city acquired by it in its descent is precisely the same as though it had fallen vertically through the same height. 74. DEFINITION. The ANGULAR VELOCITY of a body which rotates about a fixed axis is the arc which every particle of the body situated at a distance unity from the axis describes in a second of time, if the body revolves uniformly; or, if the body moves with a variable motion, it is the arc which it would describe in a second of time if (from the instant when is angular velocity is measured) its revolution were to become uniform. 75. THE ACCUMULATION OF WORK IN A BODY WHICH ROTATES ABOUT A FIXED AXIS. Propositions 68 and 69 apply to every case of the motion of a heavy body. In every such case the work accumulated or lost by the action of any moving force or pressure, whilst the body passes from any one position to another, is equal to the work which must be done in an opposite direction, to cause it to pass back from the second position into the first. Let us suppose U to represent this work in respect to a body of any given dimensions, which has rotated about a fixed axis from one given position into another, by the action of given forces. Let a be taken to represent the ANGULAR VELOCITY of the body after it has passed from one of these positions into another. Then since a is the actual velocity of a particle at distance unity from the axis, therefore the velocity of a particle at any other distance p, from the axis is ap,. Let μ represent the weight of each unit of the volume of the body, and m, the volume of any particle whose distance from the axis is p,, then will the weight of that particle be μm,; also its velocity has been shown to be ap,, therefore the amount of work accumulated in that particle is represented by 11 La22, or by 3x2mp2. μ Similarly the different amounts of work accumulated in the other particles or elements of the body whose distances. from the axis are represented by P2, P3, . . and their volumes by m, m2, m ̧ . . . ., are represented by fam,p., μ M39 am, &c.; so that the whole work accumulated is repre 2 2 2 The sum m,p,' + m2 P22 + M2 P32+ ..., or Emp2 taken in respect to all the particles or elements which compose the body, is called its MOMENT OF INERTIA in respect to the particular axis about which the rotation takes place. Let it. be represented by I; then will . (). I, represent the a2. whole amount of work accumulated in the body whilst it has been made to acquire the angular velocity a from rest. If therefore U represent the work which must be done in an opposite direction to cause the body to pass back from its last position into its first, If instead of the body's first position being one of rest, it had in its first position been moving with an angular velocity a, which had passed, in its second position, into a velocity a; and if U represent, as before, the work which must be done in an opposite direction, to bring this body back from I 2 its second into its first position, then is fa2 (“) I—«,' (5) I, or ‡ (5) (aa) I, the work accumulated between the first and second positions; therefore where the sign is to be taken according as the motion is accelerated or retarded between the first and second positions, since in the one case the angular velocity increases during the motion, so that a2 is greater than a,, whilst in the latter case it diminishes, so that a is less than a,. 76. If during one part of the motion, the work of the impressed forces tends to accelerate, and during another to retard it, and the work in the former case be represented by U1, and in the latter by U,, then From this equation it follows that when U, U1, or when the work U, done by the forces which tend to resist the motion at length, equals that done by the forces which tend to accelerate the motion, then a a,, or the revolving body then returns again to the angular velocity from which it set out. Whilst, if U, never becomes equal to U, in the course of a revolution, then the angular velocity a does not return to its original value, but is increased at each revolution; and on the other hand, if U, becomes at each revolution greater than U,, then the angular velocity is at each revolution diminished. The greater the moment of inertia I of the revolving mass, and the greater the weight of its unit of volume (that is, the heavier the material of which it is formed), the less is the variation produced in the angular velocity a by any given variation of U or U,-U, at different periods of the same revolution, or from revolution to revolution; that is, the more steady is the motion produced by any variable action of the impelling force. It is on this principle that the fly-wheel is used to equalize the motion of machinery under a variable operation of the moving power, or of the resistance. It is simply a contrivance for increasing the moment of inertia of the revolving mass, and thereby giving steadiness to its revolution, under the operation of variable impelling forces, on the principles stated above. This great moment of inertia is given to the fly-wheel, by collecting the greater part of its material on the rim, or about the circumference of the wheel, so that the distance p of each particle which composes it, from the axis about which it revolves, may be the greatest possible, and thus the sum Emp', or I, may be the greatest possible. At the same time the greatest value is given to the quantity, by constructing the wheel of the heaviest material applicable to the purpose. What has here been said will best be understood in its application to the CRANK. 77. If we conceive a constant pressure Q to act upon the B arm CB of the crank in the direction AB of the crank rod, and a constant resistance R to be opposed to the revolution of the axis C always at the same perpendicular distance from that axis, it is evident that since the perpendicular distance at which Qacts from the axis is continually varying (being at one time nothing, and at another equal to the whole length CB of the arm of the crank), the effective pressure upon the arm CB must at certain periods of each revolution exceed the constant resistance opposed to the motion of that arm, and at other periods fall short of it; so that the resultant of this pressure and this resistance, or the unbalanced pressure Р upon the arm, must at one period of each revolution have its direction in the direction of the motion, and at another time opposite to it. Representing the work done upon the arm in the one case by U,, and in the other by U,, it follows that if U-U, the arm will return in the course of each revolution, from the velocity which it had when the work U, began to be done, to that velocity again when the work U, is completed. If on the contrary U, exceed U,, then the velocity will increase at each revolution; and if U, be less than U,, it will diminish. It is evident from equation (52), that the greater the moment of inertia I of the body put in motion, and the greater the weight of its unit of volume, the less is the variation in the value of a, produced by any given variation in the value of U,-U,; the less therefore is the variation in the rotation of the arm of the crank, and of the machine to which it gives motion, produced by the varying action of the forces impressed upon it. Now the fly-wheel being fixed upon the same axis with the crank arm, and revolving with it, adds its own moment of inertia to that of the rest of the revolving mass, thereby increasing greatly the value of I, and therefore, on the principles stated above, equalizing the motion, whilst it does not otherwise increase the resistance to be overcome, than by the friction of its axis, and the resistance which the air opposes to its revolution.* *We shall hereafter treat fully of the crank and fly-wheel. 78. The rotation of a body about a fixed axis when acted upon by no other moving force than its weight. Let U represent the work necessary to raise it from its second position into the first if it be descending, or from its first into its second position if it be ascending, and let a, be its angular velocity in the first position, and a in the second; then by equation (51), a2 = a;2 + 2(?) (+)·. Now it has been shown (Art. 60.), that the work necessary to raise the body from its second position into the first if it be descending, or from its first into its second if it be ascending (its weight being the only force to be overcome), is the same as would be necessary to raise its whole weight collected in its centre of gravity from the one position into the other position of its centre of gravity. Let CA repre B sent the one, and CA, the other position of the body, and G and G, the two corresponding positions of the centre of gravity, then will the work necessary to raise the body from its position CA to its position CA,, be equal to that which is necessary to raise its whole weight W, supposed collected in G, from that point to G,; which by Article 56, is the same as that necessary to raise it through the vertical height GM. Let now CG CG,=h, let CD be a vertical line through C, let G,CD=4, and GCD=8, in the case in which the body descends, and conversely when it ascends; therefore GM=NN, CN-CN,=h cos. -h cos. 8, when the body descends, or =h cos. 4-h cos. when it ascends from the position AC to AC,, since in this last case GCD=e, and G,CD. Therefore GM=+h (cos. -cos. ), the sign± being taken according as the body ascends or descends. Now U=W. GM=±Wh (cos. 4-cos. 4), :. by equation (51) a2=a ̧2+| (2Wgh) (cos. —cos. §.). If M represent the volume of the revolving body Mu=W, :. a2=a ̧2+(2o111) (cos. 6—cos. §,) . . . . . (53). ..... When the body has descended into the vertical position, |