MECHANICS. 





MEGHAN ICS. II. 



THE UNIT OF FOECE.- 1 i'UKD TO A POINT. 



ll\\iv. hi our first lesson fxjilainud tin- meaning oft! i 



." uud shown how a force is uppli.-.! ami measured, we shall 

 next consider tho simplest khul of meelmnieal prc.l.l.-ia, that ( 

 several applied to a siiujlc point. Before I proceed, however, it 

 is advisable to fix clearly your notions of the " unit of force." 

 already laid down the rule, that a force may bo measured 

 by the number of feet it v.mtld cause the unit ivory bull, qual 

 in weight to a cubic inch of pure water, to mov over in one 

 second, when applied to it middi-nly by a blow. If the ball 

 move over seven feet, the number 7 should bo writti n fr 

 the force; if over a furlong, the number is COO, the feet in a 

 furlong. But suppose it moves over exactly one foot, then it is 

 ; hat the numeral 1 should bo written ; and that particular 

 force is the "one" of forces. And the conclusion to which 

 we thus ore led is that 



THE UNIT OF FORCE is the force which would, if applied 

 instantaneously to the unit oj mass, make it move over one foot 

 in une second. 



But you can clearly see that the force which could produce 

 no greater velocity than this in the boll which, instead of being 

 ivory, wo may take to be a ball of frozen water, a cubic inch 

 in volume cannot be a very strong force. In fact, it is equal 

 to a little loss than eight grains of weight, that is, this unit of 

 force could be balanced by that with which an eight-grain 

 weight pulls downwards. How this is ascertained I cannot 

 hero explain to yon, as you would require some little knowledge 

 of dynamics to understand tho proof. For the present, there- 

 fore, you must take my statement on credit. 



But this unit is evidently too small for practical purposes. 

 Tho strains in the mechanical powers, the lever, the wheel and 

 axle, the pulley, etc., and in roofs and bridges, cannot be 

 calculated in grains, on account of the large numbers we should 

 have to operate on. A larger unit is therefore necessary, and 

 tho pound weight exactly answers tho purpose. We can cal- 

 culate and measure forces in pounds ; or, if the figures in 

 that case bo too large, wo can calculate them in hundred- 

 weights, or even in tons. All that is necessary is to keep clearly 

 in mind what your unit is in your calculation, and to know how 

 to pass from one unit to another. If, in the same calculation, 

 you were to use different nnits in different places a pound for 

 instance, in one, and a hundred-weight in another without 

 reducing tho one to tho other, the result could, be nothing but 

 confusion and error. 



But how are you to pass from one unit to another ? This is a 

 nice point in practice, as we shall see in duo time ; but this much 

 is clear, that, if your unit ba a hundred-weight, you should 

 multiply all the numbers wliich represent your foro3S by 112 

 (tho number of pounds in a hundred- weight), and then these 

 forces \vill be expressed in pounds. If they are already ex- 

 pressed in pounds, then divide by 112, and you will have them 

 in hundreds and fractions of a hundred- weight. And so, from 

 hundred-weights you con pass to tons by dividing by 20, aud 

 reverse the operation by multiplying by that number. Thus, 

 we see that "ton," " hundred- weight," and "pound," are only 

 so many different expressions for the same unit namely, the 

 pound either singly or collectively, and that, therefore, for 

 practical purposes, we may say that a pound weight is the " unit 

 of force." 



But we cannot leave this subject without determining the 



relation between this unit and tho very small one of which I 



first made mention. I have asked you to take it on credit that 



the latter is nearly eight grains. The more correct value 



involves decimals, and is 7 '85 grains nearly, that is, seven 



grains and eighty-five parts out of a hundred of one grain. 



Hence, since there are 7,000 grains in an avoirdupois pound, if 



we divide this number by 7'85, we shall have the number of 



these small units (which henceforth we shall call the dynamical 



unit), to which one pound weight is equal. The division gives 



892 nearly for tho quotient ; and thus we learn how we may 



pass from dynamical units to pounds, or from pounds to these 



units. The result may be summed up in the following table : 



7 '85 Grains make uearly one Dynamical Unit. 



892 Dynamical Units mako Hourly one Pound. 



112 Pounds make one Hundred-weight. 



20 Hundreds make one Ton. 



Forces applied to a Point. When a single force is applied 

 56 N.E. 



to any point of a body, If the latter be free, motion will 

 etume, and the question belongs to Dynamic*. If it be not free, 

 bat fastened in any way to fixed objects, the force will be 

 communicated through its substance to the points of support or 

 connection, which will n-mut, and by resisting cause the body to 

 sustain strain. For example, suppose a beam of wood is fixed 

 at one point, round which, as on a pivot, it can torn in any 

 direction, and that a force is applied to it at some other point. 

 It is clear that thi* force will poll the beam round towards 

 itself so far as it can go, that is, until the line of direction of 

 the force passes through the fixed point. Then this point will 

 resist, and equilibrium will be produced. The case thus tionomos 

 one of two forces namely, that applied and the resistance pro- 

 duced ; and we see thus that a single force can never in Htatict 

 be the subject of study, without involving the consideration of 

 other forces which it colls into existence. A statical problem 

 must be concerned about at least two forces. 



If two forces bo applied to a point in the same direction, we 

 assume in Mechanics, an a self-evident truth, the result of 

 experience, that their joint effect is the same as that which 

 would be produced by a single force equal to their sum. If two 

 men of unequal strength pull on a rope against another man 

 stronger than either, whc succeeds in balancing their united 

 strength, wo say, witho.it hesitation, that his force is equal to 

 the sum of those put forth by the two. When two forces thus 

 act separately at a point, tho single force to which their joint 

 power is equal is called the " resultant " of these forces. We 

 therefore say, if two forces act on a point in the same direction, 

 their resultant is the sum, of these forces. If three act on it, 

 since two of them are equivalent to one equal to their sum, this 

 one with the third must be equivalent to a single force equal to 

 tho sum of the three. And so on, as to more than three, we 

 may lay it down as a general rule that 



The resultant of any number of forces acting on a point in 

 the same direction, is a single force equal to the sum of the 

 separate forces. 



When two forces act in opposite directions on a point, for the 

 amo reason as in the former case, we assume that the resultant 

 is the difference of the two. And this leads us to the moat 

 general case that can occur of such forces namely, that in 

 which any number of them are applied to a body along the 

 same line, some in one direction and others in the opposite 

 direction. To determine the resultant of all, it is evident that 

 it is sufficient to take the separate resultants of the opposing 

 sets, then take the difference of these resultants, and that thin 

 difference will bo the required restdtant of all, and its direction 

 that of the greater of tho two separate resultants. Hence the 

 following rule : 



If any number of forces be applied to a body along the same 

 line, their resultant is the difference between the sums of those 

 which act in the opposite direction, and its direction is the same 

 as that of tho greater sum. 



For example, if fifteen men pull on a rope against eleven, 

 and drag them along a road, the resultant of the twenty-six 

 forces applied to the rope along its length is the difference 

 between the united powers of the fifteen and of the elevet, 

 whatever be the particular strength of each man, and ita 

 direction is that in which the fifteen pull. 



But suppose now that two forces only are employed, and that 

 they are equal and in opposite directions ; what will be tho 

 result ? They will balance, or be in equilibrium. Now it is some- 

 times said that the body to which two such forces are applied 

 at one of its points is in the same condition as if no force had 

 been applied to it. This is not true, strictly. It is in the same 

 condition so far as equilibrium is concerned, but not otherwise. 

 It is not in the same condition as to pressure or strain. The 

 ro,!!'. wiiich at one moment is lying stretched on the ground, is 

 not in the same condition it was in a few minutes before, when 

 two strong men were pulling at opposite ends of it with balanced 

 strength. In the latter case it is strained along its whole 

 length every thread on the stretch, ready to snap. Its condi- 

 tion is very different on the two occasions different in every 

 circumstance, except that of there being no motion. So, also, 

 if two equal and opposite pressures are applied to a round ball, 

 it will be an equilibrium, but the condition of its substance will 

 be changed. Its particles will be pressed towards one another 

 inwards ; and. if it bo made of soft or elastic material, ifc form 

 will bo altered by the flattening effect of th opposing force* 



