Need an account? Click here to sign up. Download Free PDF. Dannia Zulfacozta. Beatty Beatt. We can consider statics as a special case of dynamics, in which the acceleration is zero; however, statics deserves separate treatment in engineering education since many objects are designed with the intention that they remain in equilibrium. The subject of statics developed very early in history because its principles can be formulated simply from measurements of geometry and force.
For example, the writings of Archimedes — B. Studies of the pulley, inclined plane, and wrench are also recorded in ancient writings—at times when the requirements for engineering were limited primarily to building construction. Since the principles of dynamics depend on an accurate measurement of time, this subject developed much later. Galileo Galilei — was one of the first major contributors to this field. His work consisted of experiments using pendulums and falling bodies. The most significant contributions in dynamics, however, were made by Isaac Newton — , who is noted for his formulation of the three fundamental laws of motion and the law of universal gravitational attraction.
Basic Quantities. The following four quantities are used throughout mechanics. Length is used to locate the 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.
Time is conceived as a succession of events. Although the principles of statics are time independent, this quantity plays an important role in the study of dynamics. Mass is a measure of a quantity of matter that is used to compare the action of one body with that of another. This property manifests itself as a gravitational attraction between two bodies and provides a measure of the resistance of matter to a change in velocity. This interaction can occur when there is direct contact between the bodies, such as a person pushing on a wall, 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. Models or idealizations are used in mechanics in order to simplify application of the theory. Here we will consider three important idealizations.
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.
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.
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.
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 the contact force between a wheel and the ground. Steel is a common engineering material that does not deform very much under load.
Therefore, we can consider this railroad wheel to be a rigid body acted upon by the concentrated force of the rail. Since these forces all meet at point, then for any force analysis, we can assume the ring to be represented as a particle. These laws apply to the motion of a particle as measured from a nonaccelerating reference frame.
They may be briefly stated as follows. First Law. A particle originally at rest, or moving in a straight line with constant velocity, tends to remain in this state provided the particle is not subjected to an unbalanced force, Fig. 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, Fig. The mutual forces of action and reaction between two particles are equal, opposite, and collinear, Fig.
Shortly after formulating his three laws of motion, Newton postulated a law governing the gravitational attraction between any two particles. According to Eq. In the case of a particle located at or near the surface of the earth, however, the only gravitational force having any sizable magnitude is that between the earth and the particle.
Consequently, this force, termed the weight, will be the only gravitational force considered in our study of mechanics. From Eq. Since it depends on r, then the weight of a body is not an absolute quantity. Instead, its magnitude is determined from where the measurement was made. Because of this, the units used to measure these quantities cannot all be selected arbitrarily.
The International System of units, abbreviated SI after the 1 1 kg 9. As shown in Table 1—1, the SI system defines length in meters m , time in seconds s , and mass in kilograms kg. In the U. Customary system of units FPS 1 slug Table 1—2 provides a set of direct conversion factors between FPS and SI units for the basic quantities. Therefore, we will now present some of the rules for its use and some of its terminology relevant to engineering mechanics.
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 prefixes used in the SI system are shown in Table 1—3. Each represents a multiple or submultiple of a unit which, if applied successively, moves the decimal point of a numerical quantity to every third place.
Notice that the SI system does not include the multiple deca 10 or the submultiple centi 0. Except for some volume and area measurements, the use of these prefixes is to be avoided in science and engineering. Also, m s meter-second , whereas ms milli-second.
The exponential power on a unit having a prefix refers to both the unit and its prefix. With the exception of the base unit the kilogram, in general avoid the use of a prefix in the denominator of composite units.
When performing calculations, represent the numbers in terms of their base or derived units by converting all prefixes to powers of The final result should then be expressed using a single prefix. Also, after calculation, it is best to keep numerical values between 0. It is important, however, that the answers to any problem be reported with justifiable accuracy using appropriate significant figures.
In this section we will discuss these topics together with some other important aspects involved in all engineering calculations. Computers are often used in engineering for advanced design and analysis. 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. Regardless of how this equation is evaluated, it maintains its dimensional homogeneity. Keep in mind that problems in mechanics always involve the solution of dimensionally homogeneous equations, and so this fact can then be used as a partial check for algebraic manipulations of an equation.
The number of significant figures contained in any number determines the accuracy of the number. For instance, the number contains four significant figures.
However, 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 might have three , four , or five 23 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 , such as , , or 10—9.
For instance, if 23 has five significant figures, it is written as If zeros occur at the beginning of a number that is less than one, then the zeros are not significant. For example, 0. Using engineering notation, this number is expressed as 8. Likewise, 0. 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.
As a general rule, any numerical figure ending in a number greater than five is rounded up and a number less than five is not rounded up. The rules for rounding off numbers are best illustrated by examples. Suppose the number 3. Because the fourth digit 8 is greater than 5, the third number is rounded up to 3.
Likewise 0. If we round off 1. There is a special case for any number that ends in a 5. As a general rule, if the digit preceding the 5 is an even number, then this digit is not rounded up. If the digit preceding the 5 is an odd number, then it is rounded up. For example, 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. In this text we will generally round off the answers to three significant figures since most of the data in engineering mechanics, such as geometry and loads, may be reliably measured to this accuracy.
Being neat will stimulate clear and orderly thinking, and vice versa. Tabulate the problem data and draw to a large scale any necessary diagrams. Apply the relevant principles, generally in mathematical form. When writing any equations, be sure they are dimensionally homogeneous. Solve the necessary equations, and report the answer with no more than three significant figures. Study the answer with technical judgment and common sense to determine whether or not it seems reasonable.
Weight refers to the gravitational attraction of the earth on a body or quantity of mass. Its magnitude depends upon the elevation at which the mass is located. The meter, second, and kilogram are base units. Their exponential size should be known, along with the rules for using the SI units.
Thus, 0. NOTE: Remember to round off the final answer to three significant figures. Part b mm 0. Round off the following numbers to three significant figures: a 58 m, b Wood has a density of 4.
What is its density expressed in SI units? Represent each of the following quantities in the correct SI form using an appropriate prefix: a 0. The pascal Pa is actually a very small unit of pressure. Atmosphere pressure at sea level is How many pascals is this? The specific weight wt.
Use an appropriate prefix. A rocket has a mass slugs on earth. Specify a its mass in SI units, and b its weight in SI units. Evaluate each of the following to three significant figures and express each answer in SI units using an appropriate prefix: a 0. Convert each of the following to three significant figures. Evaluate each of the following and express with an appropriate prefix: a kg 2, b 0.
Determine the mass of an object that has a weight of a 20 mN, b kN, c 60 MN. Express the answer to three significant figures. What is the weight in newtons of an object that has a mass of: a 10 kg, b 0.
Express the result to three significant figures. If an object has a mass of 40 slugs, determine its mass in kilograms. Using the SI system of units, show that Eq. Determine to three significant figures the gravitational force acting between two spheres that are touching each other. The mass of each sphere is kg and the radius is mm. Water has a density of 1. What is the density expressed in SI units?
Two particles have a mass of 8 kg and 12 kg, respectively. If they are mm apart, determine the force of gravity acting between them. Compare this result with the weight of each particle. If a man weighs lb on earth, specify a his mass in slugs, b his mass in kilograms, and c his weight in newtons. Chapter 2 This electric transmission tower is stabilized by cables that exert forces on the tower at their points of connection.
In this chapter we will show how to express these forces as Cartesian vectors, and then determined their resultant. A scalar is any positive or negative physical quantity that can be completely specified by its magnitude.
Examples of scalar quantities include length, mass, and time. 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 u 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, Fig. In print, vector quantities are represented by boldface letters such as A, and the magnitude of a vector is italicized, A. For handwritten work, it is often convenient to denote a vector quantity by simply drawing an S arrow above it, A. If a vector is multiplied by a positive scalar, its magnitude is increased by that amount.
Multiplying by a negative scalar will also change the directional sense of the vector. Graphic examples of these operations are shown in Fig. All vector quantities obey the parallelogram law of addition. From the head of B, draw a line parallel to A. Draw another line from the head of A that is parallel to B.
These two lines intersect at point P to form the adjacent sides of a parallelogram. The resultant R extends from the tail of A to the head of B. In a similar manner, R can also be obtained by adding A to B, Fig. By comparison, it is seen that vector addition is commutative; in other words, the vectors can be added in either order, i. Subtraction is therefore defined as a special case of addition, so the rules of vector addition also apply to vector subtraction.
Two common problems in statics involve either finding the resultant force, knowing its components, or resolving a known force into two components. We will now describe how each of these problems is solved using the parallelogram law. Finding a Resultant Force. The two component forces F1 and F2 The parallelogram law must be used to determine the resultant of the two forces acting on the hook.
From this construction, or using the triangle rule, Fig. Sometimes it is necessary Using the parallelogram law the supporting force F can be resolved into components acting along the u and v axes. For example, in Fig. In order to determine the magnitude of each component, a parallelogram is constructed first, by drawing lines starting from the tip of F, one line parallel to u, and the other line parallel to v. These lines then intersect with the v and u axes, forming a parallelogram. The force components Fu and Fv are then established by simply joining the tail of F to the intersection points on the u and v axes, Fig.
This parallelogram can then be reduced to a triangle, which represents the triangle rule, Fig. From this, the law of sines can then be applied to determine the unknown magnitudes of the components. If more than two forces are to be added, successive applications of the parallelogram law can be carried out in order to obtain the resultant force.
Using the parallelogram law to add more than two forces, as shown here, often requires extensive geometric and trigonometric calculation to determine the numerical values for the magnitude and direction of the resultant. The sides of the parallelogram represent the components, Fu and Fv. The magnitudes of two force components are determined from the law of sines. The formulas are given in Fig.
The sense of the vector will change if the scalar is negative. Determine the magnitude and direction of the resultant force. The parallelogram is formed by drawing a line from the head of F1 that is parallel to F2, and another line from the head of F2 that is parallel to F1.
The resultant force FR extends to where these lines intersect at point A, Fig. The two unknowns are the magnitude of FR and the angle u theta. From the parallelogram, the vector triangle is constructed, Fig. The arrow from A to B represents Fu. Similarly, the line extended from the head of the lb force drawn parallel to the u axis intersects the v axis at point C, which gives Fv.
The vector addition using the triangle rule is shown in Fig. The two unknowns are the magnitudes of Fu and Fv. NOTE: The result for Fu shows that sometimes a component can have a greater magnitude than the resultant. The magnitudes of FR and F are the two unknowns. They can be determined by applying the law of sines.
Determine this magnitude, the angle u, and the corresponding resultant force. Since the magnitudes lengths of FR and F2 are not specified, then F2 can actually be any vector that has its head touching the line of action of FR, Fig. Since the vector addition now forms the shaded right triangle, the two unknown magnitudes can be obtained by trigonometry.
It is strongly suggested that you test yourself on the solutions to these examples, by covering them over and then trying to draw the parallelogram law, and thinking about how the sine and cosine laws are used to determine the unknowns. Then before solving any of the problems, try and solve some of the Fundamental Problems given on the next page. The solutions and answers to these are given in the back of the book.
Doing this throughout the book will help immensely in developing your problem-solving skills. Determine the magnitude of the resultant force acting on the screw eye and its direction measured clockwise from the x axis. Resolve the lb force into components along the u and v axes, and determine the magnitude of each of these components.
Two forces act on the hook. Determine the magnitude of the resultant force. F2—4 F2—5. Resolve this force into components acting along members AB and AC, and determine the magnitude of each component.
Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis. Resolve the force F1 into components acting along the u and v axes and determine the magnitudes of the components. Resolve the force F2 into components acting along the u and v axes and determine the magnitudes of the components. The vertical force F acts downward at A on the twomembered frame.
Solve Prob. If the magnitude of the resultant force is to be N, directed along the positive y axis, determine the magnitude of force F and its direction u. Resolve F1 into components along the u and v axes and determine the magnitudes of these components. Resolve F2 into components along the u and v axes and determine the magnitudes of these components.
Force F acts on the frame such that its component acting along member AB is lb, directed from B towards A, and the component acting along member BC is lb, directed from B towards C. Determine the magnitude of F and its direction u. Force F acts on the frame such that its component 2 acting along member AB is lb, directed from B towards A.
The plate is subjected to the two forces at A and B as shown. Resolve this force into two components acting along the lines aa and bb. Determine the angle u for connecting member A to the plate so that the resultant force of FA and FB is directed horizontally to the right. Also, what is the magnitude of the resultant force? The component of force F acting along line aa is required to be 30 lb. Determine the magnitude of F and its component along line bb.
What is the component of force acting along member AB? Two forces act on the screw eye. Two forces F1 and F2 act on the screw eye. Two forces are applied at the end of a screw eye in order to remove the post. The chisel exerts a force of 20 lb on the wood dowel rod which is turning in a lathe. Resolve this force into components acting a along the n and t axes and b along the x and y axes.
If the resultant force of the two tugboats is 3 kN, directed along the positive x axis, determine the required magnitude of force FB and its direction u.
If the resultant force of the two tugboats is required to be directed towards the positive x axis, and FB is to be a minimum, determine the magnitude of FR and FB and the angle u. The beam is to be hoisted using two chains. Determine the magnitudes of forces FA and FB acting on each chain in order to develop a resultant force of N directed along the positive y axis.
If the resultant force is to be N directed along the positive y axis, determine the magnitudes of forces FA and FB acting on each chain and the angle u of FB so that the magnitude of FB is a minimum. Three chains act on the bracket such that they create a resultant force having a magnitude of lb. If two of the chains are subjected to known forces, as shown, determine the angle u of the third chain measured clockwise from the positive x axis, so that the magnitude of force F in this chain is a minimum.
All forces lie in the x—y plane. What is the magnitude of F? Hint: First find the resultant of the two known forces. Force F acts in this direction. For analytical work we can represent these components in one of two ways, using either scalar notation or Cartesian vector notation. The rectangular components of force F shown in Fig.
It is important to keep in mind that this positive and negative scalar notation is to be used only for computational purposes, not for graphical representations in figures. Throughout the book, the head of a vector arrow in any figure indicates the sense of the vector graphically; algebraic signs are not used for this purpose.
Thus, the vectors in Figs. It is also possible to represent the x and y components of a force in terms of Cartesian unit vectors i and j. They are called unit vectors because they have a dimensionless magnitude of 1, and so they can be used to designate the directions of the x and y axes, respectively, Fig. We can use either of the two methods just described to determine the resultant of several coplanar forces. To do this, each force is first resolved into its x and y components, and then the respective components are added using scalar algebra since they are collinear.
The resultant force is then formed by adding the resultant components using the parallelogram law. Represent each force as a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant. FR x 89 Ans: If the resultant force of the two tugboats is 3 kN, directed y along the positive x axis, determine the required magnitude A of force FB and its direction u. Two forces F1 and F2 staticcs on the screw eye.
The two mooring cables exert forces on the stern of a ship z as shown. Express z each force in Cartesian vector form and then determine the resultant force FR. Determine the magnitude of the projected component of z the lb force mnual along the axis BC of the pipe. The coordinate direction angles are FR x Enter the email address you signed up with and hibbele email you a reset link. B Applying the law of cosines to Fig. Determine the magnitude and coordinate direction angles z of F3 so that the resultant of the three forces acts along the F3 positive y axis and has a magnitude of lb.
Log In Sign Up. Determine the resultant of the two forces and express the result as a Cartesian vector. Hibbeler statics 13th edition pdf free download. Statics Beer. Google Play Editors Choice. Mechanics of Materials Solutions Manual.
Its easy to get startedall you need is a library card. An Introduction to US. Engineering mechanics combined statics and dynamics 8th edition r.
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