Mechanics
One of the most important fields in engineering
General Areas
Some Courses at FIU
 Mechanical Engineering Curriculum
• EGN 3311 Statics
• EGN 3321 Dynamics
• EGN 3365 Materials in Engineering
• EMA 3702 Mechanics and Material Science
• EMA 3702L Mechanics and Materials Science Lab
• EML 4702 Fluid Dynamics
• EML 4711 Gas Dynamics
Scalars and Vectors
 A scalar is a quantity having magnitude, but no direction.
Having magnitude only a scalar may be positive or
negative. but has no directional characteristics.
 Common scalar quantities are length, mass, temperature,
energy, volume, and density.
 A vector is a quantity having both magnitude and
direction. A vector may be positive or negative and has a
specified direction in space. Common vector quantities are
displacement, force, velocity, acceleration, stress, and
momentum.
 A scalar quantity can he fully defined by a single parameter,
its magnitude, whereas a vector requires that both its
magnitude and direction be specified.
Scalars and Vectors
Scalars and Vectors
Scalars and Vectors
Vector Operations
Vector Operations
Vector Components
 Express:
 Components
 Magnitudes
 Sum
 Magnitude of Sum
Unit Vectors
 Express:
 Components
 Magnitudes
 Sum
 Magnitude of Sum
Example
 Two vectors have
magnitudes of A = 8 and B
= 6 and directions as
shown in the figure
 Find the resultant vector,
using
a. The parallelogram law
and
b. By resolving the vectors
into their x andy
components.
Example
 For the vectors:
 A = 3i- 6i + k
 B = 5i + j – 2k
 C =-2i + 4j + 3k
 Find the resultant vector and its magnitude.
Forces
 To the engineer, force is defined as an influence that causes a body
to deform or accelerate. For example:
 Push
 Pull
 Lift
 When the forces acting on a body are unbalanced, the body
undergoes an acceleration. For example:
 The propulsive force delivered to the wheels of an automobile can
exceed the frictional forces that tend to retard the automobile’s
motion so the automobile accelerates
 Similarly. the thrust and lift forces acting on an aircraft can exceed the
weight and drag forces, thereby allowing the aircraft to accelerate
vertically and horizontally.
Forces
 Forces commonly encountered in the majority of engineering
systems may be generally categorized as:
 A contact force
 Gravitational force
 Cable force
 Pressure force, or
 fluid dynamic force
Forces
Forces
Forces are vectors, so all the mathematical operations and
expressions that apply to vectors apply to forces
Forces
 Three coplanar forces act as shown in the figure. Find the
resultant force, its magnitude and its direction with respect
to the positive x-axis.
Stabilizing a Communications
Tower with Cables

Tall slender structures often incorporate
cables to stabilize them.

The cables, which are connected at various
points around the structure and along its
length. are connected to concrete anchors
buried deep in the ground.

Shown in Figure 4.16(a) is a typical
communications tower that is stabilized with
several cables On this particular tower each
ground anchor facilitates two cables that are
connected at a common point, as shown in
Figure 4.16(b).

The upper and lower cables exert forces of
15 kN and 25 kN respectively. and their
directions are 45 and 32 respectively, as
measured from the ground (Figure 4.16(c)).
What is the resultant force exerted by the
cables on the ground anchor?
FREE-BODY DIAGRAMS
 A free-Body diagram is a diagram that shows all external
forces acting on the body. As the term implies, a free-body
diagram shows only the body in question, being isolated or
“free” from all other bodies
 The body is conceptually removed from all:
 supports
 connections, and
 regions of contact with other bodies
 All forces produced by these external influences are
schematically represented on the free- body diagram.
Procedure for Constructing FreeBody Diagrams
 The following procedure should be followed when
constructing free-body diagrams:
1.
2.
3.
4.
5.
Identify the body you wish to isolate and make a simple
drawing of it.
Draw the appropriate force vectors at all locations of
supports, connections and contacts with other bodies
Draw a force vector for the weight of the body, unless the
gravitational force is to he neglected in the analysis.
Label all forces that are known with a numerical value and
those that are unknown with a letter.
Draw a coordinate system on. or near, the tree-body to
establish directions of the forces
FREE-BODY DIAGRAMS
FREE-BODY DIAGRAMS
 A body is in static or dynamic equilibrium if the vector sum
of all external forces is zero. Consistent with this definition,
the condition of equilibrium may be stated mathematically
as:
Simple Truss
Simple Truss
More Complex Truss
More Complex Truss
More Complex Truss
http://www.jhu.edu/~virtlab/bridge/bridge.htm
Stress
P

A
Stress is used to:
•To determine if a certain structure can withstand the forces applies to I
•To compare different materials
Stress
 This mathematical definition of normal stress is actually an
average normal stress,
 because there may be a variation of stress across the cross
section of the bar.
 Stress variations are normally present only near points where
the external forces are applied however.
 The stress equation may he used in the majority of stress
calculations without regard to stress variations.
P

A
Strain
 : strain


L
Strain is dimensionless but sometimes can be expressed in
mixed units like m/m
Hooke’s Law
F  kx
  E
 The second equation is
another form (More useful
for Material Engineering)
of Hooke’s Law
 E: modulus of Elasticity or
Young’s Module, obtained
experimentally

 : stress

 : strain
L
P

A
PL

AE
Stress-Strain Diagram
Homework
 For the following homework problems, use the general
analysis procedure of
1.
2.
3.
4.
5.
6.
7.
Problem Statement
Diagram.
Assumptions
governing equations
Calculations
Solution check
Discussion
Homework
 A 250-kg cylinder rests in a long channel as shown. Find the
forces acting on the cylinder by the sides of the channel.
Homework
 A 200 kg engine block hangs from a system of cables as
shown in the Figure. Find the tension in cables AB and AC.
Cable AB is horizontal.
Homework
 A 200-kg engine block hangs from a system of cables as
shown in the Figure. Find the Normal Stress and Axial
Deformation in cables AB and AC. The cables are 0.7m long
and have a diameter of 4 mm. The cables are steel with a
modulus of elasticity of E = 200 GPa.
Homework

Tall slender structures often incorporate
cables to stabilize them.

The cables, which are connected at various
points around the structure and along its
length. are connected to concrete anchors
buried deep in the ground.

Shown in Figure (a) is a typical
communications tower that is stabilized with
several cables On this particular tower each
ground anchor facilitates two cables that are
connected at a common point, as shown in
Figure 6(b).

The upper and lower cables exert forces of
15 kN and 25 kN respectively. and their
directions are 45 and 32 respectively, as
measured from the ground (Figure (c)).
What is the resultant force exerted by the
cables on the ground anchor?