Ministry of Higher Education And Scientific Research Al-Huda University College Fuel and Energy Techniques Eng. Dep.
Engineering Mechanics
For First Stage
By Asst. Lect. Ibrahim Khudhur A. 2020-2021
Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
References:
1. "Vector Mechanics for Engineers Statics" Beer & Johnston 9th Edition.
2. ''Engineering Mechanics: Statics'' 7th Edition by Meriam, J. L., Kraige, L. G. published by Wiley.
3. ''Engineering Mechanics: Statics & Dynamics'' 12th Edition by Russell Hibbeler.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Contents of Chapter One
1.1 Introduction……………………...... 4 1.2 Basic Concepts……………...... 5 1.3 Units……………… ...... 7 1.4 Vectors and Scalars………………...... 10 1.5 Rigid-body Mechanics………………… ...... 11 1.5.1 Vector addition (parallelogram law)……………………………………… ...... 12 1.6 Two-Dimensional Force Systems…………………...... 16 1.7 Moment:……………… ...... 25 1.8 Couple…………………...... 30 1.9 Three-Dimensional Force Systems………………… ...... 32
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Chapter One Engineering Mechanics
1.1 Introduction
Mechanics is the physical science which deals with the effects of forces on objects. Mechanics is the oldest of the physical sciences. No other subject plays a greater role in engineering analysis than mechanics. Although the principles of mechanics are few, they have wide application in engineering. The principles of mechanics are central to research and development in the fields of vibrations, stability and strength of structures and machines. A thorough understanding of this subject is an essential prerequisite for work in these and many other fields.
The early history of this subject is synonymous with the very beginnings of engineering. The earliest recorded writings in mechanics are those of Archimedes (287–212 B.C.) on the principle of the lever and the principle of buoyancy. Stevinus (1548–1620) also formulated most of the principles of statics. The first investigation of a dynamics problem is credited to Galileo (1564–1642) for his experiments with falling stones. The accurate formulation of the laws of motion, as well as the law of gravitation, was made by Newton (1642–1727).
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.2 Basic Concepts
The following concepts and definitions are basic to the study of mechanics, and they should be understood at the beginning.
Space: is the geometric region occupied by bodies whose positions are described by linear and angular measurements relative to a coordinate system. For three- dimensional problems, three independent coordinates are needed. For two- dimensional problems, only two coordinates are required as shown in the figure.
Time: is the measure of the succession of events and is a basic quantity in dynamics. Time is not directly involved in the analysis of statics problems.
Mass: is a physical quantity, and it is defined as the amount of matter in an object, and it differs from weight in that it does not depend on the force of gravity, while weight depends on the force of gravity, so the weight changes with the change of location
Force: is the action of one body on another. A force tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus force is a vector quantity.
A particle: is a body of negligible dimensions. In the mathematical sense, a particle is a body whose dimensions are considered to be near zero so that we may analyze it as a mass concentrated at a point. We often choose a particle as a differential element of a body. We may treat a body as a particle when its dimensions are irrelevant to the description of its position or the action of forces applied to it.
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. This model is important because the material properties of - 5 -
Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
anybody that is assumed to be rigid will not have to be considered when studying the effects of forces acting on the body. In most cases, the actual deformations occurring in structures, machines, mechanisms, and the like are relatively small, and the rigid-body assumption is suitable for analysis. A rigid body does not deform under load.
Concentrated Force represents the effect of a loading that 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.
Newton's Three Laws of Motion: Engineering mechanics is formulated on the basis of Newton's three laws of motion. They may be briefly stated as follows: The first law states that an object at rest will stay at rest, and an object in motion will stay in motion unless acted on by a net external force. Mathematically, this is equivalent to saying that if the net force on an object is zero, then the velocity of the object is constant. See Fig. (1-1)
Figure 0-1: Newton's first law Second Law. A particle acted upon by an unbalanced force (F) experiences an acceleration that has the same direction as the force and a magnitude that is directly proportional to the force. Fig. (1-2) .If (F) is applied to a particle of mass m. This law may be expressed mathematically as:
Figure 0-2: Newton's second law. - 6 -
Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Third Law. Every action force has a reaction force, equal in magnitude and opposite in direction. See Fig (1-3).
Figure 0-3: Newton's third law. 1.3 Units
A standard quantity against which a quantity is measured [e.g. gram, meter, second, liter, pascal; which are units of the above quantities].
International System of Units (SI units): The internationally adopted system which defines or expresses all quantities in terms of seven basic units, the six used by chemists being:
Other quantities commonly used in engineering, and which have special names for the units derived from these basic units are:
Further quantities used in chemistry but without special names for the derived units are: area, m2; volume, m3; density, kg m-3.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
The sizes of these units are often unsuitable for some measurements and the decimal multiples, shown below with the name and symbol of the prefix, are used:
U.S. Customary: In the U.S. Customary System of units (FPS) length is measured in feet (ft), time in seconds (s), and force in pounds (lb), (see Table 1-1). The unit mass, called a slug, is derived from F = ma. Hence, 1 slug is equal to the amount of matter accelerated at 1 ft/s2 when acted upon by a force of 1 Ib (slug = lb· s2/ft).
Table 0-1: Systems of Units
Conversion of Units. Table 1-2 provides a set of direct conversion factors between FPS and SL units for the basic quantities.
Table 0-2: Conversion Factors
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.4 Vectors and Scalars All physical quantities in engineering mechanics are measured using either scalars or vectors.
Scalar: is any quantity in physics that has magnitude, but not a direction associated with it such as:
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Vector: is any quantity in physics that has both magnitude and direction. Vectors are typically illustrated by drawing an arrow (→ ↑ ← ↓) above the symbol. The arrow is used to convey direction and magnitude, such as:
A vector is shown graphically by an arrow. The length of the arrow represents the magnitude of the vector, and the angle (θ) between the vector and a fixed axis defines the direction of its line of action. The head or tip of the arrow indicates the sense of direction of the vector, see Fig. 1-4.
Figure 0-4: Direction of the vector.
1.5 Rigid-body Mechanics
• A basic requirement for the study of the mechanics of deformable bodies and the mechanics of fluids (advanced courses). • Essential for the design and analysis of many types of structural members, mechanical components, electrical devices, etc, encountered in engineering.
Engineering mechanics
- Deals with effect of forces on objects Mechanics principles used in vibration, spacecraft design, fluid flow, electrical, mechanical m/c design, etc.
Statics: deals with effect of force on bodies which are not moving.
Dynamics: deals with force effect on moving bodies.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.5.1 Vector addition (parallelogram law)
If we have a two vectors (A&B). These two vectors can be added to form a resultant vector R = A + B by using the ''parallelogram law''. Fig. (1-5)
To do this A & B are joined together by their tails. Parallel lines drawn from the head of each vector intersect at a common point to form a parallelogram.
The resultant R is the diagonal of the parallelogram which extends from the tail of A & B to the intersection point.
Figure 0-5: parallelogram law.
Magnitude and direction of the resultant (R): We can determine the magnitude of the resultant and the direction measured from the horizontal line by using the Sine law and the Cosine law. To solve a triangle is to find the lengths of each of its sides and all its angles. The sine rule is used when we are given either
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
a) two angles and one side, or
b) two sides and a non-included angle.
The cosine rule is used when we are given either
a) three sides or
b) two sides and the included angle.
Sine low:
Cosine low:
푐2 = √푎2 + 푏2 − 2 푏 푐 cos 퐴
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.6 Two-Dimensional Force Systems
Rectangular Components: When a force is resolved into two components along with the x and y axes, the components are then called rectangular components. For analytical work, we can represent these components in one of two ways. using either scalar notation or Cartesian vector notation.
Scalar Notation: The rectangular components of force F shown in Fig. (1-6 a) are found using the parallelogram law, so that F = Fx + Fy. Because these components form a right triangle. their magnitudes can be determined from:
퐹푥 = 퐹 ∗ cos 휃 and 퐹푦 = 퐹 ∗ sin 휃
퐹 2 2 −1 푦 퐹 = √퐹푥 + 퐹푦 and 휃 = tan 퐹푥 Instead of using the angle θ. however, the direction of F can also be defined using a small ''slope'' triangle. such as shown in Fig. (1-6 b). Since this triangle and the larger shaded triangle are similar. the proportional length of the sides gives: - 16 -
Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
퐹 푎 푎 푥 = or 퐹 = 퐹 ( ) 퐹 푐 푥 푐
퐹푦 푏 푏 and = or 퐹 = 퐹 ( ) 퐹 푐 푦 푐
(a) (b) Figure 0-6: The rectangular components of the force.
Cartesian Vector Notation: where Fx and Fy are vector components of F in the x- and y-directions. Each of the two vector components may be written as a scalar times the appropriate unit vector. In terms of the unit vectors i and j of Fig. (1-7), Fx = Fxi and
Fy= Fyj, and thus we may write:
퐹 = 퐹푥푖 + 퐹푦푗
Figure 0-7: Cartesian Vector Notation.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Resultant: We can represent the components of the resultant force of any number of coplanar forces symbolically by the algebraic sum or the x and y components of all the forces. i.e.,
퐹푅푥 = ∑ 퐹푥
퐹푅푦 = ∑ 퐹푦
Also. the angle θ. which specifies the direction of the resultant force, is determined from trigonometry: 퐹푅푦 휃 = tan−1 퐹푅푥
Figure 0-8: The components of the resultant force.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Examples:
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Ex: Find R and θ:
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.7 Moment: In addition to the tendency to move a body in the direction of its application, a force can also tend to rotate a body about an axis. The axis may be any line which neither intersects nor is parallel to the line of action of the force. This rotational tendency is known as the moment M of the force. Moment is also referred to as torque. As a familiar example of the concept of the moment, consider the pipe wrench of Fig. 1-9 a. One effect of the force applied perpendicular to the handle of the wrench is the tendency to rotate the pipe about its vertical axis. The magnitude of this tendency depends on both the magnitude F of the force and the effective length d of the wrench handle. Common experience shows that a pull that is not perpendicular to the wrench handle is less effective than the right-angle pull shown.
Moment about a Point Figure 1-9 b shows a two-dimensional body acted on by a force F in its plane, the magnitude of the moment is defined as:
Where d is the moment arm or perpendicular distance from the axis at point o to the line of action of the force. Units of moment are N.m or lb.ft. Direction: The direction of Mo is defined by its moment axis, which is perpendicular to the plane that contains the force F and its moment arm d. The right-hand rule is used to Figure 0-9 establish the sense of direction of Mo. Resultant Moment (Varignon’s Theorem): For two- dimensional problems, where all the forces lie within the x–y plane, Fig. 1-10 , the resultant moment (MR)o about point o (the z axis) can be determined by finding the algebraic sum of the moments caused by all the forces in the system. As a convention, we will generally consider positive (+) moments as counter-clockwise since they are directed along the positive z axis (out of the page). Clockwise moments will be negative (-) . Therefore:
If the numerical result of this sum is a positive scalar, Figure 0-10 (MR)o will be a Counterclockwise moment (out of the page); and if the result is negative, (MR)o will be a clockwise moment (into the page).
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.8 Couple
A Couple is defined as two parallel forces that have the same magnitude. but opposite directions. and are separated by a perpendicular distance (d). Fig. 1-11. Since the resultant force is zero.
Figure 0-11: Couple. Consider the action of two equal and opposite forces F and -F a distance d apart, as shown in Fig. (1- 11). These two forces cannot be combined into a single force because their sum in every direction is zero. Their only effect is to produce a tendency of rotation. The combined moment of the two forces about an axis normal to their plane and passing through any point such as O in their plane is the couple M. This couple has a magnitude:
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
Because the couple vector M is always perpendicular to the plane of the forces which constitute the couple, in the two-dimensional analysis, we can represent the sense of a couple of vectors as clockwise or counterclockwise by one of the conventions shown in Fig. 1-12. Later, when we deal with a couple of vectors in three-dimensional problems, we will make full use of vector notation to represent them, and the mathematics will automatically account for their sense.
Figure 0-12: clockwise or counterclockwise couples.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
1.9 Three-Dimensional Force Systems
Many problems in mechanics require analysis in three dimensions, and for such problems it is often necessary to resolve a force into its three mutually perpendicular components. The force F acting at point O in Fig. has the rectangular components Fx,
Fy, Fz, where:
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
2 2 2 푐표푠 휃푥 + 푐표푠 휃푦 + 푐표푠 휃푧 = 1
Specification by two angles which orient the line of action of the force. Consider the geometry of Fig. We assume that the angles θ and ϕ are known. First resolve F into horizontal and vertical components.
Then resolve the horizontal component Fxy into x- and y-components.
The quantities Fx, Fy, and Fz are the desired scalar components of F. The choice of orientation of the coordinate system is arbitrary, with convenience being the primary consideration. However, we must use a right-handed set of axes in our three- dimensional work to be consistent with the right-hand-rule definition of the cross product. When we rotate from the x- to the y-axis through the 90o angle, the positive direction for the z-axis in a right-handed system is that of the advancement of a right- handed screw rotated in the same sense. This is equivalent to the right-hand rule.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
PROBLEMS:
1- The force F has a magnitude of 600 N. Express F as a vector in terms of the unit vectors i and j. Identify the x and y scalar components of F.
2- The 1800-N force F is applied to the end of the I-beam. Express F as a vector using the unit vectors i.
3-
4-
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
5-
6- If θ = 30° and T = 6 kN, determine the magnitude of the resultant force acting on the eyebolt and its direction measured clockwise from the positive x axis.
7- Determine the magnitude of the resultant force acting on the bracket and its direction measured counterclockwise from the positive u axis.
8- Determine the magnitude of the resultant force acting on the pin and its direction measured clockwise from the positive x axis.
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Engineering Mechanics First Stage Asst. Lect. Ibrahim Khudhur
9- Determine the moment of the 800-N force about point A and about point O.
10- Determine the moment of the 50-N force (a) about point O by Varignon’s theorem and (b) about point C by a vector approach.
11- The 30-N force P is applied perpendicular to the portion BC of the bent bar. Determine the moment of P about point B and about point A.
12- Compute the combined moment of the two 90- lb forces about (a) point O and (b) point A.
13- Express the force as a Cartesian vector.
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