1. Statics of Rigid Bodies-Fundamental Concepts.pptx

TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES CAVITE CAMPUS Statics of Rigid Bodies...

TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES CAVITE CAMPUS Statics of Rigid Bodies Fundamental Concepts ENGR. CHRISTOPHER G. CHAN Department of Engineering  LEARNING OUTCOMES By the end of this lesson, students should be able to: 1. Define mechanics, engineering mechanics, and statics of rigid bodies, and understand their significance in engineering. 2. Differentiate between scalar and vector quantities and identify examples of each in real- world scenarios. 3. Recognize and categorize different types of units and demonstrate an understanding of the SI system of units. 4. Apply dimensional homogeneity to verify the correctness of equations and physical formulas. 5. Utilize significant figures and rounding-off techniques in numerical calculations to ensure accuracy in results. 6. Describe the concept of force, including its definition and classification based on application and direction. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Mechanics Mechanics is a branch of science dealing with the state of bodies - at rest, in motion or resulting in deformation when acted upon by a force system. Upon applying a force system to a body, at least one of the following outcomes may occur: the body may move; if there is not adequate restraint, or it may not move but noticeably deform. Generally, if the deformation is noticeable, then it is discussed under deformable-body mechanics (very often called strength of materials or mechanics of materials). If the deformation is negligible or no movement occurs, then it is discussed under rigid-body mechanics. Both are divided into two areas, statics and dynamics. Statics deals with bodies at rest, and dynamics deals with bodies in motion. The branch of mechanics that deals with liquids is called fluid mechanics. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Mechanics Engineering Mechanics is basically a branch of mechanics, associated with the study of effect of forces acting on rigid bodies. Such forces may keep a body or bodies in rest or in motion. If a body remains in rest condition, it means the net effect of all forces acting is zero; this condition of the body is termed as static or equilibrium. However, if the body moves, it means an unbalanced force is acting and causing the motion; this condition of body is termed as dynamics. Thus, engineering mechanics is mainly concerned with the effect of forces on rigid body. However, mechanics is broadly classified into various categories depending upon type (nature) of bodies influenced by forces. The bodies may be solid (rigid or deformable) or fluid. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Fundamental Concepts SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Engineering Mechanics SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Engineering The purpose of mechanics is to Mechanics explain and predict physical phenomena and thus to lay the foundations for engineering applications. You need to know statics to determine how much force will be exerted on a point in a bridge design and whether the structure can withstand that force. Determining the force, a dam needs to withstand from the water in a river requires statics. You need statics to calculate how much weight a crane can lift, how SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Engineering The purpose of mechanics is to Mechanics explain and predict physical phenomena and thus to lay the foundations for engineering applications. You need to know statics to determine how much force will be exerted on a point in a bridge design and whether the structure can withstand that force. Determining the force, a dam needs to withstand from the water in a river requires statics. You need statics to calculate how much weight a crane can lift, how SUBJECT ENGR. CHRISTOPHER G. CHAN much force a locomotive need to pull Civil Engineering Department of Engineering  Engineering Space is the geometric region Mechanics 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. Time is the measure of the succession of events and is a basic quantity in dynamics. SUBJECT ENGR. CHRISTOPHER G. CHAN Time Civil is not directly involved in Engineering Department of Engineering  Engineering Mass is a measure of the Mechanics inertia of a body, which is its resistance to a change of velocity. Mass can also be thought of as the quantity of matter in a body. The mass of a body affects the gravitational attraction force between it and other bodies. This force appears in many applications in statics. Force is the action of one body on another. A force tends to move a body in the direction of its action. The SUBJECT ENGR. CHRISTOPHER G. CHAN action Civil Engineering of a force is Department of Engineering  Engineering A particle is a body of negligible Mechanics 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. Rigid body. A body is considered rigid when the change in distance between any two of its points is SUBJECT ENGR. CHRISTOPHER G. CHAN negligible Civil Engineering for the purpose at hand. Department of Engineering  Engineering Length is used to locate the Mechanics position of a point in space and thereby describe 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 as multiples of this unit. 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 SUBJECT ENGR. CHRISTOPHER G. CHAN is Engineering Civil very small compared to the Department of Engineering  Units Types of Units Fundamental or Base Units. Units which cannot be altered and have separate entities are called fundamental or base units. there are three physical quantities i.e., length, time and mass, whose units i.e., meter, sec and kg, respectively; are called fundamental or base units and extensively used in mechanics. Derived Units. These are units which are derived or dependent on fundamental or base units. For example, the units of force, work, power, density, area, volume, velocity and acceleration; are derived units. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Units System of Units M.K.S. (Meter-Kilogram-Second) system C.G.S. (Centimeter-Gram-Second) system F.P.S. (Foot-Pound-Second) system / USCS (US Customary System) S.I. (International System) system SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  SI Unit Types of Units Physical Unit Symbol Quantity Base Units amount of mole substance ampere current meter length candela luminous kilogram intensity kelvin mass second temperature Time Supplementar plane angle radian ENGR. CHRISTOPHER G. CHAN SUBJECT y Units Civil Engineering Department of Engineering  SI Unit Types of Physical Unit Symbol Units Quantity force newton frequency hertz Derived units power watt with Distinct pressure, stress pascal name work, heat, joule energy SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  SI Unit Types of Units Physical Quantity Unit Symbol acceleration meter/sec/sec activity (radioactive) 1/sec area square meter concentration mole/cubic meter luminance candela/square meter Derived units in mass density kilogram/cubic meter terms of Base units specific volume cubic meter/kilogram velocity, speed meter/sec volume cubic meter SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Units Prefixes When a numerical quantity is either very large or very small, the SI units used to define its size may be modified by using a prefix. Exponential Form Prefix SI Symbol Multiple 1 000 000 000 000 tera 1 000 000 000 giga 1 000 000 mega 1 000 kilo Submultiple 0.001 milli 0.000 001 micro 0.000 000 001 nano 0.000 000 000 001 pico SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Numerical Calculations It is important, however, that the answers to any problem be reported with justifiable accuracy using appropriate significant figures. 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. Provided this is the case, all the terms of an equation can then be combined if numerical values are substituted for the variables. For example, SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Numerical Calculations It is important, however, that the answers to any problem be reported with justifiable accuracy using appropriate significant figures. Significant Figures The number of significant figures contained in any number determines the accuracy of the number. For example, 1079 – four significant figures SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Numerical Calculations It is important, however, that the answers to any problem be reported with justifiable accuracy using appropriate significant figures. Significant Figures If zeros occur at the end of a whole number, it may be unclear as to how many significant figures the number represents. For example, 23 400 - ??? significant figures To avoid these ambiguities, we will use engineering notation to report a result. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Numerical Calculations It is important, however, that the answers to any problem be reported with justifiable accuracy using appropriate significant figures. Significant Figures For example, 23 400 - ??? significant figures To avoid these ambiguities, we will use engineering notation to report a result. This requires that numbers be rounded off to the appropriate number of significant digits and then expressed in multiples of (103), such as (103), (106), or (10–9). For instance, if 23 400 has five significant figures, it is written as 23.400(103), but if it has only three significant SUBJECT figures,G. CHAN ENGR. CHRISTOPHER it is 3 Department of Engineering Civil Engineering  Numerical Calculations Rounding-Off Numbers Rounding off a number is necessary so that the accuracy of the result will be the same as that of the problem data. In general, when dealing with numerical figures, if the last digit is greater than five, it is rounded up, and if it is less than five, it is not rounded up. However, there is a special case for numbers ending in 5. If the digit preceding the 5 is even, it is not rounded up, but if it is odd, it is rounded up. For example, 3.5587 rounded off to three significant figures 3.56 0.1049 rounded off to two significant figures 0.10 75.25 rounded off to three significant figures ??? 0.2555 rounded off to two significant figures ??? SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Numerical Calculations Calculations When a sequence of calculations is performed, it is best to store the intermediate results in the calculator. In other words, do not round off calculations until expressing the final result. This procedure maintains precision throughout the series of steps to the final solution. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Scalar and Vector Quantities Scalar Quantities are those that can be described completely by its magnitude only, like mass, length, volume, power, temperature, and time, etc. A scalar is any positive or negative physical quantity that can be completely specified by its magnitude. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Scalar and Vector Quantities Vector Quantities are those which cannot be described completely by its magnitude only, like force, weight, moment, couple, displacement, velocity, acceleration, and momentum, etc. A vector is any physical quantity that requires both a magnitude and a direction for its complete description. Examples of vectors encountered in statics are force, position, and moment. A vector is shown graphically by an arrow. The length of the arrow represents the magnitude of the vector, and the angle 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. In print, vector quantities are represented by boldface letters such as , and the magnitude of a vector is italicized,. For handwritten SUBJECT work,G. CHAN ENGR. CHRISTOPHER it is Department of Engineering Civil Engineering  Force Force is the action of a body about another body, and it changes or tends to change the state of rest or motion of a body. Forces exist because of an interaction of one object with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. The force is a vector quantity as its effect depends on the direction as well as on the magnitude of the action. Force acting on an object may cause the object to change its shape, to start moving, to stop moving, to accelerate or decelerate. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force As shown in the figure below, the effect of the force applied to the bracket depends on 𝑃, the angle 𝜃 and the location of the point of application. If any of these specifications will change, it alters the effect on the bracket. So, magnitude, direction, sense and point of application are characteristics of the force. Magnitude represents the value of force. The magnitude can be represented graphically by drawing a vector to scale. Direction of the force is represented by line of action and angle it forms with some reference axis. The line of action is indefinitely SUBJECT long G. ENGR. CHRISTOPHER line on CHAN Department of Engineering Civil Engineering  Force As shown in the figure below, the effect of the force applied to the bracket depends on 𝑃, the angle 𝜃 and the location of the point of application. If any of these specifications will change, it alters the effect on the bracket. So, magnitude, direction, sense and point of application are characteristics of the force. Sense of the force is represented by an arrowhead; it specifies the direction in which the force moves along the line of action. The direction relates to the line of action of the force, and the sense is the way in which the force moves along that line. Point of application is the exact contact point (location) at which a force is applied to a body. Graphically it is represented by the tip of the arrowhead (it may be situated in the opposite end as the arrowhead) and it is unique to each force. It may happen that different forces share same point of application. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force Classification of Force Contact force is produced by direct physical contact of two objects, and they are distributed over a surface area of the body; examples of contact forces include applied force, frictional forces, normal forces etc. Body force is produced when body (object) is in a force field such as a gravitational, electric or magnetic field. These types of forces results when the two interacting objects are not in physical contact with each other, and they are distributed over the SUBJECT ENGR. CHRISTOPHER G. CHAN volume Civil of the body. Engineering Department of Engineering  Force Classification of Force External force is applied externally to an object; they are either applied or reactive forces. Internal force is developed inside the body to resist deformation of body. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Classification of Force When the area over which contact force is applied is very small like a point, the force may be considered as concentrated on a point. The force which is distributed over an area is considered as distributed force. ENGR. CHRISTOPHER G. CHAN SUBJECT Department of Engineering Civil Engineering  Force System Force System is a collection or pattern or group of various forces acting on a rigid body. It is of two types and its classification depends upon number of forces acting in planes. Classification of Force System The force system is classified into two categories 1. Coplanar Force System 2. Non-Coplanar Force System SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Coplanar Force System Coplanar force system is the one where a number of forces work in a single plane or common plane. It is further divided into: 1. Concurrent Coplanar Force System 2. Collinear Coplanar Force System 3. Parallel Coplanar Force System 4. Non-parallel Coplanar Force System SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Concurrent Coplanar Force System If a number of forces work through a common point in a common plane, then the force system is called concurrent coplanar force system. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Collinear Coplanar Force System If a number of forces have single line of action in a common plane, then the force system is called collinear coplanar force system. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Parallel Coplanar Force System If a number of forces have a parallel line of action in a common plane, then the force system is called parallel coplanar force system. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Non-parallel Coplanar Force System If a number of forces do not have parallel line of action in a common plane, then the force system is known as non-parallel coplanar force system. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Non-coplanar Force System It is the one where a number of forces work in different planes. It is further divided into: 1. Concurrent Non-coplanar Force System 2. Non-parallel Non-coplanar Force System 3. Parallel Non-coplanar Force System SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Concurrent Non-coplanar Force System If a number of forces work through a common point in different planes, then the force system is called concurrent non-coplanar force system. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Non-parallel Non-coplanar Force System When a number of forces are having different line of action in three different planes, then the force system is called non-parallel non-coplanar force system. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Force System Parallel Non-coplanar Force System When a number forces are having parallel line of action in two different planes, then the force system is known as parallel non-coplanar force system. This force system can exist in two, two-dimensional planes only SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics The whole mechanics relies on relatively few basic laws which lay down the foundation of mechanics. Laws of Motion Newton’s First Law. A particle remains at rest, or continuous to move in a straight line with uniform velocity, if there is no unbalanced force acting on it. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics The whole mechanics relies on relatively few basic laws which lay down the foundation of mechanics. Laws of Motion Newton’s Second Law. The acceleration of a particle is proportional to the resultant force acting on the particle and is in the direction of this force. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics The whole mechanics relies on relatively few basic laws which lay down the foundation of mechanics. Laws of Motion Newton’s Third Law. The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction, and collinear. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics The whole mechanics relies on relatively few basic laws which lay down the foundation of mechanics. Laws of Motion Newton’s Third Law. The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction, and collinear. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics Newton’s Law of Gravitation This law states that two particles of mass and are mutually attracted with equal and opposite forces and of magnitude given by the formula; where and Let g P= SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics Law of Transmissibility It states that, the point of application of a force can be transmitted to any other point along its line of action within the body. Force acting at point can be transmitted to point along its line of action as shown in the figure below. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics Parallelogram Law This law states that if two forces acting a point are represented as per magnitude and direction by two adjacent sides of a parallelogram, then the diagonal of such parallelogram will represent their resultant force in magnitude and direction. The parallelogram law can determine the resultant by two ways, one graphically and second analytically. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics Parallelogram Law In analytical method it can be determined from the figure as follows: And direction, SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics Triangle Law This law states that if two forces acting at a point are represented as per magnitude and direction taken in order than the closing side of triangle taken from starting point to last point represents the resultant force. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Laws of Mechanics Polygon Law This law states that if several forces acting at a point are represented as per their magnitude and direction in the correct order, then the closing side from the starting point of the first force to the last point of the last force represents the resulting force. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  GENERAL PROCEDURE FOR ANALYSIS To learn engineering mechanics effectively, it's helpful to attend lectures, read books, and study example problems. However, the most successful way to understand the principles is by solving problems yourself. To do this well, it's important to present your work in a clear and logical manner, following the suggested steps below: 1. Carefully examine the problem and establish connections between the real-world scenario and the theory you have learned. 2. Create a table to organize the problem's data and draw diagrams to a larger scale if needed. 3. Utilize the appropriate principles, typically in mathematical form. Ensure that any equations you write are consistent in terms of dimensions. 4. Solve the equations required to find the solution and present the answer with the required number significant figures. 5. Analyze the obtained answer using technical judgment and common SUBJECT sense to assess ENGR. CHRISTOPHER G. CHAN its reasonableness. Civil Engineering Department of Engineering  Sample Problems Convert 100 km/h to m/s and 24 m/s to km/h List down the relevant ‘conversion factors’ Multiply accordingly minding the ‘units’ SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems Convert the density of steel 7.85 g/cm3 to kg/m3 List down the relevant ‘conversion factors’ Multiply accordingly minding the ‘units’ SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems Round off the following numbers to three significant figures: SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems Round off the following numbers to three significant figures: SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems Represent each of the following combinations of units in the correct SI form using an appropriate prefix: (a) (b) (c) SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems Evaluate each of the following to two significant figures and express each answer in SI units using an appropriate prefix: 2 SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems Evaluate each of the following and express with SI units having an appropriate prefix: SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems A concrete column has a diameter of 350 mm and a length of 2 m. If the density (mass/volume) of concrete is 2.45 Mg/m3, determine the weight of the column. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  Sample Problems A rectangular timber beam is made of molave with a density of 0.47 g/cm3. Determine the weight of the beam if it has a width of 200 mm, height of 350 mm, and length of 3.50 m. Express your answer in four significant figures. ;; SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering  References Beer, F. P., Johnston, E. R., Mazurek, D. F., Cornwell, P. J., & Self, B. P. (2016). Vector Mechanics for Engineers - Statics and Dynamics (Eleventh ed.). New York, New York, USA: McGraw-Hill Education. Hibbeler, R. C. (2023). Engineering Mechanics - Statics. Pearson Prentice Hall. Islam, M., Al Faruque, M., Zoghi, B., & Kalevela, S. A. (2021). Engineering Statics. Boca Raton, Florida, USA: CRC Press, Taylor & Francis Group, LLC. Meriam, J. L., Kraige, L., & Bolton, J. N. (2016). Engineering Mechanics - Statics - SI Version (Vol. 1). John Wiley & Sons Singapore Pte. Ltd. Sharma, V., Kumar, A., Baruaole, N. S., & Kumar, M. (2018). Engineering Mechanics Statics. Oxford, United Kingdom: Alpha Science International Ltd. Siddiquee, A. N., Khan, Z. A., & Goel, P. (2018). Engineering Mechanics - Problems and Solutions. Cambridge, United Kingdom: Cambridge University Press. SUBJECT ENGR. CHRISTOPHER G. CHAN Department of Engineering Civil Engineering