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# Engineering mechanics statics and dynamics by shames pdf

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How far does the block have to move if the force F is to do I O ft-lb of work? Define a vector and a scalar. I teacher. These forces form the second couple. Of the many possible sets of basic dimensions that we could use, we will confine ourselves at present to the set that includes the dimensions of length, time, and mass. It then follows that the fundamental equations of physics are dimensionally homogeneous; and all equations derived analyti- cally from these fundamental laws must also be dimensionally homogeneous. We have now established three basic independent dimensions to describe certain physical phenomena.

Bodies restrained by identical equal. And since they are, we can conclude that the bodies have an equivalent innate property. This property of each body that manifests itself in the amount of gravitational attraction we call man.

The equivalence of these bodies, after the aforementioned grinding oper- ation, can be indicated in yet a second action.

If we move both bodies an equal distance downward, by stretching each spring, and then release them at the same time, they will begin to move in an identical manner except for small variations due to differences in wind friction and local deformations of the bodies. We have imposed, in effect, the same mechanical disturbance on each body and we have elicited the same dynamical response.

Hence, despite many obvious differences, the two bodies again show an equivalence. The pcoperry of mpcs, thn, Chomcrcrke8 a body both in the action of na1 a n r a c k and in tlu response IO a mekhnnicd. To communicate this property quantitatively, we may choose some convenient body and compare other bodies to it in either of the two above-.

The two basic units commonly used in much American engineering practice to measure mass are the pound mass, which is defined in terms of the attraction of gravity for a standard body at a standard location, and the slug, which is defined in terms of the dynamical response of a stan- dard body to a standard mechanical disturbance.

A similar duality of mass units does not exist in the SI system. There only the kilugmm is used as the basic measure of mass. The kilogram is measured in terms of response of a body to a mechanical disturbance.

Both systems of units will he discussed further in a subsequent section. We have now established three basic independent dimensions to describe certain physical phenomena. It is convenient to identify these dimen- sions in the following manner: For instance, we may wish to change feet into inches or millime- ters. In such a case, we must replace the unit in question by a physically equivalent number of new units. Thus, a foot is replaced by 12 inches or A listing of common systems of units is given in Table 1.

Such relations between units will he expressed in this way: Here is another way of expressing the relations above:. Table 1. The unity on the right side of these relations indicates that the numerator and denominator on the left side are physically equivalent, and thus have a 1: This notation will prove convenient when we consider the change of units for secondary dimensions in the next section. In Section 1. The dimensional representation of secondary quantities is given in terms of the basic dimensions that enter into the formula- tion of the concept.

The units for a secondary quantity are then given in terms of the units of the constituent basic dimensions. Thus, one scale unit of velocity in the American system is 1 foot per second, while in the SI system it is I meter per second. How may these scale units he correctly related for complicated secondary quantities?

That is, for our simple case, how many meters per second are equivalent to 1 foot per second? The formal expressions of dimensional representation may he put to good use for such an evaluation.

## Irving H. Shames-Engineering Mechanics (Statics and Dynamics).(1996)

The procedure is as follows. Express the dependent quantity dimensionally; substitute existing units for the basic dimensions; and finally, change these units to the equivalent numbers of units in the new system. The result gives the number of scale units of the quantity in the new system of units that is equivalent to 1 scale unit of the quantity in the old system.

Performing these operations for velocity, we would thus have. Another way of changing units when secondary dimensions are present is to make use of the formalism illustrated in relations 1. To change a unit in an expression, multiply this unit by a ratio physically equivalent to unity, as we discussed earlier, so that the old unit is canceled out, leaving the desired unit with the proper numerical coefficient.

It should he clear that, when we multiply by such ratios to accomplish a change of units as shown above, we do not alter the magnitude of the actual physical quantity represented by the expression.

Students are strongly urged to employ the above technique in their work, for the use of less formal meth- ods is generally an invitation to error.

In this regard, there is an important law, the law of dimen. This law states that. It then follows that the fundamental equations of physics are dimensionally homogeneous; and all equations derived analyti- cally from these fundamental laws must also be dimensionally homogeneous.

What restriction does this condition place on an equation? To answer this, let us examine the following arbitrary equation: For this equation to be dimensionally homogeneous, the numerical equality between both sides of the equation must he maintained for all systems of units.

To accomplish this, the change in the scale of measure of each group of terms must be the same when there is a change of units. That is, if the numer- ical measure of one group such as ygd is doubled for a new system 0 1 units, so must that of the quantities x and k. For 1hi. In this regard, consider the dimensional representation of the above equation expressed in the following manner:.

Mathe- matically. Inserting the constant of proportional- ity. In a later section. Throughout this book. The equation. To three significant figures. We have formulated two units of mass by two different actions. Experiment would then indicate that for a given body the acceleration is directly proportional to the applied force.

This fraction or multiple will then represent the number of units of pound mass that are equivalent to I slug. It turns out that this coefficient is go. Physi- cists prefer the former. Let us consider the FLt system of basic dimensions tor the following discussion. Using Newton's law. As was mentioned earlier. On the other hand.

In American engineering prac- tice. The unit of force may he taken to be the pound-force Ihf. The mass. Here we do not have the problem of 2 units of mass. Weight is defined as the force of gravity on a body. If the altitude is not exceedingly large.

In the SI system of units. In this age of rockets and missiles.. Examination of the units on the right side of the equation then indicates that the units of go must be 1. If we know the weight of a body at some point. Even the simpliI"ica1ion of matter into molecules. In this text. A kilonewton newtons. We must he sure. In most problems. Invariably in our deliberations. All analytical physical sciences must resort to this technique.

Without such an 'This is particularly true in the marketplace where the word "kilos" is often heard. We shall. We want to represent an action using the known laws of physics. A newton. That is.

Other cases will be presented later. The particle is defined as an object that has no size but that has a mass.

Rigid Body. Point Force. Rigid-body assumption-use the desired analysis. The most elemental case is that of a rigid body. Her rev- olutions are controlled beautifully by the orientation of the body. A finite force exerted on one body by another must cause a finite amount of local deformation. If we were to attempt a more accu- rate analysis-even though a slight increase in accuracy is not required-we would then need to know the exact position that the load assumes relative to Figure 1.

For example. Although the alternative to a rigid-body analysis here leads us to a virtually impossible calculation. In Figs. The guiding principle is to make such simplifications as are consistent with the required accuracy ojthe results. In many cases involving the action on a body by a force. In many cases where the actual area of contact io a problem is very small but is not known exactly. This simplification of a force distribution is called a point force. In this motion. It is then preferable to consider the body as rigid.

If P is small enough. Deformable body. To do this accurately is a hopelessly difficult task. Perhaps this does not sound like a very helpful definition for engineers to employ.

For the tra- jectory of a planet. You will learn later that the wiitri. The perfectly elastic body. How fast? The second question. Which way? The concept o f velocity entails the information desired in questions I and 2.

Many other simplifications pervade mechanics. The first question. For instance. Other quantities that have only magnitude. The magnitude of the displacement vector corresponds to the distance moved along a straighr line between two points. The most common example is force. There are many physical quantities that are represented by a directed line segment and thus are describable by specifying a magnitude and a direc- tion.

Certain quantities having magnitude and direction combine their effects e in a special way. Parallelogram law. F or F E or E are other possibilities. Displacement vector pAB.. Figure 1. All quantities that have magnitude and direction and that add according to the parallelogram law are called vector quaniities.

A vector quantity will be denoted with a boldface italic let. Another example is the displacement vecior between two points on the path of a particle. Successive rotations are not commutative. If a combination is not commutative. We can associate a magnitude degrees or radians and a direction the axis and a stipulation of clockwise or counterclockwise with this quantity.

One very important example will he pointed out after we reconsider Fig. In other words. In the construction of the parallelo- gram it matters not which force is laid out first. With this in mind. They are. Line of action of B vector. In Fig. If the criterion in Fig.

The easiest way to show this is to demonstrate that the combination of such rotations is not commuta- tive. The tbct that infinitesimal rotations are vectors i n accordance with our definition w i l l be an irnpoltant consideration when we discuss angular velocity in Chapter This next definition will have certain advantages as we will see later.

Finite angular rotation. This is carried out in Figs. Two vectorr are equivalent in a certain c a p a c i y if each prodnces the vev. In the next chapter. A proof of this assertion is presenlcd in Appcn- din IV. Although they have different lines of actinn.

The answer to this query is as follows. Twjo L'ecfors are equal if they have the. These operations are valid in general only if the parallelogram law is satisfied as you will see when we get to Chapter 2.

Keep in mind that the line of action involves no connotation as to sense.. Equal-velocity vectors. To sum up. X are lrce vectors a s far as total dis- Lance traveled i h concerned. The resulting motion i s lhe same in all cases. I n that capacity. For this case. In probleins o f mech.

I f the absolute height u l the parlicles above the. The point may he represented as the tail or head of the arrow iii thc graphical representation. Under such circunislainces the vectors are called truri. Fur- thermore. All such references are called inertial references. These laws were first stated by Newton as Every particle continues in a state of rest or uniform motion in a straight line upless it is compelled to change that state by forces imposed on it.

The gravitational law of attraction. F i s transmissible for towing. The c b g c of motion is proportional to the naturn1. We shall be concerned throughout this text with considerations of equivalence. The force is thus a bound vector for this problem. We may then ask: For such information to he meaningful. We shall now discuss briefly the following laws. The parallelogram law. Under such circumstances. Never- theless. Because of the rotation of the earth and the varia-.

In ciinhidci-ing the motion of high- energy elementary particles occurring i n nuclear phenomena. In addition tn the reference limitations explained above. As pointed out carlicr.

I t should he pointed out that there are actiiiiis that dii nut fiillow this law. In this case. Other imporrant actions i n which Newton's third law holds arc gravitationdl attractions to be discussed next and electrostatic forces between charged pat-ticks. It has alrcady been piiintcd out that these i s an attraction between the earth and the bodies at its surface. Many situii- rions f a l l into this category. Newton stated in his third law: To every action rhere is always opposed an equul rcucrion.

This i s illustrated graphically i n Fig. Ncar the spccd of light. The study of bodies in equilibrium i s called S I U I - i c s. Newton's Third Law. Stevinius 1 was the first to demonstrate that forces could be combined by representing them by arrows to some suitable scale. In the actions involv- ing the earth and the bodies discussed above. These forces of attraction may be given by the law of gravitational attractiun: As we pointed out.

Avoiding vector notation for now. There is also an attraction between the two bodies A and B themselves. Parallelogram Law. Express mass density dimensionally.

What are the two phenomena wherein mass plays a key role? What is its dimensional representation? If a pound force is defined by the extension of a standard spring. What are two kinds of limitations on Newtonian mechanics'? These operations are generally c: Since the equations of these laws relate vector quantities. These hasic. Define a vector and a scalar. What is a di.

What is an inertial reference? The baric laws of mechanics were thus introduccd. How many scale units of mass density mass per unit volume in the SI units are equivalent to I scale unit in the American system using a slugs. What is meant by line of action o f a vector? The proportionality constant pis called the coefficient of viscosity. It is not merely for elegance or sophisti- cation that we employ vector algebra. For vector quantities. The mathe- matical symbol for indicating the magnitude of a quantity is a set of vertical lines enclosing the quantity.

Note that the magnitude of a quantity is its absolute value. The basic algebraic operations for the handling of scalar quantities are those famil- iar ones studied in grade school.

The instmcter may wish tu assign the reading of these seclians along with the aforemen- tioned questions. A i s positivc sciiliir qu. A tI n-ci lhc final iirrou' ciiii then he iiiterpretcd in teiniir o i i t s length by cinployinf tlic chosen scale f k t o r.

Addition hy pmillclograiii iau.: The magnitude it. This cnmhin. The angle is determined by employing the law of sines for triangle OBA.

Find F and a using trigonometry To get the sum shown as F. Onc force has a magnitude of 10 Ib act- ing in the positive x direction. Thc total unstretched length of the rubber band i s 5 in. The teiisiiin i n the entire extended rubber hand i s then II. The top view 01 the slingshot i s shown in Fig. Simplc dingshut. Con- sequently. We ciin use the law Figure 2. Parsllclogram orcrs It must be emphatically pointed out that thc additiim if vectorsA and R only involves the vectors themselves and iiot thcir lines of actions or thcir posi- tions along their respective lincs 0faction.

For thc additional vector algebra that we will devclop in this chapter. The vectors are connected in Fig. The vector sum that closes the polygon is equivalent to the system of given vectors. The sum F then closes the triangle and is OB.. This process may also be used in the polygon construction. Figure 2. The extension of this proce- dure to any number of vectors is obvious. Subtraction of vectors. Addition and we proceed a s shown in Fig. The sum of the vectors then is the dashed vector that closes the polygon.

I right dirccted away from the origin. A homing pigcon i s released at point A and is observed. Add ii N force pointing in the positive r direction to a magnitude of force 8 and the direction of forcc C? For the Eim- N forcc at an nnple 45" to the. Subtract the N force in Prohlem 2. Figure P. What is the Figure P. Do this first graphically. A Figure P. A lightweight homemade plane i s bcinf ohserved LIS i t flies i.

Whal 2. At the outset. I t flier I O krn due south. Next i t goes southcast for I O k m and finally gocs due south 5 km to reach i t s destination H. See Fig. Give the magnitude of 2. If we do not change the force transmitted by the horizontal 2. Give the magnitude of the sum nf these forces using the parallelogram law twice. You will leam very shortly that the weight of M must be equal 2. In the preceding problem. Three forces act on the block. The N and the N and opposite to the vector sum of the supporting forces for forms act.

Suppose in Problem 2. Add the three vectors using the parallelogram law twice. What is the sum ofthe forces transmitted by the structural rods to the pin at A? What is the total force? If the difference between forces B and A in Fig. I f N Figure P. Using the parallelogram law. A man pulls with force Won a rope through a simple fric- force D having a magnitude of 25 N.

A mass M is supported by cables I and 2. What total force is exerted on and the direction o f D? We will do this problem differ- ently in Example S. Thc rollers on thc side u l the hlock tlo not 70 Ih crmtrihute to thc vertical support nf the hlock.

Twu soccer player5 approach a stntir. The wire5 cnnnect til the gcrrmerric center of the hlock C. Wc olten replace a vector b y its components siticc the cnniponents are alway:. Dn prohlcin 2. Does the offense score asuming lhiit the g d i c i s asleep '!

When this i s done. Two-dimensional original vector. The two-dimensional resolution citn he accomplished by graphical construction 0 1 the parallelogram. We shall discuss three- dimensional resolution involving three noncoplanar component vectors later in the section.

Form three independent equations for any given a involving the unknowns f. What i s the total furce cxertcd on the dummy hy the players'! Thc two vectors C. Two foothall player. Scalar Components The opposite action 11 addition nf vectors i s c.

The order of the subacnpts gives the sense of the vector-here going from A to 8. We leave it for you to jus- tify the various angles indicated in the diagram. I I wherein a sailboat is going from marker A to marker I3 5. Orthogonal or the orthogonal directions are used most often in engineering practice. Sailboat tacking. The resolution may be accomplished in two steps.

This is the aforementioned three-dimensional resolution. Resolve C along the z direction. What is the additional distance AL beyond 5. Now we first use the law if sines.. OOO m that the sailboat must travel to get from A to B? Clearly the displacement vectofl pARis equivalent to the vector sum of displacement vectors pAc plus pcR in that the same starting points A. It is also readily possible to find three components not in the. Enlarged parallelogram. Consider the specification of three orthopnal directions' for the resolution of C positioned in the first quadrant.

Note tlial. Rcctaogular component OF C. Now le1 us consider the right triangle. Hence C. I uhcrc x. This sh iuld spui. Next take vector C. Clearly irthogonal 01s C. The unit vector is also at times denoted ash. As a final consideration. It has no dimensions. We formu- late this vector as follows: In the preceding equation. On the diagram. The vector a is called a unit vector. In establishing C. This will be determined entirely by its use. It is always the case that the triangle formed by the vector and its scalar rec- tangular component is a right triangle.

You will write it as 6. The sum of the vectors A and B is found by the parallelogram law to be C. It is important to remember. It thus acts as a free vector. As will be seen Iaitcr. Unit vector a. That determination proceeds by first replacing the displacement vector p.

XImagineyou are "walking" from A to B hut restricting your movements to he along the coordinate directions. Rectangular parallelepiped used for specifying the direction of a vector.

We thereby can replace the vector p. Numbers for this purpose are shown along the sides of the rectan- gular parallelepiped without units.. This movement i s equivalent to going directly from A to B in that the Same endmints result. IOb as the corresponding scalar equations. AB in the diagram is such a diagonal used for the determination of the line of action of vector F. Any set of numbers can be used as long as the ratios of these numbers remain the ones required for the proper determi- nation of the orientation of the vector.

The purpose of this rectangular parallelepiped and diagonal is to allow for the easy determination of the ori- entation of the line of action and hence the orientation of a vector. There are times when the rectangular parallelpiped i s not shown explic- itly.

This would take us from initial point A to final point IT. Other ways to use the reclangular parallelepiped. In the two-dimensional case. The corresponding displacement veclor would then he p. We thus form the unit vectorbA8. The simplest procedure i s IO mcntally move from the beginning point o f the diagonal to the final point always moving along coordinatc directions Right triangle used fur sppecifying the direction ot it vector m two dirnensionq.

This i s shown i n Fig. Here we can say. Note that D is at the center of the outer edge of the crate.. OOOk N. We first express these three forces in terms of rectangular components. OWk N. What are the forces Fl. We will soon l e a n formally what our common sense tells us. A crate is supported by three forces. OOOk We have three scalar equations from the previous equation. The lorccs in thc mcmherq must have a vector w m q u a l and opposite to the vcctoi sum of E. If the component along AH i s N.

The entire Eystem i s coplanar. A simple truss to be studied later in detail supports two forces. Resolve thc h force into a set of components along the slot shown and in the vertical direction. If the forces in the members are colinear with the mem- hcrs. I n the previous pruhlem. A lamer needc to build a fence from the corner of his ham to the corner of hic chicken house 10 m away in the NE dircction. Two tughants are maneuvering an w e a n liiw The desired iota1 inrcc i s 3.

How long i s the fencc? Two men are trying to pull a crate which will not move y until a lb total force is applied in any one direction. The N force is to be resolved into components along the AC and AB directions in the xy plane measured by the angles a and p.

The orthogonal components of a force are: What IS the sum of the three forces? The 2. The 1. What force must each man exert to start the box moving as shown? If the component along AC is to be 1. What are the rectangular components of the lb force?

Man A can pull only at 45' to the desired direction of crate motion. What is the vector sum of these forces? Y x Figure P. What is the total cirmpi. What is thc orthogonal total f h x cwnponcnt in tlic. I direction 01 the ioice tiansmittcd to pin A of a roo1 t n h i h i tlic four rncrnher. What is y? How long mist rncmhel-c OA. What is the unit vector in the direction of the N force'! I 1 where a is the smaller angle between the two vectors. A vector operation that represents such operations con- cisely is the scalar product or dot product.

In effect. See the footnote o n p. The appro- priate sign must. Express the 1. These are unit vectors for cvlindricul coordinates. In other physical problems. Express the unit vectors i. The force lies along diagonal AB. Express the N force in terms of the unit vectors i. Note that the dot prod- uct may involve vectors of different dimensional representation. I I is 90".

Let us next consider the scalar product of mA and n u. How- ever. We can thus conclude that the dot product of equal orthogonal unit vectors for a given reference is unity and that of un- equal orthogonal unit vectors is zero.

BJ is independent of the order of multiplication of its terms. The scalar product between unit vectors will now be carried out. By definition. If we express the vectors A and B in Cartesian components whcn taking the dot product. From the definition. Remember in so doing we must not alter the magnitudcs and directions of the vectors.

If we carry it out according to our definitions: If you refer back to Fig. The dot product may be of immediate use in expressing the scalar rec- tangular component of a vector along a given direction as discussed in Sec- tion 2. Unit vector idirected from 0. If we carry out the dot product of C and s according to our fundamental definition.

Example 2. Kvdio transmission tiiwers. What are the length:. How far does the block have to move if the force F is to do I O ft-lb of work? The boat has a velocity component along its axis of 6 kn but because of side slip X and water currents. If we have for E: A block A is constrained to move along a 20' incline in 2. Explain why the following operations are meaningless: Whenever simply a component is asked for.

An electrostatic field E exerts a force on a charged pani- cle of qE. What is cos A. What is the projection of A where I. Given the vectors 2. A sailboat is tacking into a knot wind. Given the vectors the yz plane. The vector points away from the origin. What are the x and y components of the wind velocity and the boat velocity? What is the angle between the wind d o c - ity and the sailboat velocity'?

The force is i n thc diagonal plane GCDE. What is the anglc bctwccn the I. A radio tiiwci is held by guy wires. R to A'! I Fieure P. What is the angle bctweeii the 1. Whal is thc angle hetween them?

Find the dot prtiduct ul the vectors represented hy thz diagonal5 from A to I" arid trmn 1 to G. What is the angle hetween F arid i?

The angle a is the smaller of the two angles between the vectors. As in the previous case. The line of action of C is not determined by the cross product. To set up a convenient operation for these situations. The vector C has an orientation normal to the plane of the vectors A and B. The reader can easily ver- ify this. The sense. We can verify. The description of vector C is now complete.

One such interaction is the moment of a force.

Again we remind you that the cross product. This may he deduced from the nature of the definition. For the two vectors having possibly different dimensions shown in Fig.

If the: It will be left to the student to justil'y the given formulation for each of the vectors in Fig. To do this. Prim using A. Area vzctorc tor prism f i c m Noting that the second and third expressions cancel each olher. We can represent the area of each face of the prism as a vector whose inngnitude equals the area o l the face and whose direction is norm.

The product i X j is unity i n magnitude. Diffcrent kinds of rcfcrcnccs. Since the prism is a closed surface. We then add all six products as follows: It must be cautioned that this method of evaluating a determinant is correct only for 3 x 3 determinants.

If the cross product of two vectors involves less than six nonzero components. To get thc unit normal n I lo plane A B D. S00k ft? Accordingly A I. If the height of the pyramid i s I t. A simple geometric meaning can be associated with this operation.

We have set up an xyz reference such that the A and B vectors are in the xy plime. Substituting from Eqs. We see from this example that a plane surface can be represented as a vector. Thc vector triple product i s a vector quantity and w i l l appear quitc o l t c n i n sttidier oidynamics. A and R in p planc. Using thi. It will he left iis an cxercise Prohleni 2. The projected area then is given as A. The normal n to the infinite plane must have three equal direction cosines.

When ire simply identify- ing quantities i n il di. A correct representation o l this Iiirce i n a vector equation would he I. Thus, in Fig. If the coordinates of vertex E of the inclined pyramid are x A. What is the magnitude of the resulting vector'! What are the cross and dot products for the vectors A and B given as: If vectors A and B in the xy plane have a dot product of SO units, and if the magnitudes of these vectors are I O units and R units, respectively.

Y Explain. What is the cross product of the displacement vector from A to B times the displacement vector from C to D'! I Figure P. In Prvblem 2. Compute the determinant I. Ar Ay A: B, gv B: Compare the result with the computation of A x B C by using the dot-product and crowproduct operations. In Example 2. Give the results in kilo- Figure P. The illustra- tions needed for problem statement and solution are taken as enlarge- ments from the text. Generally, each problem is on a separate page.

The instructor will be able conveniently to select problems in order to post solutions or to form transparencies as desired. Also, there are 30 computer projects in which, for a number of cases, the student pre- pares hidher own software or engages in design.

As an added bonus, the student will be able to maintain hidher proficiency in program- ming. Carefully prepared computer programs as well as computer out- puts will be included in the manual. I normally assign one or two such projects during a semester over and above the usual course material. Also included in the manual is a disk that has the aforementioned pro- grams for each of the computer projects. Another important new feature of the fourth edition is an organi- zation that allows one to go directly to the three dimensional chapter on dynamics of rigid bodies Chapter 18 and then to easily return to plane motion Chapter Or one can go the opposite way.

Footnotes indicate how this can be done, and complimentary problems are noted in the Solutions Manual. Another change is Chapter 16 on plane motion.

It has been reworked with the aim of attaining greater rigor and clarity particu- larly in the solving of problems. There has also been an increase in the coverage and problems for hydrostatics as well as examples and problems that will preview prob- lems coming in the solids course that utilize principles from statics. It should also be noted that the notation used has been chosen to correspond to that which will be used in more advanced courses in order to improve continuity with upper division courses.

Thus, for mo- ments and products of inertia I use I, I,, lxzetc. The same notation is used for second moments and products of area to emphasize the direct relation between these and the preceding quantities. Experience indicates that there need be. The concept of the tensor is presented in a way that for years we have found to be readily under- stood by sophomores even when presented in large classes. This saves time and makes for continuity in all mechanics courses, particularly in the solid mechanics course.

For bending moment, shear force, and stress use is made of a common convention for the sign-namely the convention involving the normal to the area element and the direction of the quantity involved be it bending moment, shear force or stress component. All this and indeed other steps taken in the book will make for smooth transition to upper division course work. In overall summary, two main goals have been pursued in this edition. They are 1. To encourage working problems from first principles and thus to minimize excessive mapping from examples and to discourage rote learning of specific methodologies for solving various and sundry kinds of specific problems.

Also, the purpose is to engage the interest and curiosity of students for further study of mechanics. During the 13 years after the third edition, I have been teaching sophomore mechanics to very large classes at SUNY, Buffalo, and, after that, to regular sections of students at The George Washington University, the latter involving an international student body with very diverse backgrounds. During this time, I have been working on improv- ing the clarity and strength of this book under classroom conditions.

I believe the fourth edition as a result will be a distinct improvement over the previous editions and will offer a real choice for schools desiring a more mature treatment of engineering mechanics. I believe sophomore mechanics is probably the most important course taken by engineers in that much of the later curriculum depends heavily on this course. And for all engineering programs, this is usu- ally the first real engineering course where students can and must be creative and inventive in solving problems.

Their old habits of map- ping and rote learning of specific problem methodologies will not suf- fice and they must learn to see mechanics as an integral science. No other subject so richly involves mathematics, physics, computers, and down to earth common sense simultaneously in such an interesting and challenging way. We should take maximum advan- tage of the students exposure to this beautiful subject to get h i d h e r on the right track now so as to be ready for upper division work.

At this stage of my career, I will risk impropriety by presenting now an extended section of acknowledgments. And I want to salute the thousands about of fine students who took my courses during this long stretch. I wish to thank my eminent friend and colleague Professor Shahid Ahmad who among other things taught the sophomore mechanics sequence with me and who continues to teach it. He gave me a very thorough review of the fourth edition with many valuable suggestions.

I thank him pro- fusely. I want particularly to thank Professor Michael Symans, from Washington State University, Pullman for his superb contributions to the entire manuscript. Here, I came back into con- tact with two well-known scholars that I knew from the early days of my career, namely Professor Hal Liebowitz president-elect of the National Academy of Engineering and Professor Ali Cambel author of recent well-received book on chaos.

He has allowed me to play a vital role in the academic pro- gram of the department. I will be able to continue my writing at full speed as a result. Let me not forget. Zephra Coles in her decisive efficient way took care of all my needs even before I was aware of them.

And Ms. Joyce Jeffress was no less help- ful and always had a humorous comment to make. I was extremely fortunate in having the following professors as reviewers.

I have two people left. One is my good friend Professor Bob Jones from V. I continue to benefit in the new edition from his input of the third edition. And now, finally, the most important person of all, my dear wife Sheila.

She has put up all these years with the author of this book, an absent-minded, hope- less workaholic. Whatever I have accomplished of any value in a long and ongoing career, I owe to her. Professor Shames has written up to this point in time 10 text- books.

His first book Engineering Mechanics, Statics and Dynamics was originally published in , and it is going into its fourth edition in All of the books written by Professor Shames have been char- acterized by innovations that have become mainstays of how engineer- ing principles are taught to students. Engineering Mechanics, Statics and Dynamics was the first widely used Mechanics book based on vector principles.

It ushered in the almost universal use of vector prin- ciples in teaching engineering mechanics courses today. Other textbooks written by Professor Shames include:. Dym , Hemisphere Corp. Cozzarelli , Prentice- Hall, Inc. The programs involved the iiitegr: Since such behavior is involved in virtually all the situations that confront an engineer, mechanics lies at the core of much engi- neering analysis.

In fact, no physical science plays a greater role in engineer- ing than does mechanics, and it is the oldest of all the physical sciences. The writings of Archimedes covering buoyancy and the lever were recorded before B.

Our modem knowledge of gravity and motion was established by Isaac Newton , whose laws founded Newtonian mechanics, the subject matter of this text. In , Einstein placed limitations on Newton's formulations with his theory of relativity and thus set the stage for the development of relativistic mechanics.

The newer theories, however, give results that depart from those of Newton's formulations only when the speed of a body approaches the speed of light I 86, mileslsec. These speeds are encountered in the large- scale phenomena of dynamical astronomy.

Furthermore for small-scale phenomena involving subatomic particles, quantum mechanics must be used rather than Newtonian mechanics. Despite these limitations, it remains never- theless true that, in the great bulk of engineering problems, Newtonian mechanics still applies.

Also, the nutation t before the titles of certain sections indicates thal specific queslions concerning the contents of these sections requiring verbal answers are presented at the end of the chapler. The instructor may wish to assign these sections as a reading asignment along with the requirement to answer the aforestated asssiated questions as the author routinely daes himself.

These abstractions are called dimensions. The dimensions that we pick, which are independent of all other dimensions, are termed primary or basic dimensions, and the ones that are then developed in terms of the basic dimensions we call secondary dimensions. Of the many possible sets of basic dimensions that we could use, we will confine ourselves at present to the set that includes the dimensions of length, time, and mass.

Another convenient set will he examined later. In order to deter- mine the size of an object, we must place a second object of known size next to it.

Thus, in pictures of machinery, a man often appears standing disinter- estedly beside the apparatus. Without him, it would be difficult to gage the size of the unfamiliar machine. Although the man has served as some sort of standard measure, we can, of course, only get an approximate idea of the machine's size.

Men's heights vary, and, what is even worse, the shape of a man is too complicated to be of much help in acquiring a precise measure- ment of the machine's size. What we need, obviously, is an object that is constant in shape and, moreover, simple in concept. Thus, instead of a three- dimensional object, we choose a one-dimensional object. A straight line scratched on a metal bar that is kept at uniform thermal and physical conditions as, e. We can now readily cal- culate and communicate the distance along a cettain direction of an object by counting the number of standards and fractions thereof that can be marked off along this direction.

We commonly refer to this distance as length, although the term " length could also apply to the more general concept of size. Other aspects of size, such as volume and area, can then be formulated in terms of the standard by the methods of plane, spherical, and solid geometry.

A unit is the name we give an accepted measure of a dimension. Many systems of units are actually employed around the world, but we shall only use the two major systems, the American system and the SI system. The basic unit of length in the American system is the foot, whereas the basic unit of length in the SI system is the meter. In observing the pic- ture of the machine with the man standing close by, we can sometimes tell approximately when the picture was taken by the style of clothes the man is.

But how do we determine this? We may say to ourselves: Then, we can order the events under study by counting the num- her of these repeatable actions and fractions thereof that occur while the events transpire. The rotation of the earth gives rise to an event that serves as a good measure of time-the day.

But we need smaller units in most of our work in engineering, and thus, generally, we tie events to the second, which is an interval repeatable 86, times a day. Mass-A Property of Matter. The student ordinarily has no trouble under- standing the concepts of length and time because helshe is constantly aware of the size of things through hisher senses of sight and touch, and is always conscious of time by observing the flow of events in hisher daily life.

The concept of mass, however, is not as easily grasped since it does not impinge as directly on our daily experience. Mass is a property of matter that can be determined from two different actions on bodies. To study the first action, suppose that we consider two hard bodies of entirely different composition, size, shape, color, and so on.

If we attach the bodies to identical springs, as shown in Fig. By grinding off some of the material on the body that causes the greater extension, we can make the deflections that are induced on both springs Figure 1. Bodies restrained by identical equal. And since they are, we can conclude that the bodies have an equivalent innate property.

This property of each body that manifests itself in the amount of gravitational attraction we call man. The equivalence of these bodies, after the aforementioned grinding oper- ation, can be indicated in yet a second action.

If we move both bodies an equal distance downward, by stretching each spring, and then release them at the same time, they will begin to move in an identical manner except for small variations due to differences in wind friction and local deformations of the bodies. We have imposed, in effect, the same mechanical disturbance on each body and we have elicited the same dynamical response. Hence, despite many obvious differences, the two bodies again show an equivalence. The pcoperry of mpcs, thn, Chomcrcrke8 a body both in the action of na1 a n r a c k and in tlu response IO a mekhnnicd.

To communicate this property quantitatively, we may choose some convenient body and compare other bodies to it in either of the two above-. The two basic units commonly used in much American engineering practice to measure mass are the pound mass, which is defined in terms of the attraction of gravity for a standard body at a standard location, and the slug, which is defined in terms of the dynamical response of a stan- dard body to a standard mechanical disturbance.

A similar duality of mass units does not exist in the SI system. There only the kilugmm is used as the basic measure of mass. The kilogram is measured in terms of response of a body to a mechanical disturbance.

Both systems of units will he discussed further in a subsequent section. We have now established three basic independent dimensions to describe certain physical phenomena. It is convenient to identify these dimen- sions in the following manner: For instance, we may wish to change feet into inches or millime- ters.

In such a case, we must replace the unit in question by a physically equivalent number of new units. Thus, a foot is replaced by 12 inches or A listing of common systems of units is given in Table 1.

Such relations between units will he expressed in this way: Here is another way of expressing the relations above:. Table 1. The unity on the right side of these relations indicates that the numerator and denominator on the left side are physically equivalent, and thus have a 1: This notation will prove convenient when we consider the change of units for secondary dimensions in the next section. In Section 1. The dimensional representation of secondary quantities is given in terms of the basic dimensions that enter into the formula- tion of the concept.

The units for a secondary quantity are then given in terms of the units of the constituent basic dimensions. Thus, one scale unit of velocity in the American system is 1 foot per second, while in the SI system it is I meter per second. How may these scale units he correctly related for complicated secondary quantities?

That is, for our simple case, how many meters per second are equivalent to 1 foot per second? The formal expressions of dimensional representation may he put to good use for such an evaluation. The procedure is as follows. Express the dependent quantity dimensionally; substitute existing units for the basic dimensions; and finally, change these units to the equivalent numbers of units in the new system.

The result gives the number of scale units of the quantity in the new system of units that is equivalent to 1 scale unit of the quantity in the old system. Performing these operations for velocity, we would thus have. Another way of changing units when secondary dimensions are present is to make use of the formalism illustrated in relations 1. To change a unit in an expression, multiply this unit by a ratio physically equivalent to unity, as we discussed earlier, so that the old unit is canceled out, leaving the desired unit with the proper numerical coefficient.

It should he clear that, when we multiply by such ratios to accomplish a change of units as shown above, we do not alter the magnitude of the actual physical quantity represented by the expression. Students are strongly urged to employ the above technique in their work, for the use of less formal meth- ods is generally an invitation to error. In this regard, there is an important law, the law of dimen. This law states that. It then follows that the fundamental equations of physics are dimensionally homogeneous; and all equations derived analyti- cally from these fundamental laws must also be dimensionally homogeneous.

What restriction does this condition place on an equation? To answer this, let us examine the following arbitrary equation: For this equation to be dimensionally homogeneous, the numerical equality between both sides of the equation must he maintained for all systems of units. To accomplish this, the change in the scale of measure of each group of terms must be the same when there is a change of units. That is, if the numer- ical measure of one group such as ygd is doubled for a new system 0 1 units, so must that of the quantities x and k.

For 1hi. In this regard, consider the dimensional representation of the above equation expressed in the following manner:. Mathe- matically. Inserting the constant of proportional- ity. In a later section. Throughout this book. The equation. To three significant figures. We have formulated two units of mass by two different actions.

Experiment would then indicate that for a given body the acceleration is directly proportional to the applied force. This fraction or multiple will then represent the number of units of pound mass that are equivalent to I slug. It turns out that this coefficient is go. Physi- cists prefer the former. Let us consider the FLt system of basic dimensions tor the following discussion. Using Newton's law. As was mentioned earlier. On the other hand.

In American engineering prac- tice. The unit of force may he taken to be the pound-force Ihf. The mass. Here we do not have the problem of 2 units of mass. Weight is defined as the force of gravity on a body. If the altitude is not exceedingly large. In the SI system of units. In this age of rockets and missiles.. Examination of the units on the right side of the equation then indicates that the units of go must be 1.

If we know the weight of a body at some point. Even the simpliI"ica1ion of matter into molecules. In this text. A kilonewton newtons. We must he sure. In most problems. Invariably in our deliberations. All analytical physical sciences must resort to this technique. Without such an 'This is particularly true in the marketplace where the word "kilos" is often heard. We shall. We want to represent an action using the known laws of physics.

A newton. That is. Other cases will be presented later. The particle is defined as an object that has no size but that has a mass. Rigid Body. Point Force. Rigid-body assumption-use the desired analysis. The most elemental case is that of a rigid body. Her rev- olutions are controlled beautifully by the orientation of the body. A finite force exerted on one body by another must cause a finite amount of local deformation. If we were to attempt a more accu- rate analysis-even though a slight increase in accuracy is not required-we would then need to know the exact position that the load assumes relative to Figure 1.

For example. Although the alternative to a rigid-body analysis here leads us to a virtually impossible calculation. In Figs. The guiding principle is to make such simplifications as are consistent with the required accuracy ojthe results.

In many cases involving the action on a body by a force. In many cases where the actual area of contact io a problem is very small but is not known exactly. This simplification of a force distribution is called a point force. In this motion. It is then preferable to consider the body as rigid. If P is small enough. Deformable body. To do this accurately is a hopelessly difficult task.

Perhaps this does not sound like a very helpful definition for engineers to employ. For the tra- jectory of a planet. You will learn later that the wiitri.

The perfectly elastic body. How fast? The second question. Which way? The concept o f velocity entails the information desired in questions I and 2. Many other simplifications pervade mechanics. The first question. For instance. Other quantities that have only magnitude.

The magnitude of the displacement vector corresponds to the distance moved along a straighr line between two points. The most common example is force.

There are many physical quantities that are represented by a directed line segment and thus are describable by specifying a magnitude and a direc- tion. Certain quantities having magnitude and direction combine their effects e in a special way. Parallelogram law. F or F E or E are other possibilities. Displacement vector pAB.. Figure 1. All quantities that have magnitude and direction and that add according to the parallelogram law are called vector quaniities.

A vector quantity will be denoted with a boldface italic let. Another example is the displacement vecior between two points on the path of a particle. Successive rotations are not commutative. If a combination is not commutative. We can associate a magnitude degrees or radians and a direction the axis and a stipulation of clockwise or counterclockwise with this quantity.

One very important example will he pointed out after we reconsider Fig. In other words. In the construction of the parallelo- gram it matters not which force is laid out first. With this in mind. They are. Line of action of B vector.

In Fig. If the criterion in Fig. The easiest way to show this is to demonstrate that the combination of such rotations is not commuta- tive.

The tbct that infinitesimal rotations are vectors i n accordance with our definition w i l l be an irnpoltant consideration when we discuss angular velocity in Chapter This next definition will have certain advantages as we will see later. Finite angular rotation. This is carried out in Figs. Two vectorr are equivalent in a certain c a p a c i y if each prodnces the vev.

In the next chapter. A proof of this assertion is presenlcd in Appcn- din IV. Although they have different lines of actinn. The answer to this query is as follows. Twjo L'ecfors are equal if they have the. These operations are valid in general only if the parallelogram law is satisfied as you will see when we get to Chapter 2. Keep in mind that the line of action involves no connotation as to sense.. Equal-velocity vectors.

To sum up. X are lrce vectors a s far as total dis- Lance traveled i h concerned. The resulting motion i s lhe same in all cases. I n that capacity. For this case. In probleins o f mech. I f the absolute height u l the parlicles above the. The point may he represented as the tail or head of the arrow iii thc graphical representation. Under such circunislainces the vectors are called truri. Fur- thermore. All such references are called inertial references.

These laws were first stated by Newton as Every particle continues in a state of rest or uniform motion in a straight line upless it is compelled to change that state by forces imposed on it.

The gravitational law of attraction. F i s transmissible for towing. The c b g c of motion is proportional to the naturn1. We shall be concerned throughout this text with considerations of equivalence. The force is thus a bound vector for this problem.

We may then ask: For such information to he meaningful. We shall now discuss briefly the following laws. The parallelogram law. Under such circumstances. Never- theless. Because of the rotation of the earth and the varia-. In ciinhidci-ing the motion of high- energy elementary particles occurring i n nuclear phenomena. In addition tn the reference limitations explained above. As pointed out carlicr. I t should he pointed out that there are actiiiiis that dii nut fiillow this law.

In this case. Other imporrant actions i n which Newton's third law holds arc gravitationdl attractions to be discussed next and electrostatic forces between charged pat-ticks. It has alrcady been piiintcd out that these i s an attraction between the earth and the bodies at its surface.

Many situii- rions f a l l into this category. Newton stated in his third law: To every action rhere is always opposed an equul rcucrion. This i s illustrated graphically i n Fig. Ncar the spccd of light. The study of bodies in equilibrium i s called S I U I - i c s.

Newton's Third Law. Stevinius 1 was the first to demonstrate that forces could be combined by representing them by arrows to some suitable scale. In the actions involv- ing the earth and the bodies discussed above.

These forces of attraction may be given by the law of gravitational attractiun: As we pointed out. Avoiding vector notation for now.

There is also an attraction between the two bodies A and B themselves. Parallelogram Law. Express mass density dimensionally. What are the two phenomena wherein mass plays a key role? What is its dimensional representation? If a pound force is defined by the extension of a standard spring. What are two kinds of limitations on Newtonian mechanics'?

These operations are generally c: Since the equations of these laws relate vector quantities. These hasic. Define a vector and a scalar. What is a di.

What is an inertial reference? The baric laws of mechanics were thus introduccd. How many scale units of mass density mass per unit volume in the SI units are equivalent to I scale unit in the American system using a slugs. What is meant by line of action o f a vector? The proportionality constant pis called the coefficient of viscosity.

It is not merely for elegance or sophisti- cation that we employ vector algebra. For vector quantities. The mathe- matical symbol for indicating the magnitude of a quantity is a set of vertical lines enclosing the quantity. Note that the magnitude of a quantity is its absolute value. The basic algebraic operations for the handling of scalar quantities are those famil- iar ones studied in grade school.

The instmcter may wish tu assign the reading of these seclians along with the aforemen- tioned questions. A i s positivc sciiliir qu. A tI n-ci lhc final iirrou' ciiii then he iiiterpretcd in teiniir o i i t s length by cinployinf tlic chosen scale f k t o r.

Addition hy pmillclograiii iau.: The magnitude it. This cnmhin. The angle is determined by employing the law of sines for triangle OBA. Find F and a using trigonometry To get the sum shown as F.

Onc force has a magnitude of 10 Ib act- ing in the positive x direction. Thc total unstretched length of the rubber band i s 5 in. The teiisiiin i n the entire extended rubber hand i s then II. The top view 01 the slingshot i s shown in Fig. Simplc dingshut. Con- sequently. We ciin use the law Figure 2. Parsllclogram orcrs It must be emphatically pointed out that thc additiim if vectorsA and R only involves the vectors themselves and iiot thcir lines of actions or thcir posi- tions along their respective lincs 0faction.

For thc additional vector algebra that we will devclop in this chapter. The vectors are connected in Fig. The vector sum that closes the polygon is equivalent to the system of given vectors. The sum F then closes the triangle and is OB.. This process may also be used in the polygon construction. Figure 2.

The extension of this proce- dure to any number of vectors is obvious. Subtraction of vectors. Addition and we proceed a s shown in Fig. The sum of the vectors then is the dashed vector that closes the polygon. I right dirccted away from the origin. A homing pigcon i s released at point A and is observed. Add ii N force pointing in the positive r direction to a magnitude of force 8 and the direction of forcc C?

For the Eim- N forcc at an nnple 45" to the. Subtract the N force in Prohlem 2. Figure P. What is the Figure P. Do this first graphically. A Figure P. A lightweight homemade plane i s bcinf ohserved LIS i t flies i. Whal 2. At the outset. I t flier I O krn due south. Next i t goes southcast for I O k m and finally gocs due south 5 km to reach i t s destination H. See Fig. Give the magnitude of 2. If we do not change the force transmitted by the horizontal 2.

Give the magnitude of the sum nf these forces using the parallelogram law twice. You will leam very shortly that the weight of M must be equal 2. In the preceding problem. Three forces act on the block. The N and the N and opposite to the vector sum of the supporting forces for forms act. Suppose in Problem 2. Add the three vectors using the parallelogram law twice.

What is the sum ofthe forces transmitted by the structural rods to the pin at A? What is the total force? If the difference between forces B and A in Fig. I f N Figure P. Using the parallelogram law. A man pulls with force Won a rope through a simple fric- force D having a magnitude of 25 N.

A mass M is supported by cables I and 2. What total force is exerted on and the direction o f D? We will do this problem differ- ently in Example S. Thc rollers on thc side u l the hlock tlo not 70 Ih crmtrihute to thc vertical support nf the hlock.

Twu soccer player5 approach a stntir. The wire5 cnnnect til the gcrrmerric center of the hlock C. Wc olten replace a vector b y its components siticc the cnniponents are alway:. Dn prohlcin 2. Does the offense score asuming lhiit the g d i c i s asleep '!

When this i s done. Two-dimensional original vector. The two-dimensional resolution citn he accomplished by graphical construction 0 1 the parallelogram. We shall discuss three- dimensional resolution involving three noncoplanar component vectors later in the section.

Form three independent equations for any given a involving the unknowns f. What i s the total furce cxertcd on the dummy hy the players'! Thc two vectors C. Two foothall player. Scalar Components The opposite action 11 addition nf vectors i s c.

The order of the subacnpts gives the sense of the vector-here going from A to 8. We leave it for you to jus- tify the various angles indicated in the diagram. I I wherein a sailboat is going from marker A to marker I3 5.

Orthogonal or the orthogonal directions are used most often in engineering practice. Sailboat tacking. The resolution may be accomplished in two steps. This is the aforementioned three-dimensional resolution. Resolve C along the z direction. What is the additional distance AL beyond 5. Now we first use the law if sines..

OOO m that the sailboat must travel to get from A to B? Clearly the displacement vectofl pARis equivalent to the vector sum of displacement vectors pAc plus pcR in that the same starting points A.

It is also readily possible to find three components not in the. Enlarged parallelogram. Consider the specification of three orthopnal directions' for the resolution of C positioned in the first quadrant. Note tlial.

## Irving H. Shames-Engineering Mechanics (Statics and Dynamics).(1996)

Rcctaogular component OF C. Now le1 us consider the right triangle. Hence C. I uhcrc x. This sh iuld spui. Next take vector C. Clearly irthogonal 01s C. The unit vector is also at times denoted ash.

As a final consideration. It has no dimensions. We formu- late this vector as follows: In the preceding equation. On the diagram. The vector a is called a unit vector.

In establishing C. This will be determined entirely by its use. It is always the case that the triangle formed by the vector and its scalar rec- tangular component is a right triangle. You will write it as 6. The sum of the vectors A and B is found by the parallelogram law to be C. It is important to remember. It thus acts as a free vector. As will be seen Iaitcr. Unit vector a. That determination proceeds by first replacing the displacement vector p.

XImagineyou are "walking" from A to B hut restricting your movements to he along the coordinate directions.

## Where can I get the PDF of Engineering Mechanics by Shames? - Quora

Rectangular parallelepiped used for specifying the direction of a vector. We thereby can replace the vector p. Numbers for this purpose are shown along the sides of the rectan- gular parallelepiped without units.. This movement i s equivalent to going directly from A to B in that the Same endmints result.

IOb as the corresponding scalar equations. AB in the diagram is such a diagonal used for the determination of the line of action of vector F.

Any set of numbers can be used as long as the ratios of these numbers remain the ones required for the proper determi- nation of the orientation of the vector. The purpose of this rectangular parallelepiped and diagonal is to allow for the easy determination of the ori- entation of the line of action and hence the orientation of a vector.

There are times when the rectangular parallelpiped i s not shown explic- itly. This would take us from initial point A to final point IT. Other ways to use the reclangular parallelepiped. In the two-dimensional case. The corresponding displacement veclor would then he p. We thus form the unit vectorbA8. The simplest procedure i s IO mcntally move from the beginning point o f the diagonal to the final point always moving along coordinatc directions Right triangle used fur sppecifying the direction ot it vector m two dirnensionq.

This i s shown i n Fig. Here we can say. Note that D is at the center of the outer edge of the crate.. OOOk N. We first express these three forces in terms of rectangular components. OWk N. What are the forces Fl. We will soon l e a n formally what our common sense tells us.

A crate is supported by three forces. OOOk We have three scalar equations from the previous equation. The lorccs in thc mcmherq must have a vector w m q u a l and opposite to the vcctoi sum of E.

If the component along AH i s N. The entire Eystem i s coplanar. A simple truss to be studied later in detail supports two forces. Resolve thc h force into a set of components along the slot shown and in the vertical direction. If the forces in the members are colinear with the mem- hcrs.

I n the previous pruhlem. A lamer needc to build a fence from the corner of his ham to the corner of hic chicken house 10 m away in the NE dircction. Two tughants are maneuvering an w e a n liiw The desired iota1 inrcc i s 3.

How long i s the fencc? Two men are trying to pull a crate which will not move y until a lb total force is applied in any one direction. The N force is to be resolved into components along the AC and AB directions in the xy plane measured by the angles a and p. The orthogonal components of a force are: What IS the sum of the three forces? The 2. The 1. What force must each man exert to start the box moving as shown?

If the component along AC is to be 1. What are the rectangular components of the lb force? Man A can pull only at 45' to the desired direction of crate motion. What is the vector sum of these forces? Y x Figure P. What is the total cirmpi. What is thc orthogonal total f h x cwnponcnt in tlic. I direction 01 the ioice tiansmittcd to pin A of a roo1 t n h i h i tlic four rncrnher.

What is y? How long mist rncmhel-c OA. What is the unit vector in the direction of the N force'! I 1 where a is the smaller angle between the two vectors. A vector operation that represents such operations con- cisely is the scalar product or dot product.

In effect. See the footnote o n p. The appro- priate sign must.