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

ENGR Introduction to Engineering2 Coordinate Systems (in 2 dimensions)

ENGR 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,  )

ENGR Introduction to Engineering4 Cartesian Coordinate System

ENGR Introduction to Engineering5 Cartesian Coordinate System

ENGR Introduction to Engineering6 Polar Coordinate System

ENGR Introduction to Engineering7 Polar Coordinate System

ENGR 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(  )

ENGR Introduction to Engineering9 Cartesian ↔ Polar

ENGR Introduction to Engineering10 Scalars and Vectors

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

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

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

ENGR 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.

ENGR 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

ENGR Introduction to Engineering16 Addition and Subtraction of Vectors

ENGR 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

ENGR Introduction to Engineering18 Addition and Subtraction Exercises

ENGR Introduction to Engineering19 Multiplication and Division of Vectors

ENGR 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 <  

ENGR Introduction to Engineering21 Multiplication and Division Exercises

ENGR Introduction to Engineering22 Forces

ENGR 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.

ENGR 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 

ENGR 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.

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

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

ENGR 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.

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

ENGR 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

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

ENGR 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.

ENGR 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.

ENGR Introduction to Engineering34 Free Body Diagram Examples

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