Senior High School GENERAL PHYSICS 1 Quarter 1 – Module 2 VECTORS AND VECTOR ADDITION Department of Education ● Republic of the Philippines GENERAL PHYSICS 1 Alternative Delivery Mode Quarter 1 - Module 1: VECTORS AND SCALAR FORCES First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalty. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. 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Department of Education ● Republic of the Philippines Table of Contents How to learn from this Module ................................................................................................................ i Icons of this Module ................................................................................................................................... ii What I Know ................................................................................................................................................iii Lesson 1: VECTORS AND SCALAR FORCES What I Need to Know..................................................................................................... 1 What’s In …………………………………………………………………………1 What’s New: (Activity 1 Graphical representation of vectors) ......................... 2 What Is It ........................................................................................................................... 2 What Is It (Discussion) .................................................................................................. 4 What’s More: (Activities) ............................................................................................. 4 Lesson 2: RESULTANT OF VECTORS What’s In............................................................................................................................ 4 What I Need to Know..................................................................................................... 4 What Is It ( Discussion) .............................................................................................. ..5 What’s More: (Example: Social Media Scenario) .............................................. .9 What I Have Learned..................................................................................................... 10 Assessment: (Post-Test)………………………………………………………………………..11 Key to Answers......................................................................................................................................... 12 How to Learn from this Module To achieve the objectives cited above, you are to do the following: • Take your time reading the lessons carefully. • Follow the directions and/or instructions in the activities and exercises diligently. • Answer all the given tests and exercises. Icons of this Module What I Need to Know This part contains learning objectives that are set for you to learn as you go along the module. What I know This is an assessment as to your level of knowledge to the subject matter at hand, meant specifically to gauge prior related knowledge This part connects previous lesson with that of the current one. What’s In What’s New An introduction of the new lesson through various activities, before it will be presented to you What is It These are discussions of the activities as a way to deepen your discovery and understanding of the concept. What’s More These are follow-up activities that are intended for you to practice further in order to master the competencies. What I Have Learned Activities designed to process what you have learned from the lesson What I can do These are tasks that are designed to showcase your skills and knowledge gained, and applied into real-life concerns and situations. II What I Know (Pretest) Multiple Choice. Select the letter of the best answer from among the given choices. Direction: Multiple choices. Select the letter that correspond to the best answer. 1. Which of the following statement is true in determining the resultant of two or more vectors acting together in the same direction? a. Add the given vectors and take the common direction b. Subtract the given vectors and take the common direction c. Get the product of the given vectors and take the two direction d. Add the given vectors and take the two direction 2. All statement are true about vectors, except; a. Vectors are quantities with specified magnitude and direction b. Vectors are quantities that have magnitude only c. Vectors of two or more can be add by component method d. Vectors can be add and take the common direction 3. A quantity that can be completely described by a single value called magnitude. a. Vector b. magnitude c. scalar d. velocity 4. The following statement is an example of scalar, except; a. An ant crawl on top of the table 15 cm. b. The troop of soldier walking 20km to northward direction c. A bunch of flowers weighing 5 kilograms d. A car runs fast 50km/h 5. A quantity that have magnitude and direction. a. Vector b. scalar c. displacement d. velocity 6. What is the vector some of 3 unit east and 5 unit east? a. 5 units east b. 6 units east c. 7 units east d. 8 units east 7. To find the resultant of two vectors with opposite direction, you need to; a. Get the difference and take the direction of the vector with the greater value b. Get the sum and take the direction of the vector with the greater value. c. Get the product and take the direction of the vector with the greater value d. Get the product and take the direction of the vector with the smaller value 8. Methods which are used to determine the vector sum or resultant of two or more vectors, except; a. Parallelogram b. polygon c. component d. symmetrical 9. Which of the following is true about graphical method in calculating resultant vector? a. The vectors to be added are drawn according to a convenient scale b. The vectors to be subtracted are drawn according to a convenient scale c. The product of vectors are drawn according to a convenient scale d. The vectors to be subtracted are directly drawn. 10. Since vectors are quantities with specified magnitudes and direction, the most appropriate representation of the vector is; a. Arrow b. line c. curve d. angle theta 11. The formula used in obtaining the magnitude of the resultant in component method. a. Pythagorean theorem b. Polygon method c. arc tangent d. Trigonometric 12. Which of the following is an example of vector quantity? a. 25km/h b. jumping 20 seconds c. 50 grams d. 5cm, east 13. What is the difference of the two vectors 8 units, east and 5 units, west? a. 3 units, west b. 3 units, east c. 4units west d. 4units, east 14. What is the exact sum of vectors A= 10cm, east and B= 5cm, east? a. 5 cm, east b. 5 cm, west c. 15cm, east d.15cm, west 15. The net displacement obtained from two or more vectors. a. Scalar b. resultant c. sum d. force Lesson 1 VECTORS AND SCALAR FORCES Physical quantities are all around us, the number of hours we spend in school or in our work, the speed and direction of the jeepney that we ride on everyday, and the amount of food that we buy. How to describe accurately these physical quantities that involved? Physical quantities are divided into two groups, the first group consists of length, area, volume, and speed while the second group consists of force, acceleration, and velocity. Which group do you think gives a clearer picture of quantities? When we describe the speed of the wind, we say it 69 kmh; but when we talk about its velocity, we express it as 69 kmh, North of West of Manila. Could you find any differences between these two quantities? Which gives an accurate and precise description of the wind? Why? This module will present information regarding another physical quantity, force. This quantity is common thing to us. Pulling a table, lifting heavy baggage, moving a chair, or opening a window involves force. Pushing a table requires a greater exertion of the muscles than pushing a chair. Why is this so? Why is it easier to transport a block of wood on a cart with wheels than to carry it on one’s shoulder? In this module you will learn that force is a vector. A vector is a quantity that includes information about the size, strength and direction. What I Need to Know At the end of this module, you should be able to: 1. Differentiate vector and scalar quantities (STEM_GP12V-Ia-8); 2. Perform addition of vectors (STEM_GP12V-Ia-9); 3. Rewrite a vector in component form (STEM_GP12V-Ia-10) What’s In There are physical quantities in physics which are represented by magnitudes and units. When we say that the mass of an umbrella is 3 kilograms, such information gives the magnitude and unit of mass. This quantity is called scalar. Mass therefore is an example of scalar quantity. Other quantities however cannot be completely specified by a magnitude and unit alone. To describe the velocity of a car by saying that it is 50 kmh is incomplete. There is still a need to describe the motion of direction of the car. The type of quantities which contains direction is called the vector. What’s New Since vectors are quantities with specified magnitudes and direction, the most appropriate representation of the vectors is the arrow ( ). Why? This is because the length of the arrow with respect to some chosen scale indicates the magnitude while the arrowhead represents the direction of the vector. A vector is usually represented by a letter with an arrow above it ( A ) .The same letter without an arrow indicates the magnitude. What you will do? ACTIVITY 1: Graphical Representation of Vectors You will need Ruler protractor pencil meterstick Procedure 1. Walk around the four corners of your classroom. Record the direction of your paths. 2. Measure the lengths and the widths of the room in meters. 3. Record the measured lengths and widths with the specified direction. 4. Represent the quantities graphically by using a convenient scale. Questions: 1. What is the length? Width? 2. Give the direction of your path while walking along the lengths and widths of the room. 3. Represent your results graphically using a specified scale What Is It A scalar is a quantity that can be completely described by a single value called magnitude. Magnitude means the size or amount and always includes units of measurement. Sometimes a single number does not include enough information to describe a measurement. Giving complete directions would mean including instructions to go to two kilometres to the north, turn right, then go to two kilometres east. The information “Two kilometres to the north” is an example of a vector.(p110 Tom Hsu, Ph.D.) The direction of some vectors is given in terms used by weather forecaster, travellers, map readers and cartographers. The basic reference for angles is the following N W N S E W E North South East West NE NW SW SE Northeast Northwest Southwest Southeast S For example: we draw a vector for a wind blowing at 30km/h in the northeast direction 450. N 30km/h 45o W E S 2 SCALAR ADDITION Adding scalar quantities is similar to ordinary addition. We add together the quantities express in the same units. Look at the example. Example: If the mass (m1)=25g and another mass (m2)=50g their sum is m=m1 + m2 Thus, m= 25g + 50g m= 75g However, you must be careful about how the given magnitude are expressed. If there are two different units, you need to convert the unit before adding. Example: If mass (m1)= 25g and another mass (m2)= 5kg. What is the total mass of an object? Thus, convert g to kg so that, 25g x 1kg/1000g =25/1000 or 0.025kg Therefore, m = m1 + m2 =0.025 kg + 5kg =5.025kg (total mass of an object) VECTOR ADDITION Now that we know how to represent vectors graphically, we are now ready to add two or more vectors. For example, we are asked to determine the vector sum of the resultant of the following vectors: 1. A= 2 units, East 2. A= 2 units, East B= 4 units, East B= 4 units, West Solution: 2 units 4 units 2 units 4 units 2 units West 6 units East (R) Resultant ( R) Resultant NOTE: To determine the resultant of two or more vectors acting together in the same direction, add the given vectors and take the common direction. On the other hand, to find the resultant of two vectors with opposite direction, get the difference and take the direction of the vector with the greater value. What’s More ACTIVITY 2: ( Graphing of vectors). Represent the following vectors graphically: A: 40 units, 45o North of East B: 50 units, 30o South of West C: 40 units, 45o Counterclockwise from the (+) x-axis. D: 50 units, 30o clockwise from the (-) y-axis ACTIVITY 3: (Scalar addition) In separate paper, solve the following problem with complete solution. 1. If area A1=20cm2 and area A2=36cm2, the total is A= A1 + A2 2. If the mass (m1) = 15kg and the second mass (m2) = 250g. Find the total mass m= m1 + m2. Convert kg to grams. ACTIVITY 4. (Vector Addition) In separate paper. Solve the given vectors and find the resultant with exact direction. 1. A= 7 units, East 3. A= 2 units, East B= 3 units, East B= 8 units, West 2. A= 8 units, East 4. A= 3 units, East B= 4 units, West B= 9 units, West Lesson RESULTANT OF VECTORS 2 What’s In When you draw a force vector on a graph, distance along the x or y- axes represents the strength of the force in the x- and y- directions. A force at an angle has the same effect as two smaller forces aligned with the x- and y- direction. To determine the resultant of two or more vectors acting together in the same direction, add the given vectors and take the common direction. What I Need to Know At the end of this lesson, you should be able to: 1. Determine the vector sum of two or more vectors; 2. Apply the Pythagorean theorem formula in getting the resultant vector; 3. Determine the vector resultant using the component method 4 What Is It RESULTANT OF VECTORS How do you determine the resultant of vectors acting at certain angles? The following are the different methods which are used to determine the vector sum or resultant of two or more vectors. 1. Graphical Method a. Parallelogram b. Polygon method 2. Analytical or Mathematical method a. Trigonometrical method b. Component method In the graphical method, the vectors to be added are drawn according to a convenient scale. They also have to be drawn in their specified direction. These directions are usually indicated by an angle measured from a certain reference line. With the aid of protractor, vectors can be drawn in their respective direction and the resultant is then drawn. If the vectors are represented by rectangular coordinate system, the angle is measured with respect to the x-axis or the y-axis. In the analytical method, the vectors need not be drawn according to scale. A rough sketch is simply made to show their magnitude and direction. The figure is analysed mathematically with the application of mathematical procedures and formulas. Graphical Method (Parallelogram) The parallelogram method is a common way of adding two vectors. For example, vector A and vector B are two vectors to be added, they are drawn from a common point. The angle Φ (phi) is included angle between vector A and vector B. A parallelogram is then formed using the vectors A and B as two sides. Φ A R ΦΦ Φ Φ Φ 0 B The resultant R is the diagonal line of the parallelogram from the common point 0. The magnitude of the resultant is obtained by using the length and the established scale. The angle Φ (theta) relative to B is simply measured by a protractor. Sample Problem: (Used scientific calculator ) Carlito was observing an ant crawled along a tabletop. With a piece of chalk, he followed its path. He determine the ants displacements by using a ruler and protractor. The displacements were as followed; 4cm east and change its direction 7cm 450 north of east. R=11cm 7cm NE R= 11cm North of East 450 0 4cm E 5 The ant went 4cm east and change direction 7cm, 450 north of east (NE). The resultant displacement does not change. The order in which displacement vectors are taken does not affect the resultant (11cm NE). Analytical Method (Trigonometrical) We can also get the resultant of two vectors by analytical method that is the method of trigonometry. In the case of two perpendicular vectors, the included angle is equal to 900. The value of the resultant is determined by applying the Pythagorean theorem. Thus; R2 = A2 + B2 Example: What is the vector sum of 8cm and 5cm acting on point O at an angle of 600? Solution: R2 = A2 + B2 - 2AB cos (120) R2 = (8cm)2 + (5cm)2 - 2(8cm) (5cm) (-0.5) R2= 64cm2 + 25cm2 - 2(40cm2)(-0.5) R2= 64cm2 + 25cm2 – 2 (-20cm2) R2=64cm2 + 25cm2 + 40cm2 R2=129cm2 B=5cm R= 11.36cm 2 R = √129𝑐𝑚 600 23.180 1200 R = 11.36cm (The resultant vector) 0 A = 8cm To find the angle θ (theta) may be determined from the sine law; R B = sin θ Sin 1200 11cm 5cm = sin θ Sin 1200 (11cm)(sin θ) (5cm)(sin 1200) = 11cm 11cm Sin θ = 5 ( 0.866) 11 = 0.3936 Sin θ θ = arc sin 0.3936 θ = 23.180 Analytical (Component method) The component of a given vector makeup the set of vectors whose vector sum is the given vector. The following procedure are applied in adding several vectors in terms of their components: 1. Resolve the initial vectors into their components in the x and y directions 6 2. Add the component in the x direction to give Rx and add the components in the y direction to give Ry. The following formulas help explain the second procedure: Rx = x- component of R = Ax + Bx + Cx + …… = sum of the x-components Ry = y- component of R = Ay + By + Cy + …… = sum of the y-components 3. Find the magnitude and direction of the resultant R from the components Rx and Ry. The Pythagorean theorem is used , Thus; R = √𝑎2 + 𝑏 2 The direction of R can be found the values of the component by trigonometry. This is given by: Tan θ = Ry Rx or θ = arc tan Ry Rx The angle θ may be situated in any of the four quadrants depending on the directions of Rx and Ry. There are four possible cases and those are shown below; Rx Ry 1 2 3 4 How θ is measured Position of R st + + - + - + - 1 Quadrant 2nd Quadrant 3rd Quadrant 4th Quadrant Example: Given: Counterclockwice from (+) x-axis Clockwise from (-) x-axis Counterclockwise from (-) x-axis Clockwise from (+) x-axis A = 10 units, 300 counterclockwise from (+) x-axis B = 25 units, 450 clockwise from (-) x- axis y Solution: B A 450 300 x The figure shows A and B relatives to the rectangular coordinate system Ax = A cos 300 = 10 units (0.866) = 8.66 units Ay = A sin 300 = 10 units (0.5) = 5 units The component of B are similarly obtained; = B cos 450 = 25 units (0.707) = -17.68 units This is considered negative since Bx is to the left By = B sin 450 = 25 units (0.707) = +17.68 units Since By is upward, we consider it to be positive. Bx To summarize the components of A and B, the following data are given x-component y-component Ax = 8.66 units Ay = 5 units Bx = -17.68 units By = 17.68 units Therefore, to determine Rx and Ry, we have Rx = Ax + Bx = 8.66 units + (-17.68 units) = - 9.02 units The negative sign indicate that Rx is directed to the left Ry = Ay + By = 5 units + 17.68 units = +22.68 units The positive sign indicates that Ry is directed upward The magnitude of the resultant is obtained by using the following formula R =√(𝑅𝑥 )2 + (𝑅𝑦 )2 =√(−9.02 𝑢𝑛𝑖𝑡𝑠)2 + (22.68 𝑢𝑛𝑖𝑡𝑠)2 = √81.36 𝑢𝑛𝑖𝑡𝑠 + 514 𝑢𝑛𝑖𝑡𝑠 = √595.74 𝑢𝑛𝑖𝑡𝑠 R = 24.41 units The direction of the resultant is evaluated using the following formula Tan θ = 𝑅𝑥 𝑅𝑦 θ= arctan Ry Rx R Ry θ= arctan 22.68 units -9.02 units = arctan 2.51 68.310 θ = 68.31 measured clockwise from the (-) x-axis 0 0 Rx What’s More ACTIVITY 5: Graphical Method (Parallelogram). illustrate the given magnitude in the problem. Use separate paper and scientific calculator . Problem: A group of soldiers walked 15 km north from their camp, then covered 10 km due east. Note: Scale ( 1cm=1km). a. What was the total distance walked by the soldier? b. Determine the total displacement and exact direction of the travel. c. Give the clear illustration of the path. ACTIVITY 6: ( Analytical Method/ Trigonometrical). Calculate the vector sum and illustrate. Use separate paper and scientific calculator. Problem: What is the vector sum of 15cm East and 9cm North of east acting on point O at an angle of 800? Find: a. R resultant force b. direction of R c. angle of θ (theta) ACTIVITY 7: ( Component method) Determine the Resultant of two vectors using the component method and illustrate. Use separate paper and scientific calculator. Given: A = 15 units, 25o Counterclockwise from (+) x-axis B = 30 units, 40o clockwise from (-) x- axis 9 What I Have Learned A. Use the tail to tip method to add the given vectors 1. C= 6 cm, due north D= 8 cm, due west 2. A = 7 km, due east B = 3 km, due west 3. C = 4 cm, due south D = 3cm, due west 4. E = 10 km, northwest F = 20 km, northeast 5. G = 50 mm, northwest H = 5 mm, northwest Assessment: (Post-Test) Multiple Choice. Select the letter of the best answer from among the given choices. Direction: Multiple choices. Select the letter that correspond to the best answer. 1. Which of the following statement is true in determining the resultant of two or more vectors acting together in the same direction? a. Add the given vectors and take the common direction b. Subtract the given vectors and take the common direction c. Get the product of the given vectors and take the two direction d. Add the given vectors and take the two direction 2. All statement are true about vectors, except; a. Vectors are quantities with specified magnitude and direction b. Vectors are quantities that have magnitude only c. Vectors of two or more can be add by component method d. Vectors can be add and take the common direction 3. A quantity that can be completely described by a single value called magnitude. a. Vector b. magnitude c. scalar d.velocity 4. The following statement is an example of scalar, except; a. An ant crawl on top of the table 15 cm. b. The troop of soldier walking 20km to northward direction c. A bunch of flowers weighing 5 kilograms d. A car runs fast 50km/h 5. A quantity that have magnitude and direction. a. Vector b. scalar c. displacement d. velocity 6. What is the vector some of 3 unit east and 5 unit east? a. 5 units east b. 6 units east c. 7 units east d. 8 units east 7. To find the resultant of two vectors with opposite direction, you need to; a. Get the difference and take the direction of the vector with the greater value b. Get the sum and take the direction of the vector with the greater value. c. Get the product and take the direction of the vector with the greater value d. Get the product and take the direction of the vector with the smaller value 8. Methods which are used to determine the vector sum or resultant of two or more vectors, except; a. Parallelogram b. polygon c. component d. symmetrical 9. Which of the following is true about graphical method in calculating resultant vector? a. The vectors to be added are drawn according to a convenient scale b. The vectors to be subtracted are drawn according to a convenient scale c. The product of vectors are drawn according to a convenient scale d. The vectors to be subtracted are directly drawn. 10. Since vectors are quantities with specified magnitudes and direction, the most appropriate representation of the vector is; a. Arrow b. line c. curve d. angle theta 11. The formula used in obtaining the magnitude of the resultant in component method. a. Pythagorean theorem b. Polygon method c. arc tangent d. Trigonometric 12. Which of the following is an example of vector quantity? a. 25km/h b. jumping 20 seconds c. 50 grams d. 5cm, east 13. What is the difference of the two vectors 8 units, east and 5 units, west? a. 3 units, west b. 3 units, east c. 4units west d. 4units, east 14. What is the exact sum of vectors A= 10cm, east and B= 5cm, east? a. 5 cm, east b. 5 cm, west c. 15cm, east d.15cm, west 15. The net displacement obtained from two or more vectors. a. Scalar b. resultant c. sum d. force 11 Answer key Pretest Activity 4 1. 7units 1 .A 2. B 3. C 4. B 5. A 6. D 7. A 8. D 9. A 10. A 11.A 12. D 13.B 14. C 15. B ACTIVITY 2 3units 10 units east 2. 8units 4 Units 4nits east 3. 2units 8units 6units west 3units A. 9units 40units NE 6units west 45 0 Posttest B. 300 50 units SW C. y 40 units X D. 50 units SW 1 .A 2. B 3. C 4. B 5. A 6. D 7. A 8. D 9. A 10. A 11.A 12. D 13.B 14. C 15. B