MOLYBDir ACID. 



MOMENTUM. 



liquid becomes of a blue colour, then reddish brown, and finally Mack 

 By adding ammonia to the resulting solution and redissolving tin 

 washed protoxide in hydrochloric acid, a more pure protochloride is 

 obtained. 



KtUoridf ofmolyMenttm (Mod,) i* obtained in deliquescent crystal* 

 on evaporating a solution of the binoxide in hydrochloric n< 

 appearance it resembles iodine. 



Tke detection of nntttMrinnn is beet effected by producing th 

 colour* mentioned under protochloride of molybdenum. It is usually 

 estimated in the form of bisulphide, which contains 58-97 per cent, o: 



. i ..: 



MOLYBDIC ACID. [MOLTBDKSI-M.] 



MOMKXT. The moment of a force with respect to a point or line 

 aJxiut which it trud* to produce rotation, i" " the product of the force 

 nod the perpendicular drawn from the point or line upon the dirrrtiim 

 of the force. If P be the force, and p the perpendicular, the moment 

 of p is r /.. 



MOMKNTl M . . r MOMENT. This word has been used in various 

 Mnros It simply means a motion, the word women, from mnrimen 

 being found in several ancient authors. Momentum was originally one 

 rapid motion, whence it came to be used for a very abort time ; whence 

 our word moment, which, in common life, means an indivisible instant 

 of time. Thus an effect which requires a single second to produce it 

 would not be properly momentary. But the word has passed into 

 mechanics in its original sense of motion, and is used to signify the 

 amount of an effect of motion, actual or conceivable. Thus wo hnvc 

 one use in the artk-I<- VIRTUAL VELOCITIES, another in LKYER, a third 

 in MOMENTUM OF INHITIA, and a fourth, the most common of nil, 

 which we proceed to explain in this article. 



The English synonym of this fourth sense is " quantity of motion," 

 and we may observe that in this sense it is most usual, in our language, 

 to adopt the Latin form momentum, instead of the abbreviation moment. 

 It is impossible to give an actual definition of momentum, in simple 

 terms : but the conception is obtained by those who observe that the 

 fleets produced by matter in motion (both notions are necessary) may 

 1* augmented either by giving the same motion to more matter, or 

 greater motion to the same matter. Imagine a BALLISTIC PEXIM i i M. 

 and suppose a bullet of two pounds weight to strike it with a velocity 

 of 100 feet per second. The same oscillation which is thus produced, 

 may, it in found, be produced by a bullet of one pound weight striking 

 with a velocity of 200 feet. The same effect being produced in both 

 csti, though by different quantities of matter and different velocities, 

 there is something which wo in.iy assert to be unaltered by the substi- 

 tution of the smaller bullet with the larger velocity. This something 

 U the momentum, or quantity of motion, a notion of a cause which is 

 asserted to be the same when effects are the same. This mere defi- 

 nition would be useless except iu connection with principles observed 

 or deduced, by which it may be applied. That there is a reality in 

 connection with it, all who know the difference li'tween light and 

 heavy, as these words are frequently used, are well aware. A heavy 

 blow, for instance, does not mean a blow with a heavy body : thus the 

 fall of a poker may give a light blow, while thai of a book of one-tenth 

 part of its weight may give a henry one. The difference in these cases 

 U that of momentum. 



The velocity remaining the same, the momentum or quantity of 

 motion increases with the mass moved ; and the mass remaining the 

 name, the momentum increases proportionally to the velocity com- 

 municated. But the peculiar proposition on which the utility of the 

 term and the notion depends u this, that in all mechanical effects 

 produced by matter in motion, a diminution of the mass may be com- 

 pensated by a proportionate increase of the velocity : that is, M being 

 the number of units of mass, and v of velocity, as long as the product 

 of x and r remains the same, the effect produced is the same. Thus 

 in the preceding instance M x v is 2 x 100 in the first cose, and 1 x 200 

 in the second. And as long as Mxv = 200, the same effect will be 

 produced. 



This product, xv, is the measure of the momentum, and in generally 

 railed the momentum itself. Here (as in MASS) tacit reference is made 

 in a nnit of momentum : the equati<-n 



Momentum of M with velocity v = M x v 



implies that a unit of momentum i* that produced by a unit of ma.- 

 moving with a unit of velocity. 



When the body moves under the action of a continually applied 

 force, its mumnttum is now often expressed by the term ////. 

 r/nl, by which we mean the effect of a force with reference to its 

 time of action. Thus if a force of 1 11>., acting for one second, prodm < 

 * certain dynamical effect which we may assume to !< unity then tin- 

 dmamical effect produced by r Ibs. acting for t seconds will \- r.t. 

 Mow let w b* the weight of the body : M its mas* : v its velocity : / 

 the rate of acceleration due to r [3rd Law of Motion]. Tin ii r 



w = /: y : hence r=*. /; hence dynamical pffect = r.<= "'./( = MV. 



MOMENTUM. o/MOMENT OF INERTIA. Let ? conceive a 

 system of bodies posseming weight, and immovably attached to a 

 fixed axis, round which the whole system can turn. It is known from 

 pxperience, as well as deducible from the laws of motion, that the 

 warn- the bodies arc placrd to the axis, the more rotatory motion may 



be communicated by a given force. The moment nf inertia is a namo 

 given to a mathematical function of the masses in the system and of 

 their positions with respect to the axis, on the magnitude of whirl! the 

 rotatory motion produced by a given pressure, acting for a given time, 

 depends. This function is the sum of the products mad.- l.y multi- 

 plying the number of unite in each mass by the number of units in tin- 

 square of iU distance from the axis. Thus, if m, m', m", 4c.. be the 

 masses of material points situated at the distance* r, r', r 

 the axis, the moment of inertia ism r~ + m'r" + m" r" + ftc. Iftl 

 be continuous solid, and if rf m bo one of the elements of the mass, t a 



distance r from the axis. tli.< ni..in.-nt ! inertia in t\u-nj',-"ilm, tin- 

 integration being made throughout the whole extent of the solid. 



I AB be the axis, and let a pressure be communicated to fin- 

 tystem at the point p, and such as would, were a mass P placed I 

 cause the system which consists of that single moss only to i 

 with a velocity r, being at the distance a from the axis. The mom 

 of this velocity is pr. Let the system of m, i',and w", in const-.; 

 of this pressure, begin to revolve with an angular velocity 8 (measured 

 in theoretical units. [AXOLK.] The consequence is, that m, m 

 m" begin to revolve with velocities rt. r'S, end r"9, and momenta mrt, 

 m'r'S, m'r"8. Now, if pressures, whi.-h would just preveir 

 motion in the same time as the applied pressure generated it. 

 A 



applied in the opposite direction, the three pressures BO applied would 

 counterbalance the pressure at P. Rut [MOHKXTCM] the pressures w hi. 1 1 

 in the name time produce motions are to one another as the momenta 

 produn-d, s.. that if c.rr represent the pressure at p, those applied in 

 the contrary din-etion at , in', and >'', are cmr8, cm' r'S, and 

 cm'r'S. But the first acts perpendicularly at the extremity of tin- 

 arm (i, the others at the arms r, r, and r". Hence cmrS.r+fw'r'S.r' 

 + cm'V.r"-crr.a or 'r 



PM 



' 9= mr- 



the denominator of which is what has been called the niou.. 



uertia of the system. Hence it follows that the communication of a 

 ,,'ive!) pressure at a given distance from the axis of rotation will cause 

 an angular velocity which is inversely as the moim-nt of inertia : if tin- 

 Ms or their dix 



{stances were increased in such a way as to double 

 ;he moment of inertia, the angular velocity produced l.v i 

 M-essure would be only the half of what it would have been iicfon- tin- 

 ihange. 



The moment of inertia may be represented by Swr 8 (sum of all tin- 

 terms of the form mr : ), and the whole moss by SIR. Let t- be such a 

 distance that, if the whole moss were concentrated at that distance, the 

 nomentof inertia would not be altered: that is, M 2m * /' be = Smr 1 . 

 Then < is what was called the radiut of gyration. [GYHATIUX. | 



The property which is most important in tin- actual determination 

 of moments of inertia by the integral calculus is one in virtu.- of \\lii.-li 

 ;he moment may lie found with n-~p.-.-t to .my axis when it is knoun 

 with rc*]iect to a parallel axis passing through the centre of pi 



.- be an axis passing through r. the centre of gravity, mid let 

 \ u lie another axis parallel to r q and distant from it by en or li. 



.vhatever the moment of inertia maybe when PQ is tin- 

 hat with respect to A B i- found by adding the moment of in. . 

 In- whole system concentrated in o, or Zm x A 5 . That i-. 



Zmr ! =2mf> 5 f 2m x/< : where 'S.mff - moment of inrrtii of the 

 with renpert to P Q. 



