Unit 2 -Mechanics in Two Dimensions. Module 5 –lesson 1 Vectors Learning objectives 1) Define Vectors in Two Dimensions 2) Use Pythagorean theorem to find the net vector 3) Find the vector components 4) Apply Algebraic Addition of Vectors Focus Question How can you add forces in two dimensions? New Vocabulary components vector resolution Review Vocabulary vector: a quantity, such as position, that has both magnitude and direction. Vectors in Two Dimensions • You can add vectors by placing them tip-to-tail. • When you move a vector, do not change its length or direction. • The resultant of the vector is drawn from the tail of the first vector to the tip of the second vector. Vectors in Two Dimensions • If vectors A and B are perpendicular, use the Pythagorean theorem to find the magnitude of the resultant vector. Pythagorean Theorem R2 = A2 + B2 Vectors in Two Dimensions • If the angle between the vectors is not 90°, you can use the law of sines or the law of cosines. Law of sines Law of cosines R A B = = sin q sina sinb R2 = A2 + B2 - 2AB cos q Vectors in Two Dimensions Use with Example Problem 1. Problem Find the magnitude of the sum of two forces, one 20.0 N and the other 7.0 N, when the angle between them is 30.0°. Response SKETCH AND ANALYZE THE PROBLEM • Draw a vector diagram and add the vectors graphically. • List the knowns and unknowns. Draw the Place Drawvectors the initial resultant tip vectors. to tail. vector. Click continue. Click Clicktoto tocontinue. continue. ? 7.0 N 30.0° 20.0 N θ = 150.0° KNOWN A = 20.0 N B = 7.0 N θ = 180.0° − 30.0° = 150.0° UNKNOWN R=? θR = ? Vectors in Two Dimensions Use with Example Problem 1. Problem Find the magnitude of the sum of two forces, one 20.0 N and the other 7.0 N, when the angle between them is 30.0°. KNOWN A = 20.0 N B = 7.0 N θ = 180.0° − 30.0° = 150.0° UNKNOWN R=? θR = ? SOLVE FOR THE UNKNOWN • Use the law of cosines. R 2 A2 B 2 2 AB cos Response 20.0 N 7.0 N SKETCH AND ANALYZE THE PROBLEM • Draw a vector diagram and add the vectors graphically. • List the knowns and unknowns. 400 N2 49 N2 280 N2 0.866 2 220.0 N7.0 Ncos 150.0 R ? 7.0 N 20.0 N θ = 150.0° 2 691.49 N2 26.3 N EVALUATE THE ANSWER • This answer is consistent with this problem’s vector diagram, which shows that the resultant should indeed be slightly greater in magnitude than the 20.0-N force. Vector Components • A vector can be broken into its components, which are a vector parallel to the x-axis and another parallel to the y-axis. • The process of breaking a vector into its components is sometimes called a vector resolution. Vector Components Algebraic Addition of Vectors • Two or more vectors (A, B, C, etc.) may be added by first resolving each vector into its xand y-components. Algebraic Addition of Vectors • The x- and y-components are added to form the x- and y- components of the resultant: Rx = Ax + Bx + Cx Ry = Ay + By + Cy Algebraic Addition of Vectors • Use the Pythagorean theorem to find the magnitude of the resultant vector: R2 = Rx2 + Ry2 Algebraic Addition of Vectors • Use trigonometry to find the angle of the resultant vector: æRy æ q = tan æ æ æRx æ -1 +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem By Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest θB = 135° Response SKETCH AND ANALYZE THE PROBLEM • Establish a coordinate system. • Draw a vector diagram. Include the knowns and unknowns in your diagram. • Sketch the x-components and ycomponents of A and B. Bx θA = 270° A = 4.0 m Ay +x (East) +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem By Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest Response SKETCH AND ANALYZE THE PROBLEM • Use your vector diagram as needed. SOLVE FOR THE UNKNOWN • Find the x-components of A and B and add them to find the x-component of R. Ax 4.0 mcos 270 0 Bx 7.3 mcos 135 5.16 m Rx 5.16 m • Draw and label Rx on the vector diagram. Bx +x (East) Rx = −5.16 m A = 4.0 m Ay +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem By Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest Ry = 1.16 m Response SKETCH AND ANALYZE THE PROBLEM • Use your vector diagram as needed. SOLVE FOR THE UNKNOWN • Find the y-components of A and B and add them to find the y-component of R. Ay 4.0 msin 270 4.0 m By 7.3 msin 135 5.16 m Ry 1.16 m • Draw and label Ry on the vector diagram. Bx +x (East) Rx = −5.16 m A = 4.0 m Ay +y (North) Algebraic Addition of Vectors B = 7.3 m Use with Example Problem 2. Problem Add the following two vectors via the component method: A is 4.0 m south B is 7.3 m northwest R = 5.3 m Response θR = 167° SKETCH AND ANALYZE THE PROBLEM • Use your vector diagram as needed. SOLVE FOR THE UNKNOWN • R = 5.3 m at 13° north of west EVALUATE THE ANSWER • Check the answer graphically. +x (East) A = 4.0 m Quiz 1. How can vectors be added? A by placing them tip-to-tip B by placing them tail-to-tail C by placing them tip-to-tail D by rotating one so it is perpendicular to the other CORRECT Quiz 2. Can the Pythagorean theorem be used to calculate the length of R? A Yes, that method works for any two vectors. B Yes, because R is longer than A and B. C No, because the angle between A and B is not 90°. D No, because A and B are not the same length. CORRECT Quiz 3. During which process can a vector be broken into its x- and y-components? A axis separation C vector resolution CORRECT B vector breakdown D vector addition Quiz 4. Which can be used to find the sum of two vectors when the vectors are not perpendicular? A Pythagorean theorem C law of cosines B law of sines D either B or C CORRECT Quiz 5. What is the magnitude of the resultant of two perpendicular vectors with lengths 3 and 4? A B 5 7 CORRECT C 25 D 625