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ENGR 107 – Introduction to Engineering Coordinate Systems, Vectors, and Forces (Lecture #6) ENGR 107 - Introduction to Engineering 1 Coordinate Systems (in 2 dimensions) ENGR 107 - Introduction to Engineering 2 Coordinate Systems Cartesian Coordinate System Each point in the plane is specified by the perpendicular distance to the x-, and y- axes. P(x, y) Polar Coordinate System Each point in the plane is specified by the radial distance from the pole (or origin) and the angle to the horizontal axis. P(r, q) ENGR 107 - Introduction to Engineering 3 Cartesian Coordinate System ENGR 107 - Introduction to Engineering 4 Cartesian Coordinate System ENGR 107 - Introduction to Engineering 5 Polar Coordinate System ENGR 107 - Introduction to Engineering 6 Polar Coordinate System ENGR 107 - Introduction to Engineering 7 Cartesian ↔ Polar For a point P specified in the Cartesian Coordinate System: P(x, y) Polar Coordinate System: P(r, q) r2 = x2 + y2 → r = sqrt[ x2 + y2 ] q = arctan( y / x ) x = r.cos(q) y = r.sin(q) ENGR 107 - Introduction to Engineering 8 Cartesian ↔ Polar ENGR 107 - Introduction to Engineering 9 Scalars and Vectors ENGR 107 - Introduction to Engineering 10 Scalars and Vectors A scalar is a physical quantity that possesses only magnitude. ENGR 107 - Introduction to Engineering 11 Scalars and Vectors A vector is a physical quantity that possesses both magnitude and direction. ENGR 107 - Introduction to Engineering 12 Scalars and Vectors Which are scalars and which are vectors? Time Acceleration Force Speed Distance Temperature Mass Velocity Other examples? ENGR 107 - Introduction to Engineering 13 Vectors In the Cartesian Coordinate System A = AXi + AYj where A is the vector quantity, AX and AY are the magnitudes of the rectangular components in the x- and ydirections, respectively, And i and j are the unit vectors in the x- and ydirections, respectively. ENGR 107 - Introduction to Engineering 14 Vectors In the Polar Coordinate System A=A<q where A is the vector quantity, A is the magnitude (a scalar quantity) and q is the angle (with respect to the x-axis) note: A = |A| = magnitude of A ENGR 107 - Introduction to Engineering 15 Addition and Subtraction of Vectors ENGR 107 - Introduction to Engineering 16 Addition and Subtraction Vectors should be written in rectangular form. Cannot add or subtract vectors directly when written in polar form. Add the x- and y- components independently. R=A+B Rx = Ax + Bx Ry = Ay + By R = Rxi + Ryj A = Axi + Ayj B = Bxi + Byj ENGR 107 - Introduction to Engineering 17 Addition and Subtraction Exercises ENGR 107 - Introduction to Engineering 18 Multiplication and Division of Vectors ENGR 107 - Introduction to Engineering 19 Addition and Subtraction Vectors should be written in polar form. More difficult to multiply and divide vectors when written in rectangular form. Multiply the magnitudes and add the angles. R=A.B R=A.B qR = qA + qB R = R < qR A = A < qA B = B < qB ENGR 107 - Introduction to Engineering 20 Multiplication and Division Exercises ENGR 107 - Introduction to Engineering 21 Forces ENGR 107 - Introduction to Engineering 22 Forces A force is an action, a push or a pull, that tends to change the motion of the body acted upon. A force has both magnitude and direction Thus, it is a vector. A force may be moved along its line of action without altering the external effect. ENGR 107 - Introduction to Engineering 23 Forces y F = |F| < q F.cosq F F = F Xi + F Yj FY F.sinq q FX Fx = F.cosq Fy = F.sinq x ENGR 107 - Introduction to Engineering 24 Forces The force, F, can be resolved into its two vector components, FX and FY. FX = F.cosq i FY = F.sinq j The combined effect of the vector components of a force, FX and FY, applied to a body is equivalent to the net effect of the force F applied to the body. ENGR 107 - Introduction to Engineering 25 Mechanics The study of forces acting on physical bodies. ENGR 107 - Introduction to Engineering 26 Statics and Dynamics Branches of mechanics concerned with the analysis of forces on rigid bodies. ENGR 107 - Introduction to Engineering 27 Statics and Dynamics Statics is the study of balanced forces on a body resulting in the body remaining at rest or moving with a constant velocity. SF=0 The body is in static equilibrium. ENGR 107 - Introduction to Engineering 28 Statics and Dynamics Dynamics is the study of unbalanced forces on a body resulting in an acceleration. S F = ma ENGR 107 - Introduction to Engineering 29 Static Equilibrium A body will be in static equilibrium when the sum of all external forces and moments acting on the body is zero. Conditions of static equilibrium: S FX = 0 S FY = 0 S MP = 0 ENGR 107 - Introduction to Engineering 30 Statics To implement the analysis of a rigid body in static equilibrium, one must first draw a Free Body Diagram (FBD). ENGR 107 - Introduction to Engineering 31 Free-Body Diagrams A Free-Body Diagram (FBD) is a sketch of the body, or a portion of the body, and all of the forces acting upon the body. The body is “cut free” from all others, and only forces that act upon it are considered. Must have an understanding of the types of reactions that may occur at supports and connectors. ENGR 107 - Introduction to Engineering 32 Free-Body Diagram Steps for drawing a FBD: 1. Isolate the desired object from its surroundings. 2. Replace items cut free with appropriate forces. 3. Add known forces, including weight. 4. Establish a coordinate (xy) frame of reference. 5. Add geometric data. ENGR 107 - Introduction to Engineering 33 Free Body Diagram Examples ENGR 107 - Introduction to Engineering 34 Statics Examples To include only analysis of forces. Moments will be discussed later. ENGR 107 - Introduction to Engineering 35