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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
January 20, 2010 Basic Vector Operations 3
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
January 20, 2010 Basic Vector Operations 4
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 5
Forces, Moments, and Vectors
Vectors are mathema-cal representa-ons which have two components
Magnitude and
Direc-on
January 20, 2010 Basic Vector Operations 6
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 7
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 8
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 9
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 10
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 11
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 12
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 13
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 14
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 15
Forces, Moments, and Vectors
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.
January 20, 2010 Basic Vector Operations 16
Forces, Moments, and Vectors
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)
January 20, 2010 Basic Vector Operations 17
Forces, Moments, and Vectors
Don’t get confused if you see other axis configura-ons, the one that I have shown is just the most common
January 20, 2010 Basic Vector Operations 18
Forces, Moments, and Vectors
This will allow us to describe the forces (vectors) in reference to the axis system.
January 20, 2010 Basic Vector Operations 19
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 20
Forces, Moments, and Vectors
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.
January 20, 2010 Basic Vector Operations 21
Forces, Moments, and Vectors
To describes the force we can use the reference system (x and y axis) that we have developed.
January 20, 2010 Basic Vector Operations 22
Forces, Moments, and Vectors
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.
January 20, 2010 Basic Vector Operations 23
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 24
Forces, Moments, and Vectors
Normally, we do not show the scale. Rather we label the vector with its magnitude. However, remember that there is a scale being used.
January 20, 2010 Basic Vector Operations 25
Forces, Moments, and Vectors
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
January 20, 2010 Basic Vector Operations 26
Adding Vector Quan--es
If we use consistent references, we can use mathema-cal opera-ons to combine the effects of vector quan--es
January 20, 2010 Basic Vector Operations 27
Adding 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
January 20, 2010 Basic Vector Operations 28
Adding Vector Quan--es
No-ce that we start by drawing the vectors on the coordinate reference plane we have chosen.
January 20, 2010 Basic Vector Operations 29
Adding Vector Quan--es
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?
January 20, 2010 Basic Vector Operations 30
Adding Vector Quan--es
Which direc-on would the resultant (the sum of the two forces) act in?
January 20, 2010 Basic Vector Operations 31
Adding Vector Quan--es
With vectors that act at the same point, we have a simple method of performing the addi-on
January 20, 2010 Basic Vector Operations 32
Adding Vector Quan--es
We choose one of the vectors to remain fixed
In this case we will choose F1
January 20, 2010 Basic Vector Operations 33
Adding Vector Quan--es
We then move the other vector (F2), keeping its direc-on, un-l its tail is at the head of the first (F1)
January 20, 2010 Basic Vector Operations 34
Adding Vector Quan--es
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)
January 20, 2010 Basic Vector Operations 35
Adding Vector Quan--es
F1 and F2 are known as the components
of F
F is known as the resultant of F1 and F2
January 20, 2010 Basic Vector Operations 36
Adding Vector Quan--es
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
January 20, 2010 Basic Vector Operations 37
Adding Vector Quan--es
Page 22 in a review of this topic.
January 20, 2010 Basic Vector Operations 38
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
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