ME 221 Statics

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ME 221 Staticswww.angel.msu.eduSections 2.2
2.5
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• Announcements
• ME221 TAs and Help Sessions
• Chad Stimson stimson1_at_msu.edu
• Homework grading help room
• Tuesdays Thursdays 8am to 1pm 1522EB
• Jimmy Issa jimmy_at_msu.edu
• Quiz exam grading help room
• Tuesdays Thursdays 1pm to 5pm 2415EB

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Announcements

• HW1 Due on Friday, May 21
• Chapter 1 - 1.1, 1.3, 1.4, 1.6, 1.7
• Chapter 2 2.1, 2.2, 2.11, 2.15, 2.21

• Quiz 1 on Friday, May 21

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Last Lecture

• Chapter 1 Basics

• Vectors, vectors, vectors

• Law of Cosines

• Law of Sines

• Drawing vector diagrams

• Example 1 Addition of Vectors

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Law of Cosines

• This will be used often in balancing forces

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Law of Sines

• Again, start with the same triangle

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Example
Note resultant of two forces is the vectorial
sum of the two vectors
45o
25o
300 lb
200 lb
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155o
200 lb
? 90o25o-a
a
25o
R
45o
300 lb
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Scalar Multiplication of Vectors

• Multiplication of a vector by a scalar simply
  scales the magnitude with the direction unchanged

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Forces

• Review definition

• Shear and normal forces

• Resultant of coplanar forces

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Characteristics of a Force

• Its magnitude
• denoted by F

• Its direction

• Its point of application
• important when we discuss moments later

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Further Categorizing Forces

• Internal or external
• external forces applied outside body

• A section of the body exposes internal body

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Shear and Oblique

• Shear internal force has line of action contained
  in cutting plane

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Oblique Internal Forces

• Oblique cutting planes have both normal and shear
  components

Where N S P
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Transmissibility

• A force can be replaced by a force of equal
  magnitude provided it has the same line of action
  and does not disturb equilibrium

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Weight is a Force

• Weight is the force due to gravity

• W mg
• where m is mass and g is gravity constant
• g 32.2 ft/s2 9.81 m/s2

• English and metric

• Weight lb or N
• Mass slugs or kg

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Resultant of Coplanar Forces
A bodys motion depends on the resultant of all
the forces acting on it
In 2-D, we can use the Laws of Sines and Cosines
to determine the resultant force vector

• In 3-D, this is not practical and vector
  components must be utilized
• more on this later

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Perpendicular Vectors
y
Ax
A
Ay
Ay
qx
Ax
x
Ax is the component of vector A in the x-direction
Ay is the component of vector A in the y-direction
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Vector Components

• Vector components are a powerful way to represent
  vectors in terms of coordinates.

where
Ax A cos qx
Ay A cos qy
A sin qx
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Vector Components (continued)
Ax A cos qx
cos qx Ax / A
?x ?y
Ay A cos qy
cos qy Ay / A
A sin qx
?x and ?y are called direction cosines
?x2 ?y2 1
Note To apply this rule the two axes must be
orthogonal
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Summary

• External forces give rise to
• tension and compression internal forces
• normal and shear internal forces

• Forces can translate along their line of action
  without disturbing equilibrium

• The resultant force on a particle is the vector
  sum of the individual applied forces

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3-D Vectors Base Vectors

• Rectangular Cartesian coordinates (3-D)

• Unit base vectors (2-D and 3-D)

• Arbitrary unit vectors

• Vector component manipulation

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3-D Rectangular Coordinates

• Coordinate axes are defined by Oxyz

Coordinates can be rotated any way we like, but
...

• Coordinate axes must be a right-handed coordinate
  system.

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Writing 3-D Components

• Component vectors add to give the vector

Also,
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3-D Direction Cosines

• The angle between the vector and coordinate axis
  measured in the plane of the two

Where lx2ly2lz21
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Unit Base Vectors

• Associate with each coordinate, x, y, and z, a
  unit vector (hat). All component calculations
  use the unit base vectors as grouping vectors.

Now write vector as follows
where Ax Ax
Ay Ay
Az Az
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Vector Equality in Components

• Two vectors are equal if they have equal
  components when referred to the same reference
  frame. That is

if
Ax Bx , Ay By , Az Bz
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Additional Vector Operations

• To add vectors, simply group base vectors

• A scalar times vector A simply scales all the
  components

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General Unit Vectors

• Any vector divided by its magnitude forms a unit
  vector in the direction of the vector.

• Again we use hats to designate unit vector

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Position Vectors in Space

• Points A and B in space are referred to in terms
  of their position vectors.

A

• Relative position defined by the difference

B
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Vectors in Matrix Form

• When using MatLab or setting up a system of
  equations, we will write vectors in a matrix form

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Summary

• Write vector components in terms of base vectors

• Know how to generate a 3-D unit vector from any
  given vector

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Example Problem