Chapter 1
Introduction to Statics
Branches of Mechanics
Mechanics
(Branch of physical sciences concerned with
state of rest or motion of bodies subjected to
the action of forces).
Rigid-body Mechanics
Things that do not change shape
Statics
Dynamics
Fluid Mechanics
Incompressible
Compressible
Deformable-body Mechanics
Things that do change shape
Why Study Statics?
• A fundamental subject for every form of mechanical
engineering (and every other branch of engineering that
has ever existed)
• Static equilibrium describes the conditions where all
forces are balanced (no acceleration). Many objects are
designed with intention that they remain in equilibrium.
e.g: An electric transmission tower
•
When the weight of the transmission line (a force) is applied, how
much force does each part (beam) of the tower carry? How much
support is needed on the ground?
Fundamentals: 4 Basic Quantities
•
Length (L)
Length is used to locate the position of a point in
space and thereby describes the size of a physical
system. Once a standard unit of length is defined, one
can then use it to define distances and geometric
properties of a body.
For example, position of a point P may be defined by
three lengths measured from a reference point, or
origin in three given directions. These lengths are
known as coordinates of point P. (x, y, z)
4 Basic Quantities (continue)
•
Mass (m)
Mass is a fundamental property of a physical system or
body; a numerical measure of its inertia; a measure of
the amount (quantity) of matter in the object. Mass
gives rise to the body's resistance to being accelerated
by a force. This property also manifests itself as a
gravitational attraction between two bodies.
•
Time (t)
Time is conceived as a sequence of events, hence the
duration of an event. In statics, all events are timeindependent.This quantity is important in the study of
dynamics.
4 Basic Quantities (continue)
•
Force (F)
The action exerted by one body on another. This
interaction can occur when there is direct contact
between the bodies (eg. pushing/pulling an object) or it
can occur through a distance when the bodies are
physically separated (eg. gravitational, magnetic and
electrical forces).
A force is completely characterized by its magnitude,
its direction and its point of application. The direction of
a force is defined by the line of action and the sense of
the force. The line of action is the infinite straight line
along which the force acts; characterized by the angle
it forms with some fixed or reference axis. The force
itself is represented by a (scaled) segment of that line.
The sense of force is indicated by an arrowhead.
Force: magnitude, direction, point
of application, sense and line of
action
Point of application
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4 Basic Quantities in Statics
Quantity
Dimensional
symbol
Unit
Symbol
Mass
m
kilogram
kg
Length
L
meter
m
Time
t
second
s
Force
F
newton
N = kgms−2
• Base unit: length (meter); time (second); mass (kilogram)
• Derived unit (from Newton’s 2nd Law, F = ma): force
(Newton)
• Acceleration of gravity: 9.81 m/s2
Fundamentals: Idealizations
Models or idealizations are used to simplify application
of theories and calculations in mechanics. 3 important
idealizations:
•
•
•
Particle
Rigid body
Concentrated force
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Idealizations (continue)
•
•
Particle: A particle has a mass but a size that can be
neglected (zero dimension) so it may be idealized as one
point in space. Geometry of a particle will not be involved
in the analysis of the problem.
Rigid body: collection of particles or points in which all
points remain at fixed distance from one another, both
before and after a force acts on the body. This means a
rigid body does not change shape (deform) when force acts
on it. We assume its length or volume remains the same
before and after force application. A rigid body has mass
and size.
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Idealizations (continue)
•
By contrast, a deformable (non-rigid) body will change
shape (dimension) when a force acts on it. So, we must
include material properties of the body when analyzing
its response to force. (For eg, if we apply equal pulling
force to steel, ceramic and rubber bar, each will produce
different amount of elongation).
•
Concentrated force: We assume the force acts at a
point on a body. A load can be represented by a
concentrated force, if the area over which the load is
applied is very small compared to the overall size of the
body. Example: contact force between a wheel and the
ground.
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Example of Idealization
Steel does not deform very much under load, so we can treat the
railroad wheel as a rigid body acted upon by the concentrated force
of the rail track.
Fundamentals: Newton’s 1st Law of Motion
If the resultant force acting on a particle is zero (F = 0), the
particle will remain at rest (if originally at rest), or will move
in a straight line at constant velocity (if originally in motion).
Equilibrium, ΣF = 0
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Fundamentals: Newton’s 2nd Law of Motion
If the resultant force acting on a particle is not zero (F ≠ 0), the
particle will have an acceleration proportional to the magnitude of
the resultant force and in the direction of this resultant force. This
law may be expressed as:
F = ma
F = resultant force acting on particle
m = mass of particle
a = acceleration of particle
F
a
Accelerated motion
Fundamentals: Newton’s 3rd Law of Motion
The mutual forces of action and reaction between two bodies
or particles in contact have equal magnitude, same line of
action (collinear) and opposite sense.
A
B
Force of A on B
Force of B on A
Action-reaction
Fundamentals: Newton’s Law of Gravity
Any two particles or bodies have a mutual attractive (gravitational)
force between them. Stated mathematically:
m1m2
F  G 2
r
where:
F
G
r
m1,m2
=
=

=
=
gravitational force between two particles
universal constant of gravitation, 66.73 × 10-12 m3/kg.s2
distance between the two particles
mass of each particle
Newton’s Law of Gravity and Weight
•
Attractive force of the earth exerted on a particle/body located
on the earth’s surface is defined as the weight W of the particle.
•
Assume M as mass of the earth, and R the distance between
the earth’s center and the particle, and introducing the constant:
GM
g 
R2
the magnitude of weight W of a particle of mass m is:

W  mg
Fundamentals: Weight and Mass
• Mass is a measure of quantity of matter that does not
change from one location to another.
• Weight, W refers to the gravitational attraction of the earth
on a body or quantity of mass. Since g depends on R, then
weight, W is not an absolute quantity. Instead, magnitude of
W depends on the elevation at which the mass is located.
• By comparison with F = ma, we can see g is the acceleration
due to gravity (g = 9.80665 = 9.81 m/s2).
Basic Quantities and Units of Measurement
Quantity
Dimensional
symbol
Unit
Symbol
Mass
M
kilogram
kg
Length
L
meter
m
Time
T
second
s
Force
F
newton
N = kgm/s2
• Base unit: length (meter); time (second); mass (kilogram)
• Derived unit (from Newton’s 2nd Law, F = ma): force
(Newton)
• Acceleration of gravity: 9.81 m/s2
Units of Measurement (continue)
1 newton (N) is equal to a force required to give 1 kilogram
of mass an acceleration of 1 m/s2 (N = kgm/s2).
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SI Units: Prefixes
•
For a very large or very small numerical quantity, units
can be modified by using a prefix.
•
Each represents a multiple or sub-multiple of a unit.
Example:
4 000 000 N = 4000 kN (kilo -newton)
= 4 MN (mega -newton)
0.005 m = 5 mm (milli -meter)

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SI Units: Prefixes
Exponential Form
Prefix
SI Symbol
1 000 000 000
109
giga
G
1 000 000
106
mega
M
1 000
103
kilo
k
0.001
10−3
milli
m
0.000 001
10−6
micro
μ
0.000 000 001
10−9
nano
n
Multiple
Sub-multiple
Principal SI Units Used in Mechanics
Quantity
Unit
Acceleration
Meter per second squared
Angle
Radian
Area
Square meter
…
m2
Density
Kilogram per cubic meter
…
kg/m3
Energy
Joule
J
Nm
Force
Newton
N
kgm/s2
Frequency
Hertz
Hz
s−1
Moment of a force
Newton-meter
…
Nm
Power
Watt
W
J/s
Pressure
Pascal
Pa
N/m2
Volume (solids)
Cubic meter
m3
Velocity
Meter per second
m/s
*From Table 1.2 Beer, Johnstons
Symbol
Formula
…
m/s2
rad
SI Units: Rules for Use
•
Quantities defined by several units which are multiples of
one another are separated by a dot to avoid confusion with
prefix notation.
•
Example, N = kgm/s2 = kgms−2. Also ms (metersecond) not to be confused with ms (millisecond).
•
Exponential power on a unit having a prefix refers to both
the unit and its prefix. Example: mm2 = (mm)2 = mmmm.
Likewise, μN2 = (μN)2 = μNμN.
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SI Units: Rules for Use (continue)
•
When doing calculations, represent the numbers in terms
of their base or derived units by converting all prefixes to
powers of 10. Give the final answer using a single prefix.
•
Example:
(50 kN)(60 nm)
=
=
=
=
(50 × 103 N)(60 × 10−9 m)
3000 × 10−6 Nm
3 × 10-3 Nm
3 mm.Nm
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Parallelogram Law for Addition of Forces
•
Vectors can be added using the parallelogram law.
•
This law states that two forces acting on a particle may be
replaced by a single force called their resultant, obtained by
drawing the diagonal of a parallelogram whose sides are
equal to the given forces A and B.
•
Parallelogram law is a variation of the triangle law. In both
cases we are putting vectors head to tail.
Principle of Transmissibility
This state that the conditions of equilibrium or of motion of a rigid
body will remain unchanged if a force acting at a given point of the
rigid body is replace by a force of the same magnitude and same
direction, but acting at a different point, provided that the two forces
have the same line of action.
F
A
F
B
Sliding a force along its line of action to a new point on a rigid body
does not change the body’s motion or equilibrium state.
Numerical Calculations
Dimensional Homogeneity
•
•
•
•
•
Each term must be expressed in the same units.
Example: s  vt 21 at 2. In SI unit, s is the position in meters,
t is the time (seconds), v is velocity in m/s and a is
acceleration in m/s2.
Regardless of how this equation is evaluated, the equation

maintains its dimensional homogeneity.
For example, in the form given, each of the three terms is
expressed in meters: [m, (m/s) s, (m/s2) s2]
Never mix SI units with US units.
Numerical Calculations
Significant Figures
•
Accuracy of a number is specified by the number of
significant figures it contains.
•
A significant figure is any digit including zero
e.g. 5604 and 34.52 have four significant numbers.
•
When numbers begin or end with zero, we make use of
prefixes to clarify the number of significant figures
e.g. 400 as one significant figure would be 0.4(103)
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Problem-Solving Procedure
 Carry out analysis as follows:
–
–
–
–
List all information stated in the problem.
Draw necessary diagram (free-body diagram)
Put all known data and dimensions on FBD
Write down relevant equations, make sure they are
dimensionally homogeneous.
– Always include units in your computation. Use prefix to
simplify answer.
– Check UNITS to see if your answer is logical or wrong. For
eg, if question asks for ‘force’, your answer should be in
Newton or kgms-2.
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