PDT 180 ENGINEERING SCIENCE Vectors And Scalars MUNIRA MOHAMED NAZARI SCHOOL OF BIOPROCESS ENGINEERING UNIMAP

COURSE OUTCOMES CO 1 To analyze problems related to units of measurement, and scalar and vector quantities. 2session 2012/1013

TOPIC OUTLINE 4 What is the difference between a vector and a number (scalar)? 4 How can I add vectors? 4 How can vectors be subtracted? 4 Can vector components help me use vectors? 4 What is projectile motion? 4 How do you describe the motion of a projectile? 4 Around and around, how do you describe circular motion? 4 What is relative motion? session 2012/10133

INTRODUCTION 4 Scalar quantities –Any physical quantities that are specified completely by giving a number and units. –Being describe only by its magnitude. 4 Vector quantities –Any physical quantities that are specified by giving their magnitude and direction. –Direction can be either compass directions (North, East, South & West) or angular directions (n °). session 2012/10134

Introduction 4 Examples for scalar and vector quantities. session 2012/10135 ScalarVector Mass14 kgVelocity100 min/hr North Time10 secAcceleration10 m/s² at 35° with respect to East Volume1 literForce980 N straight down Temperature 25 °C Momentum100 kg m/sec at 90°

Vector Quantities Notation 4 We can draw an arrow to some scale to represent a vector quantity or putting an arrow over a font. 4 The arrow indicates that the quantity has a direction. session 2012/10136 V V or

ADDITION OF VECTORS session 2012/ Methods Graphical method Tail to tip method Parallelogram Vector components method

session 2012/ Additions of scalar quantities are straight forward since it involves only addition of magnitude or numbers. 4 For instance, to add the length of field A with magnitude of 100 m and field B with magnitude of 150 m, we only add the magnitude. The resultant of the addition is 250 m long. 4 However, same method can not be applied for vector quantities addition.

Example: Combining vectors in one dimension. 4 If a person walks 8 km east one day, and 6 km east the next day, obviously the resultant of his movement is 14 km (8 km + 6 km) east from his original place. 4 So, we can say that the resultant displacement is 14 km to the east. session 2012/10139

Example: Combining vectors in one dimension. 4 If on the other hand, the person walks 8 km east on the first day, and 6 km west on the second day, then the person will end up 2 km from the origin. 4 So the resultant displacement is 2 km to the east. 4 The resultant displacement is obtained by subtraction (8 km – 6 km = 2 km). session 2012/101310

Graphical Method 4 In the graphical method, each vector has to be drawn into scale and then the resultant of the vector can be obtained. Figure below shows the summary for adding vector using graphical method. session 2012/101311

Example: Combining vectors in two dimension. 4 Here the actual travel path are at right angles to one another; –We can find the displacement by using the Pythagorean Theorem. session 2012/101312

Example: Combining vectors in two dimension. 4 Adding the vectors in the opposite order gives the same result. session 2012/101313

Graphical Method of Vector Addition Tail to Tip Method 4 Even if the vector are not at right angles, they can be added graphically by using “tail-to-tip” method. session 2012/101314

Subtraction of Vectors 4 In order to subtract vectors, we define the negative of a vector, which has the same magnitude but point in the opposite direction. 4 Then, we add the negative vector; session 2012/101315

Multiplication by a Scalar 4 A vector V can be multiplied by a scalar c; the result is a vector cV that has the same direction as V but a magnitude cV. If c is negative, the resultant vector points in the opposite direction. session 2012/101316

Vector Components Method 4 Adding vectors graphically using a ruler and protector is often not sufficiently accurate and is not useful for vectors in three dimensions. 4 In order to solve the problems, –A precise and numerical method of adding vectors is use. Adding vectors by components session 2012/101317

Adding vector by component 4 Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other. session 2012/101318

Adding vector by component 4 If the components are perpendicular, they can be found using trigonometric functions. session 2012/ Opposite Adjacent Hypotenuse

Adding vector by component session 2012/101320

Adding vector by component 4 The components are effectively one-dimensional, so they can be added arithmetically: session 2012/101321

Adding vector by component 4 Signs of Components session 2012/101322

Adding vector by component 4 Adding vectors: 1. Draw a diagram; add the vectors graphically. 2. Choose x and y axes. 3. Resolve each vector into x and y components. 4. Calculate each component using sines and cosines. 5. Add the components in each direction. 6. To find the length and direction of the vector, use: session 2012/101323

EXAMPLES session 2012/101324

Example 1: Mail carrier’s displacement A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction to the next town. He then drives in a direction 60.0° south of east for 47.0 km to another town. Calculate the displacement from the post office. session 2012/101325

Example 2: Three short trips An airplane trip involves three journeys as shown below. The first journey is to the east for 620 km, the second journey is 45° southeast for 440 km and the third journey is 53° southwest for 550 km. Calculate the total displacement or resultant of the vectors. session 2012/101326

THANK YOU To be continue… session 2012/101327