1. ENGR 107 - Introduction to Engineering1 ENGR 107 – Introduction to Engineering Coordinate Systems, Vectors, and Forces (Lecture #6)

2. ENGR 107 - Introduction to Engineering2 Coordinate Systems (in 2 dimensions)

3. ENGR 107 - Introduction to Engineering3 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,  )

4. ENGR 107 - Introduction to Engineering4 Cartesian Coordinate System

5. ENGR 107 - Introduction to Engineering5 Cartesian Coordinate System

6. ENGR 107 - Introduction to Engineering6 Polar Coordinate System

7. ENGR 107 - Introduction to Engineering7 Polar Coordinate System

8. ENGR 107 - Introduction to Engineering8 Cartesian ↔ Polar For a point P specified in the  Cartesian Coordinate System: P(x, y)  Polar Coordinate System:P(r,  )  r 2 = x 2 + y 2 → r = sqrt[ x 2 + y 2 ]   = arctan( y / x )  x = r.cos(  )  y = r.sin(  )

9. ENGR 107 - Introduction to Engineering9 Cartesian ↔ Polar

10. ENGR 107 - Introduction to Engineering10 Scalars and Vectors

11. ENGR 107 - Introduction to Engineering11 A scalar is a physical quantity that possesses only magnitude. Scalars and Vectors

12. ENGR 107 - Introduction to Engineering12 Scalars and Vectors A vector is a physical quantity that possesses both magnitude and direction.

13. ENGR 107 - Introduction to Engineering13 Scalars and Vectors Which are scalars and which are vectors? TimeAcceleration ForceSpeed DistanceTemperature MassVelocity Other examples?

14. ENGR 107 - Introduction to Engineering14 Vectors In the Cartesian Coordinate System  A = A X i + A Y j  where A is the vector quantity,  A X and A Y are the magnitudes of the rectangular components in the x- and y- directions, respectively,  And i and j are the unit vectors in the x- and y- directions, respectively.

15. ENGR 107 - Introduction to Engineering15 Vectors In the Polar Coordinate System  A = A <   where A is the vector quantity,  A is the magnitude (a scalar quantity)  and  is the angle (with respect to the x-axis) note: A = |A| = magnitude of A

16. ENGR 107 - Introduction to Engineering16 Addition and Subtraction of Vectors

17. ENGR 107 - Introduction to Engineering17 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  R x = A x + B x  R y = A y + B y  R = R x i + R y j A = A x i + A y j B = B x i + B y j

18. ENGR 107 - Introduction to Engineering18 Addition and Subtraction Exercises

19. ENGR 107 - Introduction to Engineering19 Multiplication and Division of Vectors

20. ENGR 107 - Introduction to Engineering20 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  R = R <  R A = A <   B = B <  

21. ENGR 107 - Introduction to Engineering21 Multiplication and Division Exercises

22. ENGR 107 - Introduction to Engineering22 Forces

23. ENGR 107 - Introduction to Engineering23 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.

24. ENGR 107 - Introduction to Engineering24 Forces F = F X i + F Y j F = |F| <  x F y FXFX FYFY  F.cos  F.sin  F x = F.cos  F y = F.sin 

25. ENGR 107 - Introduction to Engineering25 Forces The force, F, can be resolved into its two vector components, F X and F Y.  F X = F.cos  i  F Y = F.sin  j The combined effect of the vector components of a force, F X and F Y, applied to a body is equivalent to the net effect of the force F applied to the body.

26. ENGR 107 - Introduction to Engineering26 Mechanics The study of forces acting on physical bodies.

27. ENGR 107 - Introduction to Engineering27 Statics and Dynamics Branches of mechanics concerned with the analysis of forces on rigid bodies.

28. ENGR 107 - Introduction to Engineering28 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.   F = 0  The body is in static equilibrium.

29. ENGR 107 - Introduction to Engineering29 Statics and Dynamics Dynamics is the study of unbalanced forces on a body resulting in an acceleration.   F = ma

30. ENGR 107 - Introduction to Engineering30 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:   F X = 0   F Y = 0   M P = 0

31. ENGR 107 - Introduction to Engineering31 Statics To implement the analysis of a rigid body in static equilibrium, one must first draw a Free Body Diagram (FBD).

32. ENGR 107 - Introduction to Engineering32 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.

33. ENGR 107 - Introduction to Engineering33 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.

34. ENGR 107 - Introduction to Engineering34 Free Body Diagram Examples

35. ENGR 107 - Introduction to Engineering35 Statics Examples To include only analysis of forces. Moments will be discussed later.