CIVL%2131%)%Stacs% - Department of Civil Engineering - … Two.pdf · ·...

CIVL 2131 -‐ Sta-cs Basic Vector Opera-ons Addi-on and Subtrac-on of Coplanar Forces

Phrase Transla+on

It has been long known I haven't bothered to check the references

It is known I believe

It is believed I think

It is generally believed My group and I think

There has been some discussion Nobody agrees with me

It can be shown Take my word for it

It is proven It agrees with something mathema-cal

Objec-ves

  Understand and be able to u-lize vectors to represent forces

  Understand how forces represented as vectors can be added together to find the resultant of a series of coplanar forces

  Understand how a force can be resolved into components using the parallelogram law

January 20, 2010 Basic Vector Operations 2

Tools

  Law of Sines

  Law of Cosines

  Basic Trigonometry

  Pythagorean Theorem


Forces, Moments, and Vectors

  In Sta-cs we will be concerned with two basic elements of mechanics

  Forces – the tendency to cause movement along an axis and

 Moments – the tendency to cause movement around an axis



  Both Forces and Moments can be mathema-cally represented as vectors which allows us to have both a consistent representa-on and a convenient set of tools for manipula-ng the elements for analysis



  Vectors are mathema-cal representa-ons which have two components

 Magnitude and

 Direc-on



  If a quan-ty can be represented by magnitude only, then it is a scalar and doesn’t follow the rules of vector manipula-on



  Vectors are convenient representa-on because we have both graphical and analy-cal tools to work with them

  Remember, a vector has both magnitude and direc-on



 Graphically, we can represent our vector quan--es (forces and moments) by using an arrow

  If we are going to use a graphical solu-ons, the length of the arrow will represent the magnitude of the quan-ty and the direc-on of the arrow will represent the direc-on of the quan-ty



  For example if we have two forces, Force 1 (F1) and Force 2 (F2) and they are ac-ng at some point in space, we could draw them



  From the drawing, we can see that F1 has a greater magnitude than F2

  This is only possible if we draw the vectors to scale



 We could describe the direc-ons as sort of up and right and more up than right but some right

 Not a lot of informa-on there



 While the representa-on is quite fine and the direc-ons are evident to someone looking at the drawing, a more consistent representa-on requires that we generate some reference



  The most convenient reference would one that would allow for consistent opera-ons on what was being represented

 One of the ways u-lized is by the placing of an axis on the system



 Now there is no fixed coordinate axis in space, what we are dealing with is a mathema-cal representa-on so we get to decide on where the axis are going to be and how they are going to be oriented



  Conven-onally, we choose an x axis that is parallel with the boZom of the page and with a posi-ve direc-on directed to the right.

  The posi-ve direc-on of the axis is from the origin to the axis label.



  Also by conven-on, we usually have the y axis parallel with the sides of the page and the + y from the origin to the axis label (upward)



 Don’t get confused if you see other axis configura-ons, the one that I have shown is just the most common



  This will allow us to describe the forces (vectors) in reference to the axis system.



 When you use vectors in your wriZen work, indicate that a variable is a vector by pu]ng a small arrow over the top of the symbol you are using to represent the vector



  You will probably see textbooks and presenta-ons using bold face type to represent vectors but when you write the work out, you don’t have that op-on so use the arrow.



  To describes the force we can use the reference system (x and y axis) that we have developed.



  Assuming that we have drawn the vectors to some scale, the length of the line represen-ng the vector would give the magnitude of the vector.



  If the length of the line represen-ng the vector was 2 inches long and the scale selected was 1 inch = 50 N, then the 2 inch vector would have a magnitude of 100 N



 Normally, we do not show the scale. Rather we label the vector with its magnitude. However, remember that there is a scale being used.



 We can also describe the direc-on of the vector in terms of the coordinate system used.

  Typically, we describe the direc-on by giving the angle the vector makes CCW from the +x axis


Adding Vector Quan--es

  If we use consistent references, we can use mathema-cal opera-ons to combine the effects of vector quan--es



  For example, if we could add these two forces together to see what the net effect of their ac-on(s) would be



 No-ce that we start by drawing the vectors on the coordinate reference plane we have chosen.



  A reasonable first guess at the magnitude of the combined forces might be to just add the magnitudes of the forces together

 What do you think the problem would be with doing that?



 Which direc-on would the resultant (the sum of the two forces) act in?



 With vectors that act at the same point, we have a simple method of performing the addi-on



 We choose one of the vectors to remain fixed

  In this case we will choose F1



 We then move the other vector (F2), keeping its direc-on, un-l its tail is at the head of the first (F1)



  Then we construct the resultant (F) by drawing a new vector from the tail of the sta-onary vector (F1) to the head of the vector we moved (F2)



  F1 and F2 are known as the components

of F

  F is known as the resultant of F1 and F2



  If you knew the magnitude and direc-on of F1 and F2, you could find the magnitude and direc-on of F using trigonometry and geometry

  This is because of the consistent representa-on



Page 22 in a review of this topic.


An Example Problem

January 20, 2010 Basic Vector Operations

F2-1. Determine the magnitude of the resultant force acting on the screw eye and its direction measured from the x-axis

39

An Example Problem

January 20, 2010 Basic Vector Operations

F2-3. Detemine the magnitude of the resultant force and its direction measured counterclockwise from the positive x-axis.

40

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