Technological Impacts

1
Statics
Using 2 index cards Create a structure or
system of structures that will elevate two
textbooks at least 1.5cm off your desk
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Statics
What is Statics? Branch of Mechanics that
deals with objects/materials that are stationary
or in uniform motion. Forces are
balanced. Examples 1. A book lying on a table
(statics) 2. Water being held behind a dam
(hydrostatics)
Kentucky Indiana Bridge
Chicago
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Dynamics
Dynamics is the branch of Mechanics that
deals with objects/materials that are
accelerating due to an imbalance of
forces. Examples 1. A rollercoaster executing
a loop (dynamics) 2. Flow of water from a hose
(hydrodynamics)
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• Total degrees in a triangle
• Three angles of the triangle below
• Three sides of the triangle below
• Pythagorean Theorem
• x2 y2 r2

180
A, B, and C
x, y, and r
B
r
y
HYPOTENUSE
A
C
x
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Trigonometric functions are ratios of the
lengths of the segments that make up angles.
r
y
Q
x
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For ltA below, calculate Sine, Cosine, and
Tangent
B
opposite adjacent
1 2
tan A
sin A
A
C
1 v3
tan A
v3 2
cos A
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Law of Cosines c2 a2 b2 2ab cos C Law of
Sines sin A sin B sin C a
b c
B
c
a
C
A
b

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• Scalar a variable whose value is expressed only
  as a magnitude or quantity
• Height, pressure, speed, density, etc.
• Vector a variable whose value is expressed both
  as a magnitude and direction
• Displacement, force, velocity, momentum, etc.
• 3. Tensor a variable whose values are
  collections of vectors, such as stress on a
  material, the curvature of space-time (General
  Theory of Relativity), gyroscopic motion, etc.

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• Properties of Vectors
• Magnitude
• Length implies magnitude of vector
• Direction
• Arrow implies direction of vector
• Act along the line of their direction
• No fixed origin
• Can be located anywhere in space

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Bold type and an underline F also identify vectors
Vectors - Description
Magnitude, Direction
Hat signifies vector quantity
F 40 lbs 45o
F 40 lbs _at_ 45o
direction
magnitude
40 lbs
45o
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Vectors Scalar Multiplication

1. We can multiply any vector by a whole number.
2. Original direction is maintained, new magnitude.

2
½
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Vectors Addition

1. We can add two or more vectors together.
2. 2 methods
3. Graphical Addition/subtraction redraw vectors
   head-to-tail, then draw the resultant vector.
   (head-to-tail order does not matter)

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Vectors Rectangular Components

1. It is often useful to break a vector into
   horizontal and vertical components (rectangular
   components).
2. Consider the Force vector below.
3. Plot this vector on x-y axis.
4. Project the vector onto x and y axes.

y
F
Fy
x
Fx
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Vectors Rectangular Components
This means vector F vector Fx
vector Fy Remember the addition of
vectors
y
F
Fy
x
Fx
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Unit vector
Vectors Rectangular Components
Vector Fx Magnitude Fx times vector i
F Fx i Fy j
Fx Fx i
i denotes vector in x direction
y
Vector Fy Magnitude Fy times vector j
F
Fy Fy j
Fy
j denotes vector in y direction
x
Fx
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Vectors Rectangular Components
Each grid space represents 1 lb force. What is
Fx? Fx (4 lbs)i What is Fy? Fy (3
lbs)j What is F? F (4 lbs)i (3 lbs)j
y
F
Fy
x
Fx
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Vectors Rectangular Components
If vector V a i b j c k then
the magnitude of vector V V
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Vectors Rectangular Components
What is the relationship between Q, sin Q, and
cos Q?
cos Q Fx / F Fx F cos Qi sin Q Fy /
F Fy F sin Qj
F
Fy
Q
Fx
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Vectors Rectangular Components
When are Fx and Fy Positive/Negative?
Fy
Fy
y
F
Fx
Fx -
F
x
F
F
Fx -
Fx
Fy -
Fy -
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Vectors Rectangular Components
Complete the following chart in your notebook
I
II
III IV
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Vectors

• Vectors can be completely represented in two
  ways
• Graphically
• Sum of vectors in any three independent
  directions
• Vectors can also be added/subtracted in either of
  those ways
• F1 ai bj ck F2 si tj uk
• F1 F2 (a s)i (b t)j (c u)k

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Vectors
A third way to add, subtract, and otherwise
decompose vectors Use the law of sines or
the law of cosines to find R.
R
45o
30o
F1
F2
105o
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Vectors

• Brief note about subtraction
• If F ai bj ck, then F ai bj
  ck
• Also, if
• F
• Then,
• F

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Resultant Forces
Resultant forces are the overall combination of
all forces acting on a body. 1) find sum
of forces in x-direction 2) find sum of forces
in y-direction 3) find sum of forces in
z-direction 3) Write as single vector in
rectangular components
R SFxi SFyj SFzk
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Resultant Forces - Example

• A satellite flies without friction in space.
  Earths gravity pulls downward on the satellite
  with a force of 200 N. Stray space junk hits the
  satellite with a force of 1000 N at 60o to the
  horizontal. What is the resultant force acting
  on the satellite?
• Sketch and label free-body diagram (all external
  and reactive forces acting on the body)
• Decompose all vectors into rectangular components
  (x, y, z)
• Add vectors

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Statics
Now on to the point

• Newtons 3 Laws of Motion
• A body at rest will stay at rest, a body in
  motion will stay in motion, unless acted upon by
  an external force
• This is the condition for static equilibrium
• In other wordsthe net force acting upon a body
  is
• Zero

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• Newtons 3 Laws of Motion
• Force is proportional to mass times acceleration
• F ma
• If in static equilibrium, the net force acting
  upon a body is
• Zero
• What does this tell us about the acceleration of
  the body?
• It is Zero

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• Newtons 3 Laws of Motion
• Action/Reaction

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Statics

• Two conditions for static equilibrium

Since Force is a vector, this implies
Individually.
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Two conditions for static equilibrium 1.
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Two conditions for static equilibrium Why isnt
sufficient?
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• Two conditions for static equilibrium
• About any point on an object,
• Moment M (or torque t) is a scalar quantity that
  describes the amount of twist at a point.
• M (magnitude of force perpendicular to moment
  arm) (length of moment arm) (magnitude of
  force) (perpendicular distance from point to
  force)

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• Two conditions for static equilibrium
• MP F x MP Fy x
• M (magnitude of force perpendicular to moment
  arm) (length of moment arm) (magnitude of
  force) (perpendicular distance from point to
  force)

F
F
P
P
x
x
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• Moment Examples
• Tension test apparatus unknown and reaction
  forces?
• If a beam supported at its endpoints is given a
  load F at its midpoint, what are the supporting
  forces at the endpoints?
• Find sum of moments about a or b.

Ra
Rb
Watch your signs identify positive
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• Moment Examples
• An L lever is pinned at the center P and holds
  load F at the end of its shorter leg. What force
  is required at Q to hold the load? What is the
  force on the pin at P holding the lever?
• What is your method for solving this problem?
  Remember,

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Trusses
Trusses A practical and economic solution to
many structural engineering challenges Simple
truss consists of tension and compression
members held together by hinge or pin joints
Rigid truss will not collapse
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Trusses
Joints Pin or Hinge (fixed)
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Trusses
Supports Pin or Hinge (fixed) 2
unknowns
Reaction in x-direction Reaction in y-direction
Rax
Ray
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Trusses
Supports Roller - 1 unknown
Reaction in y-direction only
Ray
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• Assumptions to analyze simple truss
• Joints are assumed to be frictionless, so
  forces can only be transmitted in the direction
  of the members.
• Members are assumed to be massless.
• Loads can be applied only at joints (or
  nodes).
• Members are assumed to be perfectly rigid.
• 2 conditions for static equilibrium
• Sum of forces at each joint (or node) 0
• Moment about any joint (or node) 0
• Start with Entire Truss Equilibrium Equations

41
Truss Analysis Example Problems 1. A force F is
applied to the following equilateral truss.
Determine the force in each member of the truss
shown and state which members are in compression
and which are in tension.
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Truss Analysis Example Problems 2. Using the
method of joints, determine the force in each
member of the truss shown. Assume equilateral
triangles.
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Static determinacy and stability Statically
Determinant All unknown reactions and forces
in members can be determined by the methods of
statics all equilibrium equations can be
satisfied. m 2j r (Simple
Truss) Static Stability The truss is rigid it
will not collapse.
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Conditions of static determinacy and stability of
trusses
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• Materials Lab Connections
• Tensile Strength Force / Area
• Compression is Proportional to 1 / R4
• Problem Sheet solutions due Monday