Search

The science of mechanics ignores many features of the bodies whose motions and interactions are the object of study. The color, temperature, and chemical composition of the bodies, for example, are irrelevant for mechanics. The chief types of bodies in theoretical mechanics are (i) a particle; (ii) a rigid body; (iii) an elastic body; and (iv) an inextensible or inelastic string. To discuss the motions and interactions of these bodies, systems of units are needed. The different sets or systems of units used in mechanics are identified by three basic units of length, mass, and time, in terms of which other units are defined. These include the Systeme Internationale or MKS system, which uses the meter, kilogram, and second as the respective basic units; the CGS system, most widely used in basic physics, based on the centimeter, gram, and second; and the FPS system, which is not used outside the British Commonwealth and the United States, based on the foot, pound, and second.

The triangle or polygon of forces is a set of concurrent forces that, when represented in magnitude and direction by the sides of a polygon, which is closed, are in equilibrium. The fact that the polygon is closed is a necessary and sufficient condition for equilibrium. This forms the basis of graphical solutions of statical problems. When there are only three forces the closed polygon is a triangle and the magnitudes of the forces are given by Lami's Theorem—that is, each force has a magnitude proportional to the sine of the angle between the other two. The laws of motion refer to a particle. When a particle is in contact with a surface, or when two surfaces are in contact at a point, the action of the surface on the particle or the action between the surfaces is a force that is inclined to the normal to the surface(s).

Publisher Summary This chapter is concerned with the description of the motion of particles and rigid bodies—kinematics. It focuses on the useful methods of expressing algebraically and geometrically by vectors, components, etc., the motion of a point or of a rigid body. The chapter also discusses the motion of a particle and the uniplanar motion of a rigid body, or lamina. A point has a velocity when its position is altering, and it has acceleration when its velocity is altering. The velocity of a point moving in one dimension is obtained by differentiation of the position coordinate, the acceleration by differentiating the velocity. The concept of angular velocity is of more importance in connection with the motion of a lamina in its own plane.

This chapter explains the equilibrium of a perfectly flexible string or chain. Such a body is regarded as a continuous line of particles in which the only action one element can exert on a neighboring element is a force along the line joining the elements—that is, along the tangent. There can be no couple or transverse force, so the string offers no resistance to bending. The chapter discusses the equilibrium of a uniform inelastic string hanging under gravity. If an inextensible string hangs in equilibrium under gravity and carries a load or is of variable line density so that the mass per unit length is not uniform, the curve in which it hangs can be determined by methods similar to those used for the uniform string. The investigation of the equilibrium of heavy elastic strings is based on the assumption that Hooke's law applies to every element of the string, each element being in equilibrium under the action of its neighboring elements and its own weight.

This chapter discusses the general motion in its own plane of a lamina of constant mass. The methods and results are applicable also to a three-dimensional rigid body that moves so that each of its constituent particles moves parallel to a fixed plane. As with particle dynamics, in many cases it is also possible to use the energy equation first and then use the equations of motion to find unknown forces. The chapter illustrates these points. It also illustrates the use of the energy equation for motions in which there is no fixed axis of rotation. The motion of rough bodies such as spheres, cylinders, and hoops on rough planes is also discussed.

The motions of particles, bodies, and systems are motions for which the velocities vary as continuous functions of time. This chapter focuses on the limiting cases of such motions in which for certain short intervals of time the accelerations are so large that the velocity, effectively, changes discontinuously. In collisions between small bodies, or particles, the relative velocity is altered both in magnitude and direction; in an extreme case the relative velocity after a collision might be zero. This alteration of magnitude occurs because the bodies are deformed during the collision and not all the energy used to deform the bodies is released when they separate. Consequently, the value of velocity after the collision differs from its initial value, the difference depending on the elastic properties of the bodies. It is almost impossible to determine the variation of the force between two bodies during impact, particularly in the limiting case where the duration of the impact is infinitesimally short. Therefore, information is needed from some other source to determine what would happen after the collision. This information is given in the form of Newton's experimental law of impact.

This chapter focuses on differentiation, which is the process of finding the derivative. Most functions of physical interest can be differentiated a number of times. The convention for the representation of a sequence of operations is to write the operators in the opposite order to that in which the operations are carried out so that the last symbol represents the first operation to be carried out. The chapter discusses the method of “logarithmic differentiation,” which might be used directly for the differentiation of products and complicated surds. The chapter also discusses “hyperbolic functions,” which are certain combinations of the exponential functions with properties and bear a close formal analogy with those of the trigonometric functions. Identities involving hyperbolic functions resemble closely corresponding identities for trigonometric functions. A hyperbolic identity might be obtained from a trigonometric identity. The chapter presents an example of the process of mathematical induction, which is employed to prove formulae involving an integer that takes positive integral values.

Publisher Summary The difference between a stable and an unstable structure or between a stable and an unstable state of motion is known. Roughly speaking, a heavy object hanging from a hook is stable, whereas a bottle standing upside down on a horizontal plane is unstable; a slight disturbance in the first case causes little deviation from the steady position, but a slight disturbance of the bottle causes it to fall over. If there is any small disturbance that causes the system to deviate largely from the steady state, that state is unstable. It might be possible to find certain disturbances that would not upset an unstable state; therefore, it is essential that the disturbance applied to a steady state is arbitrary. This chapter discusses stability using the equations of motion. Therefore, stability is a matter of dynamics and not strictly of statics, although it is associated closely with positions of equilibrium.

If a particle is not constrained to move on a curve, then there are no unknown forces of constraint. In contrast, two parameters, usually coordinates, are required to specify the position of the particle. The vector equation of motion has two component equations that, on integration, determine the coordinates as functions of time; these are the parametric equations of trajectory. This chapter discusses the motion of a particle in a resisting medium and the motion of a particle under the action of a force directed toward a fixed point. In these motions the resultant force on the particle varies; in the case of resisted motion, the force depends upon the velocity and is not conservative; in the case of the central force, the force is almost always conservative and the energy equation can usually be written down. The projectile is a particle subject to gravity, which acts vertically downward and is constant; the resistance to the motion of the projectile is a force acting in a direction opposite to that of the velocity of the particle relative to the medium in which the particle moves. Usually the magnitude of the resistance is assumed to vary as some power of the velocity, and the method adopted to determine the trajectory depends upon the relation between resistance and velocity.

This chapter focuses on the motion of a particle that has one degree of freedom and includes motion along a straight line and motion in which the particle is constrained to move along some prescribed curve. Such a constraint is achieved by restricting a bead to slide on a fixed wire, by attaching a particle to an inextensible string, or by similar devices. The chapter discusses the solution of a second order ordinary differential equation, but in some cases, it might be convenient to start the solution by introducing more than one variable and then reducing to one variable by elimination. As in other branches of physics, the most convenient coordinate system for use in any one problem is essentially determined by the nature of that problem. In general, frictional forces, or forces that depend upon the magnitude or direction of the velocity of the particle, cannot be conservative.

Publisher Summary This chapter provides an overview of complex numbers. A single relation between complex numbers is equivalent to two relations between real numbers. The real parts of the two sides of an equation and the imaginary parts of each side must be equated. The algebraic operations of addition, subtraction, and multiplication are performed with complex numbers in exactly the same way as with real numbers. The representation of addition on the Argand diagram leads to a parallelogram law for the addition of two complex numbers. The multiplication of two complex numbers can be carried out by multiplying their moduli and adding their arguments. Division of one complex number by another can be carried out by dividing the modulus of the numerator by the modulus of the denominator and subtraction of the argument of the denominator from the argument of the numerator. When applied to complex numbers, the operations of algebra, such as addition, subtraction, multiplication, division, and taking square roots result in complex numbers. The chapter presents some of the techniques that are useful in finding the real and imaginary parts of some standard functions. Results show that the general fact that one relation between complex numbers is equivalent to two relations between real numbers.

Publisher Summary This chapter explores some of the mathematical concepts for engineers and scientists. It discusses functional notation and presents fundamental definitions. If a variable number, “y,” is determined when a second variable number, “x,” is specified, “y” is said to be a function of “x.” “x” is called the independent variable and “y” is the dependent variable. The chapter presents some examples of the functions that are defined in specified intervals or ranges of values of “x.” It discusses Horner's method, which is a systematic method of obtaining to any desired degree of accuracy the real roots of a polynomial equation. For any particular root, this method consists of repeated applications of a cycle of three operations; each cycle providing one integer of the decimal expression of the root. Continuity is represented graphically by a curve without gaps. The idea of a limit is associated with the behavior of a function “f(x)” as “x” approaches a given value from above or below.

This chapter focuses on the technique of integration. The definite integral might be regarded as the limit of a sum. This chapter presents an account of the case in which the integrand is continuous. It lists simple rules and techniques that follow directly from the definition of integration. The result of any integration might be rapidly checked by differentiation. In applications to differential equations and physical problems, an arbitrary constant must be added at each integration. The chapter also presents an account of resolution of a rational function into partial fractions. It further discusses Heaviside's rule and presents some examples to show that the rule always gives the coefficients corresponding to the unrepeated linear factors.

Publisher Summary The geometry of plane figures is the basis of much advanced mathematics and acts as a guide to the development of physical ideas. This chapter focuses on the application of calculus to the geometrical investigation of curves and discusses conic sections, which is an important class of curves. It discusses the techniques that are used for obtaining the geometrical properties of the curves whose equations are given in polar coordinates. Some problems necessitate an idea of the shape of a curve as a whole; it is best obtained from a sketch and exhibits the major features of the curve. It is impracticable to find this shape by plotting a large number of points on the curve. Each individual problem must be treated on its merits. The chapter discusses the properties of a curve at a point. It describes some techniques that are used in curve sketching.