ME 221 Statics Lecture #5 Sections 2.6 – 2.7 ME 221 Lecture 5 1 Announcements • Quiz #1 - 15 minutes before the end of the lecture • HW #2 due Wednesday 9/10 2.22, 2.23, 2.25, 2.27, 2.29, 2.32, 2.37, 2.45, 2.47 & 2.50 • Exam # 1 will be on Wednesday 9/17 ME 221 Lecture 5 2 Summary of Last Lecture • Writing vector components in terms of base vectors • Generating a 3-D unit vector from any given vector ME 221 Lecture 5 3 •Resolving Vectors into Components Using Angle Notation •Nonorthogonal Bases; Linear Equations • Resolving vectors onto nonorthogonal directions • Setting up and solving linear systems of algebraic equations ME 221 Lecture 5 4 Resolving vector into components using angle notation y Ay A a O z Az Ax x b A=A sina sinb i+A cosa j+A sina cosb k ME 221 Lecture 5 5 Vector Components in Nonorthogonal Coordinate System y A: Using Trig: v sin(90 + a + b ) sin(q y b ) sin(q x a ) A Au Av Ay A Av qy b Au A qy-b qx 90+a+b u a Au sin(90 + a + b ) sin(q x a ) Av A sin(90 + a + b ) x Ax ME 221 sin(q y b ) Lecture 5 6 B: Using Vector Addition Case 1: One Base Vector Known y y B -A P P a b A x x When vector A is known, subtract A from P B = P-A ME 221 Lecture 5 7 Case 2: Two Directions Known y P a b x Write unit vectors: ME 221 Lecture 5 8 Write the components of P: and write the vector sum equation. Next, write the x and y component equations: x-components y-components Here, we have two equations in two unknowns, A and B. Solve the equations. ME 221 Lecture 5 9 For example, using the numerical values: P = 100 lb, a = 10º, b = 20º Set up the system of equations to solve: P cos β = 93.97 = A + 0.866 B P sin β = 34.20 = 0 A + 0.5 B x-components y-components Solving yields: B = 68.4 lb and A = 34.7 lb ME 221 Lecture 5 10 Linear Algebraic Systems Write the x- and y-component equations in matrix form as follows: Solve with your calculator. ME 221 Lecture 5 11 Summary • Be able to resolve a vector onto nonorthogonal directions • Write the matrix form of the x-, y-, and zcomponent equations • Be able to solve a 2 x 2 and 3 x 3 system of equations on your calculator ME 221 Lecture 5 12 Quiz #1 ME 221 Lecture 5 13