ENGR 107 – Introduction to Engineering

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

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

3. ENGR 107 - Introduction to Engineering 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)

4. ENGR 107 - Introduction to Engineering Cartesian Coordinate System

5. ENGR 107 - Introduction to Engineering Cartesian Coordinate System

6. ENGR 107 - Introduction to Engineering Polar Coordinate System

7. ENGR 107 - Introduction to Engineering Polar Coordinate System

8. ENGR 107 - Introduction to Engineering 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)

9. ENGR 107 - Introduction to Engineering Cartesian ↔ Polar

10. ENGR 107 - Introduction to Engineering Scalars and Vectors

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

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

13. ENGR 107 - Introduction to Engineering Scalars and Vectors • Which are scalars and which are vectors? Time Acceleration Force Speed Distance Temperature Mass Velocity • Other examples?

14. ENGR 107 - Introduction to Engineering 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 y-directions, respectively, • And i and j are the unit vectors in the x- and y-directions, respectively.

15. ENGR 107 - Introduction to Engineering 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

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

17. ENGR 107 - Introduction to Engineering 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

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

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

20. ENGR 107 - Introduction to Engineering 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

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

22. ENGR 107 - Introduction to Engineering Forces

23. ENGR 107 - Introduction to Engineering 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 Engineering y F.cosq F FY F.sinq q x FX Forces F = |F| < q F = FXi + FYj Fx = F.cosq Fy = F.sinq

25. ENGR 107 - Introduction to Engineering Forces • The force, F, can be resolved into its two vector components, FX and FY. • FX = F.cosqi • FY = F.sinqj • 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.

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

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

28. ENGR 107 - Introduction to Engineering 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. • S F = 0 • The body is in static equilibrium.

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

30. ENGR 107 - Introduction to Engineering 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

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

32. ENGR 107 - Introduction to Engineering 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 Engineering Free-Body Diagram Steps for drawing a FBD: • Isolate the desired object from its surroundings. • Replace items cut free with appropriate forces. • Add known forces, including weight. • Establish a coordinate (xy) frame of reference. • Add geometric data.

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

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