Dynamics 1

urements will be considered absolute. Time is a measure of the succession of events and is considered an absolute quantity in Newtonian mechanics

urements will be considered absolute. Time is a measure of the succession of events and is considered an absolute quantity in Newtonian mechanics

A particle is a body of negligible dimensions. When the dimensions of a body are irrelevant to the description of its motion or the action of forces on it, the body may be treated as a particle. An airplane, for example, may be treated as a particle for the description of its flight path.

A particle is a body of negligible dimensions. When the dimensions of a body are irrelevant to the description of its motion or the action of forces on it, the body may be treated as a particle. An airplane, for example, may be treated as a particle for the description of its flight path.

A reference frame attached to the surface of the earth has a somewhat complicated motion in the primary system, and a correction to the basic equations of mechanics must be applied for measurements made 4 Chapter 1 Introduction to Dynamics Artificial hand *The original formulations of Sir Isaac Newton may be found in the translation of his Principia (1687), revised by F. Cajori, University of California Press, 1934. James King-Holmes/PhotoResearchers, Inc. c01.qxd 2/8/12 7:02 PM Page 4 relative to the reference frame of the earth. In the calculation of rocket and space-flight trajectories, for example, the absolute motion of the earth becomes an important parameter. For most engineering problems involving machines and structures which remain on the surface of the earth, the corrections are extremely small and may be neglected. For these problems the laws of mechanics may be applied directly with measurements made relative to the earth, and in a practical sense such measurements will be considered absolute.

A reference frame attached to the surface of the earth has a somewhat complicated motion in the primary system, and a correction to the basic equations of mechanics must be applied for measurements made 4 Chapter 1 Introduction to Dynamics Artificial hand *The original formulations of Sir Isaac Newton may be found in the translation of his Principia (1687), revised by F. Cajori, University of California Press, 1934. James King-Holmes/PhotoResearchers, Inc. c01.qxd 2/8/12 7:02 PM Page 4 relative to the reference frame of the earth. In the calculation of rocket and space-flight trajectories, for example, the absolute motion of the earth becomes an important parameter. For most engineering problems involving machines and structures which remain on the surface of the earth, the corrections are extremely small and may be neglected. For these problems the laws of mechanics may be applied directly with measurements made relative to the earth, and in a practical sense such measurements will be considered absolute.

A rigid body is a body whose changes in shape are negligible compared with the overall dimensions of the body or with the changes in position of the body as a whole. As an example of the assumption of rigidity, the small flexural movement of the wing tip of an airplane flying through turbulent air is clearly of no consequence to the description of the motion of the airplane as a whole along its flight path. For this purpose, then, the treatment of the airplane as a rigid body is an acceptable approximation. On the other hand, if we need to examine the internal stresses in the wing structure due to changing dynamic loads, then the deformation characteristics of the structure would have to be examined, and for this purpose the airplane could no longer be considered a rigid body.

A rigid body is a body whose changes in shape are negligible compared with the overall dimensions of the body or with the changes in position of the body as a whole. As an example of the assumption of rigidity, the small flexural movement of the wing tip of an airplane flying through turbulent air is clearly of no consequence to the description of the motion of the airplane as a whole along its flight path. For this purpose, then, the treatment of the airplane as a rigid body is an acceptable approximation. On the other hand, if we need to examine the internal stresses in the wing structure due to changing dynamic loads, then the deformation characteristics of the structure would have to be examined, and for this purpose the airplane could no longer be considered a rigid body.

Accurate values of the gravitational acceleration as measured relative to the surface of the earth account for the fact that the earth is a rotating oblate spheroid with flattening at the poles. These values may g g0 R2 (R h)2 g Gme R2 Article 1/5 Gravitation 9 *It can be proved that the earth, when taken as a sphere with a symmetrical distribution of mass about its center, may be considered a particle with its entire mass concentrated at its center. c01.qxd 2/8/12 7:02 PM Page 9 10 Chapter 1 Introduction to Dynamics *You will be able to derive these relations for a spherical earth after studying relative motion in Chapter 3. Standard Value of g The standard value which has been adopted internationally for the gravitational acceleration relative to the rotating earth at sea level and at a latitude of 45 is 9.806 65 m/s2 or 32.1740 ft/sec2 . This value differs very slightly from that obtained by evaluating the International Gravity Formula for 45. The reason for the small difference is that the earth is not exactly ellipsoidal, as assumed in the formulation of the International Gravity Formula. The proximity of large land masses and the variations in the density of the crust of the earth also influence the local value of g by a small but detectable amount. In almost all engineering applications near the surface of the earth, we can neglect the difference between the absolute and relative values of the gravitational acceleration, and the effect of local Figure 1/1 be calculated to a high degree of accuracy from the 1980 International Gravity Formula, which is where is the latitude and g is expressed in meters per second squared. The formula is based on an ellipsoidal model of the earth and also accounts for the effect of the rotation of the earth

Accurate values of the gravitational acceleration as measured relative to the surface of the earth account for the fact that the earth is a rotating oblate spheroid with flattening at the poles. These values may g g0 R2 (R h)2 g Gme R2 Article 1/5 Gravitation 9 *It can be proved that the earth, when taken as a sphere with a symmetrical distribution of mass about its center, may be considered a particle with its entire mass concentrated at its center. c01.qxd 2/8/12 7:02 PM Page 9 10 Chapter 1 Introduction to Dynamics *You will be able to derive these relations for a spherical earth after studying relative motion in Chapter 3. Standard Value of g The standard value which has been adopted internationally for the gravitational acceleration relative to the rotating earth at sea level and at a latitude of 45 is 9.806 65 m/s2 or 32.1740 ft/sec2 . This value differs very slightly from that obtained by evaluating the International Gravity Formula for 45. The reason for the small difference is that the earth is not exactly ellipsoidal, as assumed in the formulation of the International Gravity Formula. The proximity of large land masses and the variations in the density of the crust of the earth also influence the local value of g by a small but detectable amount. In almost all engineering applications near the surface of the earth, we can neglect the difference between the absolute and relative values of the gravitational acceleration, and the effect of local Figure 1/1 be calculated to a high degree of accuracy from the 1980 International Gravity Formula, which is where is the latitude and g is expressed in meters per second squared. The formula is based on an ellipsoidal model of the earth and also accounts for the effect of the rotation of the earth

Additional quantities used in mechanics and their equivalent base units will be defined as they are introduced in the chapters which follow. However, for convenient reference these quantities are listed in one place in the first table inside the front cover of the book. Professional organizations have established detailed guidelines for the consistent use of SI units, and these guidelines have been followed throughout this book. The most essential ones are summarized inside the front cover, and you should observe these rules carefully

Additional quantities used in mechanics and their equivalent base units will be defined as they are introduced in the chapters which follow. However, for convenient reference these quantities are listed in one place in the first table inside the front cover of the book. Professional organizations have established detailed guidelines for the consistent use of SI units, and these guidelines have been followed throughout this book. The most essential ones are summarized inside the front cover, and you should observe these rules carefully

Along with the development of the principles and analytical tools needed for modern dynamics, one of the major aims of this book is to provide many opportunities to develop the ability to formulate good mathematical models. Strong emphasis is placed on a wide range of practical problems which not only require you to apply theory but also force you to make relevant assumptions.

Along with the development of the principles and analytical tools needed for modern dynamics, one of the major aims of this book is to provide many opportunities to develop the ability to formulate good mathematical models. Strong emphasis is placed on a wide range of practical problems which not only require you to apply theory but also force you to make relevant assumptions.

An effective method of attack is essential in the solution of dynamics problems, as for all engineering problems. Development of good habits in formulating problems and in representing their solutions will be an invaluable asset. Each solution should proceed with a logical sequence of steps from hypothesis to conclusion. The following sequence of steps is useful in the construction of problem solutions. 1. Formulate the problem: (a) State the given data. (b) State the desired result. (c) State your assumptions and approximations. 2. Develop the solution: (a) Draw any needed diagrams, and include coordinates which are appropriate for the problem at hand. (b) State the governing principles to be applied to your solution. (c) Make your calculations. (d) Ensure that your calculations are consistent with the accuracy justified by the data. (e) Be sure that you have used consistent units throughout your calculations. (f) Ensure that your answers are reasonable in terms of magnitudes, directions, common sense, etc. (g) Draw conclusions. The arrangement of your work should be neat and orderly. This will help your thought process and enable others to understand your work. The discipline of doing orderly work will help you to develop skill in problem formulation and analysis. Problems which seem complicated at first often become clear when you approach them with logic and discipline

An effective method of attack is essential in the solution of dynamics problems, as for all engineering problems. Development of good habits in formulating problems and in representing their solutions will be an invaluable asset. Each solution should proceed with a logical sequence of steps from hypothesis to conclusion. The following sequence of steps is useful in the construction of problem solutions. 1. Formulate the problem: (a) State the given data. (b) State the desired result. (c) State your assumptions and approximations. 2. Develop the solution: (a) Draw any needed diagrams, and include coordinates which are appropriate for the problem at hand. (b) State the governing principles to be applied to your solution. (c) Make your calculations. (d) Ensure that your calculations are consistent with the accuracy justified by the data. (e) Be sure that you have used consistent units throughout your calculations. (f) Ensure that your answers are reasonable in terms of magnitudes, directions, common sense, etc. (g) Draw conclusions. The arrangement of your work should be neat and orderly. This will help your thought process and enable others to understand your work. The discipline of doing orderly work will help you to develop skill in problem formulation and analysis. Problems which seem complicated at first often become clear when you approach them with logic and discipline

As another example, the retarding effect of bearing friction on the motion of a machine may often be neglected if the friction forces are small compared with the other applied forces. However, these same friction forces cannot be neglected if the purpose of the inquiry is to determine the decrease in efficiency of the machine due to the friction process. Thus, the type of assumptions you make depends on what information is desired and on the accuracy required

As another example, the retarding effect of bearing friction on the motion of a machine may often be neglected if the friction forces are small compared with the other applied forces. However, these same friction forces cannot be neglected if the purpose of the inquiry is to determine the decrease in efficiency of the machine due to the friction process. Thus, the type of assumptions you make depends on what information is desired and on the accuracy required

As shown in the table, in SI the units for mass, length, and time are taken as base units, and the units for force are derived from Newton's second law of motion, Eq. 1/1. In the U.S. customary system the units for force, length, and time are base units and the units for mass are derived from the second law

As shown in the table, in SI the units for mass, length, and time are taken as base units, and the units for force are derived from Newton's second law of motion, Eq. 1/1. In the U.S. customary system the units for force, length, and time are base units and the units for mass are derived from the second law

Because the gravitational attraction or weight of a body is a force, it should always be expressed in force units, newtons (N) in SI units and pounds force (lb) in U.S. customary units. To avoid confusion, the word "weight" in this book will be restricted to mean the force of gravitational attraction.

Because the gravitational attraction or weight of a body is a force, it should always be expressed in force units, newtons (N) in SI units and pounds force (lb) in U.S. customary units. To avoid confusion, the word "weight" in this book will be restricted to mean the force of gravitational attraction.

Both the International System of metric units (SI) and the U.S. customary system of units are defined and used in Vol. 2 Dynamics, although a stronger emphasis is placed on the metric system because it is replacing the U.S. customary system. However, numerical conversion from one system to the other will often be needed in U.S. engineering F ma ¨x ˙x 6 Chapter 1 Introduction to Dynamics *To some it is preferable to interpret Newton's second law as meaning that the resultant force acting on a particle is proportional to the time rate of change of momentum of the particle and that this change is in the direction of the force. Both formulations are equally correct when applied to a particle of constant mass. c01.qxd 2/8/12 7:02 PM Page 6 Article 1/4 Units 7 practice for some years to come. To become familiar with each system, it is necessary to think directly in that system. Familiarity with the new system cannot be achieved simply by the conversion of numerical results from the old system.

Both the International System of metric units (SI) and the U.S. customary system of units are defined and used in Vol. 2 Dynamics, although a stronger emphasis is placed on the metric system because it is replacing the U.S. customary system. However, numerical conversion from one system to the other will often be needed in U.S. engineering F ma ¨x ˙x 6 Chapter 1 Introduction to Dynamics *To some it is preferable to interpret Newton's second law as meaning that the resultant force acting on a particle is proportional to the time rate of change of momentum of the particle and that this change is in the direction of the force. Both formulations are equally correct when applied to a particle of constant mass. c01.qxd 2/8/12 7:02 PM Page 6 Article 1/4 Units 7 practice for some years to come. To become familiar with each system, it is necessary to think directly in that system. Familiarity with the new system cannot be achieved simply by the conversion of numerical results from the old system.

Mass is the quantitative measure of the inertia or resistance to change in motion of a body. Mass may also be considered as the quantity of matter in a body as well as the property which gives rise to gravitational attraction

Mass is the quantitative measure of the inertia or resistance to change in motion of a body. Mass may also be considered as the quantity of matter in a body as well as the property which gives rise to gravitational attraction

Construction of an idealized mathematical model for a given engineering problem always requires approximations to be made. Some of these approximations may be mathematical, whereas others will be physical. For instance, it is often necessary to neglect small distances, angles, or forces compared with large distances, angles, or forces. If the change in velocity of a body with time is nearly uniform, then an assumption of constant acceleration may be justified. An interval of motion which cannot be easily described in its entirety is often divided into small increments, each of which can be approximated.

Construction of an idealized mathematical model for a given engineering problem always requires approximations to be made. Some of these approximations may be mathematical, whereas others will be physical. For instance, it is often necessary to neglect small distances, angles, or forces compared with large distances, angles, or forces. If the change in velocity of a body with time is nearly uniform, then an assumption of constant acceleration may be justified. An interval of motion which cannot be easily described in its entirety is often divided into small increments, each of which can be approximated.

Dynamics involves the frequent use of time derivatives of both vectors and scalars. As a notational shorthand, a dot over a symbol will frequently be used to indicate a derivative with respect to time. Thus, means dx/dt and stands for d2 x/dt2 .

Dynamics involves the frequent use of time derivatives of both vectors and scalars. As a notational shorthand, a dot over a symbol will frequently be used to indicate a derivative with respect to time. Thus, means dx/dt and stands for d2 x/dt2 .

Dynamics is a relatively recent subject compared with statics. The beginning of a rational understanding of dynamics is credited to Galileo (1564-1642), who made careful observations concerning bodies in free fall, motion on an inclined plane, and motion of the pendulum. He was largely responsible for bringing a scientific approach to the investigation of physical problems. Galileo was continually under severe criticism for refusing to accept the established beliefs of his day, such as the philosophies of Aristotle which held, for example, that heavy bodies fall more rapidly than light bodies. The lack of accurate means for the measurement of time was a severe handicap to Galileo, and further significant development in dynamics awaited the invention of the pendulum clock by Huygens in 1657.

Dynamics is a relatively recent subject compared with statics. The beginning of a rational understanding of dynamics is credited to Galileo (1564-1642), who made careful observations concerning bodies in free fall, motion on an inclined plane, and motion of the pendulum. He was largely responsible for bringing a scientific approach to the investigation of physical problems. Galileo was continually under severe criticism for refusing to accept the established beliefs of his day, such as the philosophies of Aristotle which held, for example, that heavy bodies fall more rapidly than light bodies. The lack of accurate means for the measurement of time was a severe handicap to Galileo, and further significant development in dynamics awaited the invention of the pendulum clock by Huygens in 1657.

Dynamics is that branch of mechanics which deals with the motion of bodies under the action of forces. The study of dynamics in engineering usually follows the study of statics, which deals with the effects of forces on bodies at rest. Dynamics has two distinct parts: kinematics, which is the study of motion without reference to the forces which cause motion, and kinetics, which relates the action of forces on bodies to their resulting motions. A thorough comprehension of dynamics will provide one of the most useful and powerful tools for analysis in engineering

Dynamics is that branch of mechanics which deals with the motion of bodies under the action of forces. The study of dynamics in engineering usually follows the study of statics, which deals with the effects of forces on bodies at rest. Dynamics has two distinct parts: kinematics, which is the study of motion without reference to the forces which cause motion, and kinetics, which relates the action of forces on bodies to their resulting motions. A thorough comprehension of dynamics will provide one of the most useful and powerful tools for analysis in engineering

In SI units, the kilogram should be used exclusively as a unit of mass and never force. Unfortunately, in the MKS (meter, kilogram, second) gravitational system, which has been used in some countries for many years, the kilogram has been commonly used both as a unit of force and as a unit of mass

In SI units, the kilogram should be used exclusively as a unit of mass and never force. Unfortunately, in the MKS (meter, kilogram, second) gravitational system, which has been used in some countries for many years, the kilogram has been commonly used both as a unit of force and as a unit of mass

Every object which falls in a vacuum at a given height near the surface of the earth will have the same acceleration g, regardless of its mass. This result can be obtained by combining Eqs. 1/1 and 1/2 and canceling the term representing the mass of the falling object. This combination gives where me is the mass of the earth and R is the radius of the earth.* The mass me and the mean radius R of the earth have been found through experimental measurements to be 5.976(1024) kg and 6.371(106 ) m, respectively. These values, together with the value of G already cited, when substituted into the expression for g, give a mean value of g 9.825 m/s2

Every object which falls in a vacuum at a given height near the surface of the earth will have the same acceleration g, regardless of its mass. This result can be obtained by combining Eqs. 1/1 and 1/2 and canceling the term representing the mass of the falling object. This combination gives where me is the mass of the earth and R is the radius of the earth.* The mass me and the mean radius R of the earth have been found through experimental measurements to be 5.976(1024) kg and 6.371(106 ) m, respectively. These values, together with the value of G already cited, when substituted into the expression for g, give a mean value of g 9.825 m/s2

In SI units, by definition, one newton is that force which will give a one-kilogram mass an acceleration of one meter per second squared. In the U.S. customary system a 32.1740-pound mass (1 slug) will have an acceleration of one foot per second squared when acted on by a force of one pound. Thus, for each system we have from Eq. 1/

In SI units, by definition, one newton is that force which will give a one-kilogram mass an acceleration of one meter per second squared. In the U.S. customary system a 32.1740-pound mass (1 slug) will have an acceleration of one foot per second squared when acted on by a force of one pound. Thus, for each system we have from Eq. 1/

In U.S. customary units, the pound is unfortunately used both as a unit of force (lbf) and as a unit of mass (lbm). The use of the unit lbm is especially prevalent in the specification of the thermal properties of liquids and gases. The lbm is the amount of mass which weighs 1 lbf under standard conditions (at a latitude of 45 and at sea level). In order to avoid the confusion which would be caused by the use of two units for mass (slug and lbm), in this textbook we use almost exclusively the unit slug for mass. This practice makes dynamics much simpler than if the lbm were used. In addition, this approach allows us to use the symbol lb to always mean pound force.

In U.S. customary units, the pound is unfortunately used both as a unit of force (lbf) and as a unit of mass (lbm). The use of the unit lbm is especially prevalent in the specification of the thermal properties of liquids and gases. The lbm is the amount of mass which weighs 1 lbf under standard conditions (at a latitude of 45 and at sea level). In order to avoid the confusion which would be caused by the use of two units for mass (slug and lbm), in this textbook we use almost exclusively the unit slug for mass. This practice makes dynamics much simpler than if the lbm were used. In addition, this approach allows us to use the symbol lb to always mean pound force.

In applying the laws of dynamics, we may use numerical values of the involved quantities, or we may use algebraic symbols and leave the answer as a formula. When numerical values are used, the magnitudes of all quantities expressed in their particular units are evident at each stage of the calculation. This approach is useful when we need to know the magnitude of each term.

In applying the laws of dynamics, we may use numerical values of the involved quantities, or we may use algebraic symbols and leave the answer as a formula. When numerical values are used, the magnitudes of all quantities expressed in their particular units are evident at each stage of the calculation. This approach is useful when we need to know the magnitude of each term.

In describing the relations between forces and the motions they produce, it is essential to define clearly the system to which a principle is to be applied. At times a single particle or a rigid body is the system to be isolated, whereas at other times two or more bodies taken together constitute the system.

In describing the relations between forces and the motions they produce, it is essential to define clearly the system to which a principle is to be applied. At times a single particle or a rigid body is the system to be isolated, whereas at other times two or more bodies taken together constitute the system.

Law I. A particle remains at rest or continues to move with uniform velocity (in a straight line with a constant speed) if there is no unbalanced force acting on it.

Law I. A particle remains at rest or continues to move with uniform velocity (in a straight line with a constant speed) if there is no unbalanced force acting on it.

Law II. The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force.*

Law II. The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force.*

Law III. The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction, and collinear.

Law III. The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction, and collinear.

Newton (1642-1727), guided by Galileo's work, was able to make an accurate formulation of the laws of motion and, thus, to place dynamics Galileo Galilei Portrait of Galileo Galilei (1564-1642) (oil on canvas), Sustermans, Justus (1597-1681) (school of)/Galleria Palatina, Florence, Italy/Bridgeman Art Library 1/1 History and Modern Applications 1/2 Basic Concepts 1/3 Newton's Laws 1/4 Units 1/5 Gravitation 1/6 Dimensions 1/7 Solving Problems in Dynamics 1/8 Chapter Review CHAPTER OUTLINE 1 Introduction to Dynamics © Fine Art Images/SuperStock c01.qxd 2/8/12 7:02 PM Page 3 on a sound basis. Newton's famous work was published in the first edition of his Principia,* which is generally recognized as one of the greatest of all recorded contributions to knowledge. In addition to stating the laws governing the motion of a particle, Newton was the first to correctly formulate the law of universal gravitation. Although his mathematical description was accurate, he felt that the concept of remote transmission of gravitational force without a supporting medium was an absurd notion. Following Newton's time, important contributions to mechanics were made by Euler, D'Alembert, Lagrange, Laplace, Poinsot, Coriolis, Einstein, and others.

Newton (1642-1727), guided by Galileo's work, was able to make an accurate formulation of the laws of motion and, thus, to place dynamics Galileo Galilei Portrait of Galileo Galilei (1564-1642) (oil on canvas), Sustermans, Justus (1597-1681) (school of)/Galleria Palatina, Florence, Italy/Bridgeman Art Library 1/1 History and Modern Applications 1/2 Basic Concepts 1/3 Newton's Laws 1/4 Units 1/5 Gravitation 1/6 Dimensions 1/7 Solving Problems in Dynamics 1/8 Chapter Review CHAPTER OUTLINE 1 Introduction to Dynamics © Fine Art Images/SuperStock c01.qxd 2/8/12 7:02 PM Page 3 on a sound basis. Newton's famous work was published in the first edition of his Principia,* which is generally recognized as one of the greatest of all recorded contributions to knowledge. In addition to stating the laws governing the motion of a particle, Newton was the first to correctly formulate the law of universal gravitation. Although his mathematical description was accurate, he felt that the concept of remote transmission of gravitational force without a supporting medium was an absurd notion. Following Newton's time, important contributions to mechanics were made by Euler, D'Alembert, Lagrange, Laplace, Poinsot, Coriolis, Einstein, and others.

Newton's law of gravitation, which governs the mutual attraction between bodies, is (1/2) where F the mutual force of attraction between two particles G a universal constant called the constant of gravitation m1, m2 the masses of the two particles r the distance between the centers of the particles

Newton's law of gravitation, which governs the mutual attraction between bodies, is (1/2) where F the mutual force of attraction between two particles G a universal constant called the constant of gravitation m1, m2 the masses of the two particles r the distance between the centers of the particles

One of the greatest difficulties encountered by students is the inability to make this transition freely. You should recognize that the mathematical formulation of a physical problem represents an ideal and limiting description, or model, which approximates but never quite matches the actual physical situation. In Art. 1/8 of Vol. 1 Statics we extensively discussed the approach to solving problems in statics. We assume therefore, that you are familiar with this approach, which we summarize here as applied to dynamics.

One of the greatest difficulties encountered by students is the inability to make this transition freely. You should recognize that the mathematical formulation of a physical problem represents an ideal and limiting description, or model, which approximates but never quite matches the actual physical situation. In Art. 1/8 of Vol. 1 Statics we extensively discussed the approach to solving problems in statics. We assume therefore, that you are familiar with this approach, which we summarize here as applied to dynamics.

Only since machines and structures have operated with high speeds and appreciable accelerations has it been necessary to make calculations based on the principles of dynamics rather than on the principles of statics. The rapid technological developments of the present day require increasing application of the principles of mechanics, particularly dynamics. These principles are basic to the analysis and design of moving structures, to fixed structures subject to shock loads, to robotic devices, to automatic control systems, to rockets, missiles, and spacecraft, to ground and air transportation vehicles, to electron ballistics of electrical devices, and to machinery of all types such as turbines, pumps, reciprocating engines, hoists, machine tools, etc

Only since machines and structures have operated with high speeds and appreciable accelerations has it been necessary to make calculations based on the principles of dynamics rather than on the principles of statics. The rapid technological developments of the present day require increasing application of the principles of mechanics, particularly dynamics. These principles are basic to the analysis and design of moving structures, to fixed structures subject to shock loads, to robotic devices, to automatic control systems, to rockets, missiles, and spacecraft, to ground and air transportation vehicles, to electron ballistics of electrical devices, and to machinery of all types such as turbines, pumps, reciprocating engines, hoists, machine tools, etc

Solutions to the various equations of dynamics can be obtained in one of three ways. 1. Obtain a direct mathematical solution by hand calculation, using either algebraic symbols or numerical values. We can solve the large majority of the problems this way. 2. Obtain graphical solutions for certain problems, such as the determination of velocities and accelerations of rigid bodies in twodimensional relative motion. 3. Solve the problem by computer. A number of problems in Vol. 2 Dynamics are designated as Computer-Oriented Problems. They appear at the end of the Review Problem sets and were selected to illustrate the type of problem for which solution by computer offers a distinct advantage.

Solutions to the various equations of dynamics can be obtained in one of three ways. 1. Obtain a direct mathematical solution by hand calculation, using either algebraic symbols or numerical values. We can solve the large majority of the problems this way. 2. Obtain graphical solutions for certain problems, such as the determination of velocities and accelerations of rigid bodies in twodimensional relative motion. 3. Solve the problem by computer. A number of problems in Vol. 2 Dynamics are designated as Computer-Oriented Problems. They appear at the end of the Review Problem sets and were selected to illustrate the type of problem for which solution by computer offers a distinct advantage.

Space is the geometric region occupied by bodies. Position in space is determined relative to some geometric reference system by means of linear and angular measurements. The basic frame of reference for the laws of Newtonian mechanics is the primary inertial system or astronomical frame of reference, which is an imaginary set of rectangular axes assumed to have no translation or rotation in space. Measurements show that the laws of Newtonian mechanics are valid for this reference system as long as any velocities involved are negligible compared with the speed of light, which is 300 000 km/s or 186,000 mi/sec. Measurements made with respect to this reference are said to be absolute, and this reference system may be considered "fixed" in space

Space is the geometric region occupied by bodies. Position in space is determined relative to some geometric reference system by means of linear and angular measurements. The basic frame of reference for the laws of Newtonian mechanics is the primary inertial system or astronomical frame of reference, which is an imaginary set of rectangular axes assumed to have no translation or rotation in space. Measurements show that the laws of Newtonian mechanics are valid for this reference system as long as any velocities involved are negligible compared with the speed of light, which is 300 000 km/s or 186,000 mi/sec. Measurements made with respect to this reference are said to be absolute, and this reference system may be considered "fixed" in space

Tables defining the SI units and giving numerical conversions between U.S. customary and SI units are included inside the front cover of the book. Charts comparing selected quantities in SI and U.S. customary units are included inside the back cover of the book to facilitate conversion and to help establish a feel for the relative size of units in both systems. The four fundamental quantities of mechanics, and their units and symbols for the two systems, are summarized in the following table:

Tables defining the SI units and giving numerical conversions between U.S. customary and SI units are included inside the front cover of the book. Charts comparing selected quantities in SI and U.S. customary units are included inside the back cover of the book to facilitate conversion and to help establish a feel for the relative size of units in both systems. The four fundamental quantities of mechanics, and their units and symbols for the two systems, are summarized in the following table:

The SI system is termed an absolute system because the standard for the base unit kilogram (a platinum-iridium cylinder kept at the International Bureau of Standards near Paris, France) is independent of the gravitational attraction of the earth. On the other hand, the U.S. customary system is termed a gravitational system because the standard for the base unit pound (the weight of a standard mass located at sea level and at a latitude of 45) requires the presence of the gravitational field of the earth. This distinction is a fundamental difference between the two systems of units

The SI system is termed an absolute system because the standard for the base unit kilogram (a platinum-iridium cylinder kept at the International Bureau of Standards near Paris, France) is independent of the gravitational attraction of the earth. On the other hand, the U.S. customary system is termed a gravitational system because the standard for the base unit pound (the weight of a standard mass located at sea level and at a latitude of 45) requires the presence of the gravitational field of the earth. This distinction is a fundamental difference between the two systems of units

The absolute acceleration due to gravity as determined for a nonrotating earth may be computed from the relative values to a close approximation by adding 3.382(102 ) cos2 m/s2 , which removes the effect of the rotation of the earth. The variation of both the absolute and the relative values of g with latitude is shown in Fig. 1/1 for sea-level conditions.*

The absolute acceleration due to gravity as determined for a nonrotating earth may be computed from the relative values to a close approximation by adding 3.382(102 ) cos2 m/s2 , which removes the effect of the rotation of the earth. The variation of both the absolute and the relative values of g with latitude is shown in Fig. 1/1 for sea-level conditions.*

The acceleration due to gravity as determined from the gravitational law is the acceleration which would be measured from a set of axes whose origin is at the center of the earth but which does not rotate with the earth. With respect to these "fixed" axes, then, this value may be termed the absolute value of g. Because the earth rotates, the acceleration of a freely falling body as measured from a position attached to the surface of the earth is slightly less than the absolute value

The acceleration due to gravity as determined from the gravitational law is the acceleration which would be measured from a set of axes whose origin is at the center of the earth but which does not rotate with the earth. With respect to these "fixed" axes, then, this value may be termed the absolute value of g. Because the earth rotates, the acceleration of a freely falling body as measured from a position attached to the surface of the earth is slightly less than the absolute value

The apparent weight of a body as determined by a spring balance, calibrated to read the correct force and attached to the surface of the earth, will be slightly less than its true weight. The difference is due to the rotation of the earth. The ratio of the apparent weight to the apparent or relative acceleration due to gravity still gives the correct value of mass. The apparent weight and the relative acceleration due to gravity are, of course, the quantities which are measured in experiments conducted on the surface of the earth.

The apparent weight of a body as determined by a spring balance, calibrated to read the correct force and attached to the surface of the earth, will be slightly less than its true weight. The difference is due to the rotation of the earth. The ratio of the apparent weight to the apparent or relative acceleration due to gravity still gives the correct value of mass. The apparent weight and the relative acceleration due to gravity are, of course, the quantities which are measured in experiments conducted on the surface of the earth.

The concepts basic to mechanics were set forth in Art. 1/2 of Vol. 1 Statics. They are summarized here along with additional comments of special relevance to the study of dynamics.

The concepts basic to mechanics were set forth in Art. 1/2 of Vol. 1 Statics. They are summarized here along with additional comments of special relevance to the study of dynamics.

The definition of the system to be analyzed is made clear by constructing its free-body diagram. This diagram consists of a closed outline of the external boundary of the system. All bodies which contact and exert forces on the system but are not a part of it are removed and replaced by vectors representing the forces they exert on the isolated system. In this way, we make a clear distinction between the action and reaction of each force, and all forces on and external to the system are accounted for. We assume that you are familiar with the technique of drawing free-body diagrams from your prior work in static

The definition of the system to be analyzed is made clear by constructing its free-body diagram. This diagram consists of a closed outline of the external boundary of the system. All bodies which contact and exert forces on the system but are not a part of it are removed and replaced by vectors representing the forces they exert on the isolated system. In this way, we make a clear distinction between the action and reaction of each force, and all forces on and external to the system are accounted for. We assume that you are familiar with the technique of drawing free-body diagrams from your prior work in static

The force of gravitational attraction of the earth on a body depends on the position of the body relative to the earth. If the earth were a perfect homogeneous sphere, a body with a mass of exactly 1 kg would be attracted to the earth by a force of 9.825 N on the surface of the earth, 9.822 N at an altitude of 1 km, 9.523 N at an altitude of 100 km, 7.340 N at an altitude of 1000 km, and 2.456 N at an altitude equal to the mean radius of the earth, 6371 km. Thus the variation in gravitational attraction of high-altitude rockets and spacecraft becomes a major consideration.

The force of gravitational attraction of the earth on a body depends on the position of the body relative to the earth. If the earth were a perfect homogeneous sphere, a body with a mass of exactly 1 kg would be attracted to the earth by a force of 9.825 N on the surface of the earth, 9.822 N at an altitude of 1 km, 9.523 N at an altitude of 100 km, 7.340 N at an altitude of 1000 km, and 2.456 N at an altitude equal to the mean radius of the earth, 6371 km. Thus the variation in gravitational attraction of high-altitude rockets and spacecraft becomes a major consideration.

The gravitational attraction of the earth on a body of mass m may be calculated from the results of a simple gravitational experiment. The body is allowed to fall freely in a vacuum, and its absolute acceleration is measured. If the gravitational force of attraction or true weight of the body is W, then, because the body falls with an absolute acceleration g, Eq. 1/1 gives

The gravitational attraction of the earth on a body of mass m may be calculated from the results of a simple gravitational experiment. The body is allowed to fall freely in a vacuum, and its absolute acceleration is measured. If the gravitational force of attraction or true weight of the body is W, then, because the body falls with an absolute acceleration g, Eq. 1/1 gives

The proximity of large land masses and the variations in the density of the crust of the earth also influence the local value of g by a small but detectable amount. In almost all engineering applications near the surface of the earth, we can neglect the difference between the absolute and relative values of the gravitational acceleration, and the effect of local

The proximity of large land masses and the variations in the density of the crust of the earth also influence the local value of g by a small but detectable amount. In almost all engineering applications near the surface of the earth, we can neglect the difference between the absolute and relative values of the gravitational acceleration, and the effect of local

The standard value which has been adopted internationally for the gravitational acceleration relative to the rotating earth at sea level and at a latitude of 45 is 9.806 65 m/s2 or 32.1740 ft/sec2 . This value differs very slightly from that obtained by evaluating the International Gravity Formula for 45. The reason for the small difference is that the earth is not exactly ellipsoidal, as assumed in the formulation of the International Gravity Formula.

The standard value which has been adopted internationally for the gravitational acceleration relative to the rotating earth at sea level and at a latitude of 45 is 9.806 65 m/s2 or 32.1740 ft/sec2 . This value differs very slightly from that obtained by evaluating the International Gravity Formula for 45. The reason for the small difference is that the earth is not exactly ellipsoidal, as assumed in the formulation of the International Gravity Formula.

The study of dynamics concerns the understanding and description of the motions of bodies. This description, which is largely mathematical, enables predictions of dynamical behavior to be made. A dual thought process is necessary in formulating this description. It is necessary to think in terms of both the physical situation and the corresponding mathematical description. This repeated transition of thought between the physical and the mathematical is required in the analysis of every problem.

The study of dynamics concerns the understanding and description of the motions of bodies. This description, which is largely mathematical, enables predictions of dynamical behavior to be made. A dual thought process is necessary in formulating this description. It is necessary to think in terms of both the physical situation and the corresponding mathematical description. This repeated transition of thought between the physical and the mathematical is required in the analysis of every problem.

The subject of dynamics is based on a surprisingly few fundamental concepts and principles which, however, can be extended and applied over a wide range of conditions. The study of dynamics is valuable partly because it provides experience in reasoning from fundamentals. This experience cannot be obtained merely by memorizing the kinematic and dynamic equations which describe various motions. It must be obtained through exposure to a wide variety of problem situations which require the choice, use, and extension of basic principles to meet the given conditions.

The subject of dynamics is based on a surprisingly few fundamental concepts and principles which, however, can be extended and applied over a wide range of conditions. The study of dynamics is valuable partly because it provides experience in reasoning from fundamentals. This experience cannot be obtained merely by memorizing the kinematic and dynamic equations which describe various motions. It must be obtained through exposure to a wide variety of problem situations which require the choice, use, and extension of basic principles to meet the given conditions.

The symbolic solution, however, has several advantages over the numerical solution: 1. The use of symbols helps to focus attention on the connection between the physical situation and its related mathematical description. 2. A symbolic solution enables you to make a dimensional check at every step, whereas dimensional homogeneity cannot be checked when only numerical values are used. 3. We can use a symbolic solution repeatedly for obtaining answers to the same problem with different units or different numerical values.

The symbolic solution, however, has several advantages over the numerical solution: 1. The use of symbols helps to focus attention on the connection between the physical situation and its related mathematical description. 2. A symbolic solution enables you to make a dimensional check at every step, whereas dimensional homogeneity cannot be checked when only numerical values are used. 3. We can use a symbolic solution repeatedly for obtaining answers to the same problem with different units or different numerical values.

The value of the gravitational constant obtained from experimental data is . Except for some spacecraft applications, the only gravitational force of appreciable magnitude in engineering is the force due to the attraction of the earth. It was shown in Vol. 1 Statics, for example, that each of two iron spheres 100 mm in diameter is attracted to the earth with a gravitational force of 37.1 N, which is called its weight, but the force of mutual attraction between them if they are just touching is only 0.000 000 095 1 N

The value of the gravitational constant obtained from experimental data is . Except for some spacecraft applications, the only gravitational force of appreciable magnitude in engineering is the force due to the attraction of the earth. It was shown in Vol. 1 Statics, for example, that each of two iron spheres 100 mm in diameter is attracted to the earth with a gravitational force of 37.1 N, which is called its weight, but the force of mutual attraction between them if they are just touching is only 0.000 000 095 1 N

The variation of g with altitude is easily determined from the gravitational law. If g0 represents the absolute acceleration due to gravity at sea level, the absolute value at an altitude h is where R is the radius of the earth.

The variation of g with altitude is easily determined from the gravitational law. If g0 represents the absolute acceleration due to gravity at sea level, the absolute value at an altitude h is where R is the radius of the earth.

These laws have been verified by countless physical measurements. The first two laws hold for measurements made in an absolute frame of reference, but are subject to some correction when the motion is measured relative to a reference system having acceleration, such as one attached to the surface of the earth. Newton's second law forms the basis for most of the analysis in dynamics. For a particle of mass m subjected to a resultant force F, the law may be stated as (1/1) where a is the resulting acceleration measured in a nonaccelerating frame of reference. Newton's first law is a consequence of the second law since there is no acceleration when the force is zero, and so the particle is either at rest or is moving with constant velocity. The third law constitutes the principle of action and reaction with which you should be thoroughly familiar from your work in statics.

These laws have been verified by countless physical measurements. The first two laws hold for measurements made in an absolute frame of reference, but are subject to some correction when the motion is measured relative to a reference system having acceleration, such as one attached to the surface of the earth. Newton's second law forms the basis for most of the analysis in dynamics. For a particle of mass m subjected to a resultant force F, the law may be stated as (1/1) where a is the resulting acceleration measured in a nonaccelerating frame of reference. Newton's first law is a consequence of the second law since there is no acceleration when the force is zero, and so the particle is either at rest or is moving with constant velocity. The third law constitutes the principle of action and reaction with which you should be thoroughly familiar from your work in statics.

Thus, facility with both forms of solution is essential, and you should practice each in the problem work. In the case of numerical solutions, we repeat from Vol. 1 Statics our convention for the display of results. All given data are taken to be exact, and results are generally displayed to three significant figures, unless the leading digit is a one, in which case four significant figures are displayed.

Thus, facility with both forms of solution is essential, and you should practice each in the problem work. In the case of numerical solutions, we repeat from Vol. 1 Statics our convention for the display of results. All given data are taken to be exact, and results are generally displayed to three significant figures, unless the leading digit is a one, in which case four significant figures are displayed.

Vector and scalar quantities have been treated extensively in Vol. 1 Statics, and their distinction should be perfectly clear by now. Scalar quantities are printed in lightface italic type, and vectors are shown in boldface type. Thus, V denotes the scalar magnitude of the vector V. It is important that we use an identifying mark, such as an underline V, for all handwritten vectors to take the place of the boldface designation in print. For two nonparallel vectors recall, for example, that V1 V2 and V1 V2 have two entirely different meanings

Vector and scalar quantities have been treated extensively in Vol. 1 Statics, and their distinction should be perfectly clear by now. Scalar quantities are printed in lightface italic type, and vectors are shown in boldface type. Thus, V denotes the scalar magnitude of the vector V. It is important that we use an identifying mark, such as an underline V, for all handwritten vectors to take the place of the boldface designation in print. For two nonparallel vectors recall, for example, that V1 V2 and V1 V2 have two entirely different meanings

We assume that you are familiar with the geometry and algebra of vectors through previous study of statics and mathematics. Students who need to review these topics will find a brief summary of them in Appendix C along with other mathematical relations which find frequent use in mechanics. Experience has shown that the geometry of mechanics is often a source of difficulty for students. Mechanics by its very nature is geometrical, and students should bear this in mind as they review their mathematics. In addition to vector algebra, dynamics requires the use of vector calculus, and the essentials of this topic will be developed in the text as they are needed.

We assume that you are familiar with the geometry and algebra of vectors through previous study of statics and mathematics. Students who need to review these topics will find a brief summary of them in Appendix C along with other mathematical relations which find frequent use in mechanics. Experience has shown that the geometry of mechanics is often a source of difficulty for students. Mechanics by its very nature is geometrical, and students should bear this in mind as they review their mathematics. In addition to vector algebra, dynamics requires the use of vector calculus, and the essentials of this topic will be developed in the text as they are needed.

You should be constantly alert to the various assumptions called for in the formulation of real problems. The ability to understand and make use of the appropriate assumptions when formulating and solving engineering problems is certainly one of the most important characteristics of a successful engineer.

You should be constantly alert to the various assumptions called for in the formulation of real problems. The ability to understand and make use of the appropriate assumptions when formulating and solving engineering problems is certainly one of the most important characteristics of a successful engineer.

fferent units such as meters, millimeters, or kilometers. Thus, a dimension is different from a unit. The principle of dimensional homogeneity states that all physical relations must be dimensionally homogeneous; that is, the dimensions of all terms in an equation must be the same. It is customary to use the symbols L, M, T, and F to stand for length, mass, time, and force, respectively. In SI units force is a derived quantity and from Eq. 1/1 has the dimensions of mass times acceleration or One important use of the dimensional homogeneity principle is to check the dimensional correctness of some derived physical relation. We can derive the following expression for the velocity v of a body of mass m which is moved from rest a horizontal distance x by a force F:

fferent units such as meters, millimeters, or kilometers. Thus, a dimension is different from a unit. The principle of dimensional homogeneity states that all physical relations must be dimensionally homogeneous; that is, the dimensions of all terms in an equation must be the same. It is customary to use the symbols L, M, T, and F to stand for length, mass, time, and force, respectively. In SI units force is a derived quantity and from Eq. 1/1 has the dimensions of mass times acceleration or One important use of the dimensional homogeneity principle is to check the dimensional correctness of some derived physical relation. We can derive the following expression for the velocity v of a body of mass m which is moved from rest a horizontal distance x by a force F:

nal attraction. Force is the vector action of one body on another. The properties of forces have been thoroughly treated in Vol. 1 Statics.

nal attraction. Force is the vector action of one body on another. The properties of forces have been thoroughly treated in Vol. 1 Statics.

where the is a dimensionless coefficient resulting from integration. This equation is dimensionally correct because substitution of L, M, and T gives Dimensional homogeneity is a necessary condition for correctness of a physical relation, but it is not sufficient, since it is possible to construct [MLT2][L] [M][LT1] 2 1 2 Fx 1 2mv2 F ML/T2 W mg Article 1/6 Dimensions 11 c01.qxd 2/8/12 7:02 PM Page 11 an equation which is dimensionally correct but does not represent a correct relation. You should perform a dimensional check on the answer to every problem whose solution is carried out in symbolic form.

where the is a dimensionless coefficient resulting from integration. This equation is dimensionally correct because substitution of L, M, and T gives Dimensional homogeneity is a necessary condition for correctness of a physical relation, but it is not sufficient, since it is possible to construct [MLT2][L] [M][LT1] 2 1 2 Fx 1 2mv2 F ML/T2 W mg Article 1/6 Dimensions 11 c01.qxd 2/8/12 7:02 PM Page 11 an equation which is dimensionally correct but does not represent a correct relation. You should perform a dimensional check on the answer to every problem whose solution is carried out in symbolic form.