764 MECHANICS [APPLIED MECHANICS. position P 2 and its extreme positions P 1( P 3 shall be in the same straight line SS, perpendicular to CT 2 , c< 2 , and so to place the axes C, c on. the lines CT 2 , ct 2 that the path of P between the positions J j, P 2 , P 3 shall be as near as possible to a straight line. Fig. 29. The axes C, c are to be so placed that the middle M of the versed sine VT 2 , and the middle m of the versed sine vt. 2 , of the respective arcs whose equal chords TjT,,, tjt a represent the stroke, may each be in the line SS. Then T x and T 3 will be as far to one. side of SS as T 2 is to the other, and ^ and t 3 will be as far to the latter side of SS as t. 2 is to the former ; consequently, the two extreme positions of the links T^j, T 3 3 are parallel to each other, and inclined to SS at the same angle in one direction that the middle position of the link T 2 2 is inclined to that line in the other direction, and the three intersections P 1} P 2 , P 3 are at the same point on the link. The position of the point P on the link is found by the following proportional equation : Tt : PT : Tt
- : TV+ to:TV:to
- : CM + cm: cm : CM
Suppose the axes C, c to be given, the line of stroke SS, and the length of stroke L = T^T., = t^, and that it is required to find the dimensions of the levers and link. Let fall CM and cm perpen dicular to SS ; then T 2 T 2 TV tv = - 8CM 8cm, CT = CM + 4TV; c = e> -*-!> ^ (4S) (47; If C and c are at the same side of SS, the smaller of the two perpendiculars is to be treated as negative in the formulae, and the difference of the versed sines used instead of their sum ; and the point P will lie in the prolongation of the link beyond Tt to the side of the longer lever. When the arcs of oscillation of the levers on either side of their middle positions do not exceed 20, the intermediate portions of the path of P between P 1( P 2 , and P 3 are near enough to a straight line for practical purposes ; and that point may be used to guide a sliding piece, such as the piston-rod of a steam-engine, for which purpose this parallel motion was originally invented by Watt. CHAPTER II. ON APPLIED DYNAMICS. 89. Laws of Motion. The action of a machine in transmitting force and motion simultaneously, or performing work, is governed, in common with the phenomena of moving bodies in general, by two " laws of motion," for which see pp. 676 sq. 90. Comparison of Deviating Force with Gravity. See pp. 698, 699, 104-106. 91. Deviating Forces Classed Deflecting Force Accelerating and Retarding Forces. See p. 701, 114-119. 92. Division of the Subject. On this classification of the deviat ing forces in machines is founded the following division of the subject of dynamics as applied to machines : Division 1. Balanced forces in machines of uniform velocity. Division 2. Deflecting forces in such machines. Division 3. Working of machines of varying velocity. Division 1. Balanced Forces in Machines of Uniform Velocity. 93. Application of Force to Mechanism. Forces are applied in units of weight ; and the unit most commonly employed in Britain is the pound avoirdupois. The action of a force applied to a body is always in reality distributed over some definite space, either a volume of three dimensions or a surface of two. An example of a force distributed throughout a volume is the weight of the body itself, which acts on every particle, however small. The pressure exerted between two bodies at their surface of contact, or between the two parts of one body on either side of an ideal surface of separation, is an example of a force distributed over a surface. The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body con sidered as a whole, the resultant of the distributed force, deter mined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed. Thus, the weight of each separate piece in a machine is treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a point called the centre of pressure of the surface to which the pressure is really applied. 94. Forces applied to Mechanism Classed. If 6 be the obliquity of a force F applied to a piece of a machine, that is, the angle made by the direction of the force with the direction of motion of its point of application, then by the principles of Statics, F may be resolved into two rectangular components, viz. : Along the direction of motion, P = Fcos0 ( ,.(.-. Across the direction of motion, Q = Fsin0 * " If the component along the direction of motion acts with the motion, it is called an effort ; if against the motion, a resistance. The component across the direction of motion is a lateral pressure ; the unbalanced lateral pressure on any piece, or part of a piece, is deflecting force. A lateral pressure may increase resistance by causing friction ; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resist ance, resistances are distinguished into useful and prejudicial, according as they arise from the useful effect produced by the machine or from other causes. 95. Work. Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is measured by the product of the resistance into the distance through which its point of application is moved. The unit of work commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called & foot-pound. Work is distinguished into useful work and prejudicial or lost work, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance. 96. Energy Potential Energy. Energy means capacity for performing work. The energy of an effort, or potential energy, is measiired by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work. When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved. 97. Variable Effort and Resistance. If an effort hns different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length As of the corresponding portion of the path of the point of application ; the sum 2. PAs (50) is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum appioaches continually as the divisions of the path are made smaller and more numerous, and is expressed by /Tils .... (51). of the work pcr- Similar processes are applicable to the findinf formed in overcoming a varying resistance. 98. Dynamometer or Indicator. A. dynamometer or indicator is an instrument which measures and records the energy exerted by an effort. It usually consists essentially, first, of a piece of paper moving with a velocity proportional to that of the point of application of the effort, and having a straight line marked on it parallel to its direction of motion, called the zero line ; and, secondly, of a spring acted upon and bent by the effort, and carry ing a pencil whose perpendicular distance from the zero line, as regulated by the bending of the spring, is proportional to the effort. The pencil traces on the piece of paper a line such that its ordinate perpendicular to the zero line at a given point represents the effort P for the corresponding point in the path of the point of effort, and the area between two ordinatcs represents the energy exerted, /Pds, for the corresponding portion of the path of the point of effort. 99. Principle of the Equality of Energy and Work. From the first law of motion it follows that in a machine whose pieces move
with uniform velocities the efforts and resistances must balance each