ME 205 STATICS

Assoc. Prof. Dr. Serkan Dağ

Section 1
ME 205 Section 1
Instructor:
Assoc. Prof. Dr. Serkan DAĞ (B318)
Schedule: Monday, 09:40-11:30, B103
Wednesday, 14:40-15:30, B103
Textbook: Engineering Mechanics, Statics by
R.C. Hibbeler (TA351.H5 2007)
Web Site:
http://www.me.metu.edu.tr/courses/me205
Choose the link for Section 1

Assoc. Prof. Dr. Serkan Dağ 2

ME 205 Section 1
 Grading: 2 Midterms (22.5 % Each)
Final (40 %)
Homework (10 %)
Attendance (5 %)
 Office Hours of Dr. Dağ:
Monday, 11:40–12:30, B 318
 Reference Books:
◦ Engineering Mechanics, Statics by I.H. Shames
(TA350.S491)
◦ Engineering Mechanics, Statics by J.L. Meriam and
L.G. Kraige (TA350.M458 2007)
◦ Vector Mechanics for Engineers, Statics by F.P. Beer
and E.R. Johnston Jr. (TA350.B3556 2009)

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COURSE
OUTLINE
1) General Principles
2) Force Vectors
3) Equilibrium of a Particle
4) Force System Resultants
5) Equilibrium of a Rigid Body
6) Structural Analysis
7) Internal Forces
8) Friction
9) Center of Gravity and Centroid
10) Moments of Inertia

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Chapter 1 General Principles

The design of this rocket and the gantry structure

requires a basic knowledge of both statics
and dynamics, which form the subject matter
of engineering mechanics.

Mechanics can be defined as that branch of physical sciences concerned

with the state of rest or motion of bodies that are subjected to the action of
forces.

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Branches of Mechanics
M e c h a n ic s

R ig id B o d y M e c h a n ic s D e f o r m a b le B o d y M e c h a n ic s F lu id M e c h a n ic s
( T h in g s t h a t d o n o t c h a n g e s h a p e ) ( T h in g s t h a t d o c h a n g e s h a p e ) M E 305, M E 306
M E 206

S t a t ic s D y n a m ic s I n c o m p r e s s i b le C o m p r e s s i b le
M E 205 M E 208

• In this course we consider only rigid-body mechanics since it forms a suitable

basis for the design and analysis of many types of structural, mechanical or
electrical devices encountered in engineering.

• Statics deals with the equilibrium of bodies, that is, those that are either at rest or
move with a constant velocity.

• Dynamics is concerned with the accelerated motion of bodies.

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Basic Quantities
 The following four quantities are used throughout
mechanics
 Length. Length is needed to locate the position of a
point in space and thereby describe the size of a physical
system.
 Time. Time is a component of the measuring system
used to sequence events, to compare the durations of
events and the intervals between them, and to quantify
the motions of objects. Although, the principles of
statics are time independent, this quantity does play an
important role in the study of dynamics.

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Basic Quantities (Cont’d)
 Mass. Mass can be thought of as the quantity of matter in a body. It
is the total sum of the masses of all the sub-atomic particles in a
body.
 Matter occupies space and can be solid, liquid or gaseous.
 Unless mass is added to or removed from a body, the mass of that
body never varies. It is constant under all conditions. There are as
many atoms in a kilogram of metal on the moon as there are in the
same kilogram of metal on earth. However, that kilogram of metal
will weigh less on the moon than on the earth.
 The basic unit of mass is kilogram (kg). The most commonly used
multiple of this basic unit is the tonne (1000 kg) and the most
commonly used sub-multiple is the gram (0.001 kg).
 This property manifests itself as a gravitational attraction between
two bodies and provides a quantitative measure of the resistance of
the matter to a change in velocity.

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Basic Quantities (Cont’d)
 Force. Force is considered as a “push” or “pull” exerted by one
body on another. This interaction can occur when there is direct
contact between the bodies, or it can occur through a distance when
the bodies are physically separated. Examples of the latter type
include gravitational, electrical and magnetic forces. In any case, a
force is completely characterized by its magnitude, direction and
point of application.
 A force or a system of forces cannot be seen; only the effect of a
force or a system of forces can be seen.
 A force can change or try to change the shape of an object.
 A force can move or try to move a body that is at rest. If the
magnitude of the force is sufficiently great it will cause the body to
move in the direction of the application of the force. If the force is
of insufficient magnitude to overcome the resistance to movement
it will still try to move the body albeit unsuccessfully.

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Idealizations
 Models or idealizations are used in mechanics in order
to simplify the application of the theory.
 Particle. A particle has a mass, but a size that can be
neglected. For example, the size of the earth is
insignificant compared to the size of its orbit, and
therefore the earth can be modeled as a particle when
studying its orbital motion. When a body is idealized as
a particle, the principles of mechanics reduce to a rather
simplified form since the geometry of the body will not
be involved in the analysis of the problem.

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Idealizations (Cont’d)
 Rigid Body. A rigid body can be considered as a
combination of a large number of particles in which all
the particles remain at a fixed distance from one another
both before and after applying a load. As a result, the
material properties of any body that is assumed to be
rigid will not have to be considered when analyzing the
forces acting on the body. In most cases the actual
deformations occuring in structures, machines,
mechanisms and the like are relatively small, and the
rigid body assumption is suitable for analysis.

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Idealizations (Cont’d)
 Concentrated Force. A concentrated force represents
the effect of a loading which is assumed to act at a point
on a body. We can represent a load by a concentrated
force provided the area over which the load is applied is
very small compared to the overall size of the body. An
example would be contact force between a wheel and
the ground.

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Newton’s Three Laws of Motion
 The entire subject of rigid-body mechanics is formulated
on the basis of Newton’s three laws of motion, the
validity of which is based on experimental observation.
They apply to the motion of a particle as measured from
a nonaccelerating reference frame.
 First Law. A particle originally at rest, or moving in a
straight line with constant velocity, will remain in this
state provided the particle is not subjected to an
unbalanced force.

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Newton’s Three Laws of Motion
 Second Law. A particle acted upon by an unbalanced
force F experiences an acceleration a that has the same
direction as the force and a magnitude that is directly
proportional to the force. If F is applied to a particle of
mass m, this law may be mathematically expressed as

F = ma

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Newton’s Three Laws of Motion
 ThirdLaw. The mutual forces of action and reaction
between two particles are equal, opposite and collinear.

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Newton’s Law of Gravitational
Attraction
 Shortly after formulating his three laws of motion,
Newton postulated a law governing the gravitational
attraction between any two particles. Stated
mathematically
m1m2
F G 2
r
where
F = force of gravitation between two particles
G = universal constant of gravitation according to
experimental evidence, G = 66.73(10-12) m3/(kg . s2)
m1, m2 = mass of each of the two particles
r = distance between the two particles

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Weight
 Gravitational forces exist between every pair of bodies.
On the surface of the earth the only gravitational force
of appreciable magnitude is the force due to the
attraction of the earth. Consequently, this force, termed
the weight, will be the only gravitational force
considered in our study of mechanics.

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Weight (Cont’d)
 Let’s consider the weight W of a particle having a mass m. If
we assume the earth to be a nonrotating sphere of constant
density and having a mass Me, then if r is the distance
between the earth’s center and the particle we have

mM e
W G
r2
Letting yields,
 By comparison M e F = ma, we term g the acceleration due to
g  G 2to W  mg
r
gravity. (g = 9.80665 m/s2)
 Since it depends on r, it can be seen that the weight of a
body is not an absolute quantity. Instead, its magnitude is
determined from where the measurement was made. For
most engineering calculations, however, g is determined at
sea level.

Assoc. Prof. Dr. Serkan Dağ 18

Units of Measurement
 SI Units. The International System of units, abbreviated SI is a
modern version of the metric system.
 The four basic quantities – force, mass, length and time – are not all
independent from one another; in fact, they are related by Newton’s
second law of motion F = ma. Because of this, the units used to
measure these quantities cannot all be selected arbitrarily.

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The International System of Units
 The SI system of units will be used throughout the course.
 When a numerical quantity is either very large or very small, the
units used to define its size may be modified by using a prefix.
Some of the prefexes used in the SI system are shown in the table
given below.

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Rules for Use
 A symbol is never written with a plural “s,” since it may be confused with the
unit for second (s).
 When a quantity is defined by several units, the units are separated by a dot to
avoid confusion with prefix notation. For example,
N = kg . m/s2
m . s : (meter-second)
ms : (milli-second)
 The exponential power represented for a unit having a prefix refers to both the
unit and its prefix. For example,
m = (m )2 = m . m
 When performing calculations, represent the numbers in terms of their base or
derived units by converting all prefixes to powers of 10. The final result should
then be expressed using a single prefix. Also, after calculation, it is best to keep
numerical values between 0.1 and 1000; otherwise a suitable prefix should be
chosen.

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Rules for Use (Cont’d)

( 50 kN ) ( 60 nm ) = ( 50(10)3 N) ( 60(10)-9 m)
= 3000(10)-6 N.m = 3(10)-3 N.m
= 3 mN.m

 Compound prefixes should not be used; e.g. ks (kilo-micro-

second) should be expressed as ms.

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Numerical Calculations
 Dimensional Homogeneity. The terms of any equation used to
describe a physical process must be dimensionally homogeneous;
that is, each term must be expressed in the same units. For example
consider the following equation:
s = vt + at2 / 2
where s is the position in meters, m, t is time in seconds, s, v is
velocity in m/s and a is acceleration in m/s2.
 Since problems in mechanics involve the solution of dimensionally
homogeneous equations, the fact that all terms of an equation are
represented by a consistent set of units can be used as a partial
check for algebraic manipulations of an equation.

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Numerical Calculations (Cont’d)
 Significant Figures. The accuracy of a number is specified by the number
of significant figures it contains. A significant figure is any digit, including
a zero, provided it is not used to specify the location of the decimal point
for the number.
5604  contains 4 significant figures
34.52  contains 4 significant figures
 When numbers begin or end with zero, however, it is difficult to tell how
many significant figures are in the number. In order to clarify this
situation, the number should be reported using powers of 10. Using
engineering notation, the exponent is displayed in multiples of three in
order to facilitate conversion of SI units to those having an appropriate
prefix.
400 = 0.4(10)3  1 significant figure
0.00546 = 5.46(10)-3  3 significant figures

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Numerical Calculations (Cont’d)
 Rounding Off Numbers. For numerical calculations, the accuracy
obtained from the solution of a problem generally can never be
better than the accuracy of the problem data. But, often calculators
or computers involve more figures in the answer than the number
of significant figures used for the data. For this reason, a calculated
result should always be rounded off to an appropriate number of
significant figures.
 To convey appropriate accuracy, the following rules for rounding
off a number to n significant figures apply:
 If the n +1 digit is less than 5, the n + 1 digit and others following it
are dropped. For example, 2.326 and 0.451 rounded off to n = 2
significant figures would be
2.326  2.3
0.451  0.45

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Numerical Calculations (Cont’d)
 If the n + 1 digit is equal to 5 with zeros following it, then round off
the nth digit to an even number. For example 1.245(10)3 and 0.8655
rounded off to n = 3 significant figures become
1.245(10)3  1.24(10)3
0.8655  0.866
 If the n + 1 digit is greater than 5 or equal to 5 with any nonzero
digits following it, then increase the nth digit by 1 and drop the n +
1 digit and others following it. For example, 0.72387 and 565.5003
rounded off to n = 3 significant figures become
0.72387  0.724
565.5003  566

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Numerical Calculations (Cont’d)
 Calculations. As a general rule, to ensure accuracy of a final result
when performing calculations on a calculator, always retain a
greater number of digits than the problem data.
 In engineering, we generally round off final answers to three
significant figures since the data for geometry, loads and other
measurements are often reported with this accuracy.

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Examples
 Evaluate each of the following and express with SI units having an
appropriate prefix.
 (50 mN) (6 GN)
= (50 (10)-3 N) (6 (10)9 N)
= 300 (10)6 N2 (1 kN / (10)3 N) (1 kN / (10)3 N)
=300 kN2
 (400 mm) (0.6 MN)2
= (400 (10)-3 m) (0.6 (10)6 N)2 = (400 (10)-3 m) (0.36 (10)12) N2
= 144 (10)9 m.N2 = 144 Gm.N2
 45 MN3 / 900 Gg
= (45 (10)18 ) N3 / (900 (10)6 kg) = 0.05 (10)12 N3 / kg
= 0.05(10)12 N3 / kg (1 kN / 1000 N)(1 kN / 1000 N)(1kN / 1000 N)
= 0.05 (10)3 kN3 / kg = 50 kN3 / kg

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