𝒃𝒙 𝒂𝒙 𝒄𝒙 𝒅𝒙 𝜃3 𝜃4 𝒄𝒚 𝒅𝒚 ՜ ՜ 𝒄𝒙 : −𝒗𝒆, 𝒄𝒚 : −𝒗𝒆 𝑐 𝑑 𝒅𝒙 : +𝒗𝒆, 𝒅𝒚 : −𝒗𝒆 𝒄𝒙 = −𝑐 sin 𝜃3 𝒅𝒙 = 𝑑 sin 𝜃4 𝒄𝒚 = −𝑐 cos 𝜃3 𝒅𝒚 = −𝑑 cos 𝜃4 Chapter 1 10  Resolve a vector and find its components Chapter 1 11  Resolve a vector and find its components ϴ- ϴ+ Chapter 1 11 Vector Length and Direction Knowing the components of a vector, we can calculate its length and direction. Vectors in 2 dimensions (most important case) – The length (using Pythagorean theorem) is: – The direction is defined by: Chapter 1 13  Vector Length and Direction Finding the components. Writing a vector in magnitude - angle notation Vector resolving Chapter 1 14  Unit Vectors (write a vector in the unit vector notation) Unit vectors are a set of special vectors that make the math associated with vectors easier Unit vectors have magnitude 1 and are directed along the main axes of the coordinate system. In Cartesian coordinates, the unit vectors are: 𝑖Ƹ = 𝑗Ƹ = 𝑘෠ = We can write the vector in a unit vector notation 𝑜𝑟 𝐴റ = 𝐴𝑥 𝑖Ƹ + 𝐴𝑦 𝑗Ƹ + 𝐴𝑧 𝑘෠ Scalar components ▪ Another way to write a vector 𝐴Ԧ = (𝐴𝑥 , 𝐴𝑦 , 𝐴𝑧 ) Chapter 1 15 Exercise 1.77 (Page 30) A vector 𝐶Ԧ has components 𝐶𝑥 = 34.6 m and 𝐶𝑦 = − 53.5 m. Find the vector’s length and angle with the x-axis. 𝐶Ԧ = 𝐶𝑥 𝑥ො + 𝐶𝑦 𝑦ො 𝐶Ԧ = 34.6𝑥ො − 53.5 𝑦ො Chapter 1 16  Extra Exercise 71° 63° Chapter 1 17  Extra Exercise 71° 63° 1) The unit vector notation is represented as: 𝐴റ = 𝐴 cos 30 𝑥ො + A sin 30 ෝ𝑦 𝐴റ = 75 cos 30 𝑥ො + 75 sin 30 ෝ𝑦 = 64.9 𝑥ො + 37.5 ෝ𝑦 𝐵 = −𝐵 sin 71 𝑥ො + B cos 71 ෝ𝑦 𝐵 = −60 sin 71 𝑥ො + 60 cos 71 ෝ𝑦 = −56.7 𝑥ො + 19.5 ෝ𝑦 𝐶റ = −𝐶 cos 52 𝑥ො − C sin 52 ෝ𝑦 𝐶 = −25cos(52) 𝑥ො − 25 sin 52 ෝ𝑦 = −15.3 𝑥ො − 19.7 ෝ𝑦 𝐷 = 𝐷 sin 63 𝑥ො − D cos 63 ෝ𝑦 𝐷 = 90 sin 63 𝑥ො − 90 cos 63 ෝ𝑦 = 80.19 𝑥ො − 40.85 ෝ𝑦 Chapter 1 18 Vector Addition using Components For most practical purposes, we will add vectors using the component method 𝐶Ԧ = 𝐶𝑥 𝑥ො + 𝐶𝑦 𝑦ො + 𝐶𝑧 𝑧Ƹ ▪ In the same way, we can find the difference (vector subtraction) 𝐷 = 𝐴Ԧ − 𝐵 𝐷 = 𝐷𝑥 𝑥ො + 𝐷𝑦 𝑦ො + 𝐷𝑧 𝑧Ƹ Chapter 1 19  Vector Addition using Components Adding vectors using the component method: Note: this can also be applied to vector subtraction Chapter 1 20  Exercise: 1.68 (Page 30) 71° 63° Vector x-component y-component 𝐴റ 64.9 37.5 𝐵 – 56.7 19.5 𝐶റ – 15.3 – 19.7 𝐷 80.19 – 40.85 Chapter 1 21  Exercise: 1.68 (Page 30) 71° 63° Solution: a) 𝐴Ԧ + 𝐵 + 𝐶Ԧ + 𝐷 = Vector x-component y-component 𝐴Ԧ 64.9 37.5 + + 𝐵 – 56.7 19.5 + + 𝐶Ԧ – 15.3 – 19.7 + + 𝐷 80.19 – 40.85 73.09 – 3.55 𝐴Ԧ + 𝐵 + 𝐶Ԧ + 𝐷 = 73.09 𝑥ො − 3.55 𝑦ො Chapter 1 22  Exercise: 1.68 (Page 30) 71° 63° Solution: a) 𝐴Ԧ − 𝐵 + 𝐷 = Vector x-component y-component 𝐴Ԧ 64.9 37.5 – – 𝐵 – 56.7 19.5 + + 𝐷 80.19 – 40.85 201.79 – 22.85 𝐴Ԧ − 𝐵 + 𝐷 = 201.79 𝑥ො − 22.85 𝑦ො Chapter 1 23 b) 𝐴Ԧ − 𝐵 + 𝐷 = 201.79 𝑥ො − 22.85 𝑦ො 𝐴Ԧ − 𝐵 + 𝐷 = (201.79)2 +(−22.85)2 = 203.08 −22.85 𝜃 = 𝑡𝑎𝑛−1 = −6.46𝑜 201.79 71° 63° 9/12/2020 Chapter 1 24 Multiplication of a Vector with a Scalar Multiplication of a vector with a Multiplication of a vector with a positive scalar negative scalar results in another vector that results in another vector that points points in the same direction but in the opposite direction but has a has a magnitude that is the magnitude that is the absolute value product of the scalar and the of the product of the negative scalar magnitude of the original vector. and the magnitude of the original vector. Chapter 1 25 Multiplication of a Vector with a Vector Multiplying a vector by a vector Scalar product Vector product (or Dot product) (or Cross product) will produce a will produce a scalar new vector Chapter 1 26  Scalar Product (dot product) It is sometimes called the dot product. The scalar vector product is defined as: In terms of Cartesian coordinates: If the two vectors form a 90°angle, the scalar product is zero. If 𝛼 = 90o ➔ 𝐴Ԧ ⋅ 𝐵 = 0 We can use the scalar product to find the angle between two vectors in terms of their Cartesian coordinates: Chapter 1 27  Exercise: 1.15 (Page 28) Answer: The two vectors are given in Cartesian Coordinate and we do not have the angle between the two vectors, hence we will use; 𝐴Ԧ ⋅ 𝐵 = 2 × 0 + 1 × 1 + 0 × 2 = 1 Chapter 1 28  Example: 1.5 (Page 23) Chapter 1 29  Vector Product of Vectors The vector (or sometimes called cross) product of two vectors is defined as: 𝐴Ԧ 𝐵 𝛼 𝐴Ԧ × 𝐵 = 𝐴Ԧ 𝐵 sin 𝛼 = AB sin 𝛼 Chapter 1 30 Vector Product of Vectors – Right-Hand Rule Chapter 1 31  ❖. Equation summary Vector Notation Vector Length and direction Vector resolving, angle is measured from the x direction in first quadrant Dot Vector resolving, product angle is measured from the y direction in first quadrant Angle between cross two vectors product Chapter 1 32  The END OF CHAPTER (1) Chapter 1 33