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resistance of the axis and its bearings. Now the resistance of the axis is evidently equal and opposite to the resultant R of all the forces P, P2, &c. impressed upon the body. This resistance acts, therefore, in the direction LR, and is inclined to CL at an angle equal to the limiting angle of resistance. Therefore, &c.

BA

THE WHEEL AND AXLE.

154. The pressures P, and P, applied ver tically by means of parallel cords to a wheel and axle are in the state bordering upon motion by the preponderance of P1, it is required to determine a relation between P, and P,.

The direction LR of the resistance of the axis is on that side of the centre which is towards P,, and is inclined to the perpendicular CL at the point L, where it intersects the axis at an angle CLR equal to the limiting angle of resistance. Let this angle be represented by, and the radius CL of the axis by p; also the radius CA of the wheel by a,, and that CB of the axle by a,; and let W be the weight of the wheel and axle, whose centre of gravity is supposed to be C. Now, the pressures P, P, the weight W of the wheel and axle, and the resistance R of the axis, are pres sures in equilibrium. Therefore, by the principle of the equality of moments (Art. 7.), neglecting the rigidity of the cord, and observing that the weight W may be supposed to act through C, we have,

P1. CA=P,. CB+R. Cm.

If, instead of P, preponderating, it had been on the point of yielding, or P, had been in the act of preponderating, then R would have fallen on the other side of C, and we should have obtained the relation P, . CA=P, . CBR. Cm; so that, generally, P, . CA=P, . CB±R. Cm; the sign being taken according as P, is in the superior or inferior state bordering upon motion.

Now CA=a,, CB=a,, Cm=CL sin. CLR=p sin. 9, and

The side of C on which RL falls is manifestly determined by the direction towards which the motion is about to take place. In this case it is supposed about to take place to the right of C. If it had been to the left, the direc tion of R would have been on the opposite side of C.

R=P,+P,± W; the sign being taken according as the weight W of the wheel and axle acts in the same direction with the pressures P, and P,, or in the opposite direction; that is, according as the pressures P, and P, act vertically downwards (as shown in the figure) or upwards;

..P,a=P,a,+(P,+P,±W) p sin. 9,

..P,(a,—p sin. )=P,(a,+p sin. ¢)±W p sin. ❤.

Now the effect (Art. 142.) of the rigidity of the cord BP, is the same as though it increased the tension upon that cord

(P2+D+E.

from P, to (P,+

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P

P): allowing, therefore, for the

rigidity of the cord, we have finally

P,(a,—p sin. q)=(P,+D+E. P.) (a,+p sin. 4)±Wp sin. 9,

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which is the required relation between P, and P, in the state bordering upon motion.

P

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P
A

sin. and sin. are in all cases exceedingly small; we may therefore omit, without materially affecting the result, all terms involving powers of these quantities above the first, we shall thus obtain by reduction

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155. The modulus of uniform motion in the wheel and axle.

It is evident from equation (122), that, in the case of the wheel and axle, the relation assumed in equation (119)

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and

D+(W)p sin.

1

a-p sin.

P

1

Now observing that represents the value of $, when the prejudicial resistances vanish (or when =0 and E=0),

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which is the modulus of the wheel and axle.

Omitting terms involving dimensions of

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1

}

S,D

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U.=U, {1+E+ (+), sin. } + {1+

α

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Φ

sin. 9 } . . (125).

156. The modulus of variable motion in the wheel and acle.

If the relation of P, and P, be not that of either state bordering upon motion, then the motion will be continually accelerated or continually retarded, and work will continually accumulate in the moving parts of the machine, or the work already accumulated there will continually expend

itself until the whole is exhausted, and the machine is brought to rest. The general expression for the modulus in this state of variable motion is (equation 116)

P

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Now in this case of the wheel and axle, if V, and V, re

P2

2

present the velocities of P, at the commencement and completion of the space S,, and a the angular velocity of the revolution of the wheel and axle; if, moreover, the pressures P, and P, be supposed to be supplied by weights suspended from the cords; then, since the velocity of P, is reprea,V,

sented by

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we have Zwv,'=

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I, represent the moment of inertia of the revolving wheel, and I, that of the revolving axle, (Art. 75.), and if, represent the weight of a unit of the wheel and ", of the axle; since Zwv, represents the sum of the weights of all the moving elements of the machine, each being multiplied by the square of its velocity, and that (by Art. 75.) au I, represents this sum in respect to the wheel, and a'u I, in respect to the axle. Now, V,=aa1,

V.

1

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=

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απ

V. {Pa2+Pa+,I,++,I,}.

Similarly £wv,'=V,' { P.,a,'+P,a,; +m. I, +m, L2 }

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2

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Substituting in the general expression (equation 116), we

have

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which is the modulus of the machine in the state of variable motion, the co-efficients A and B being those already determined (equation 124), whilst the co-efficient

2

P‚a‚2+P‚«‚2+~‚I,+I is the co-efficient w (equation

a,1

2

117) of equable motion. If the wheel and axle be each of them a solid cylinder, and the thickness of the wheel be b,, and the length of the axle b,, then (Art. 85.) I,=,a,, I=1′′b,a,'. Now if W, and W, represent the weights of the wheel and axle respectively, then Wa,b,,, W,a,b,,; therefore μ‚I,=¿W,a,2, μ,I,W,a,. Therefore the co-efficient of equable motion is represented by the equation

£wx3_P‚a‚'+P‚a‚'+¿W,a,'++W,a,", or

Σωλ

2 a1

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157. To determine the velocity acquired through a given space when the relation of the weights P, and P, suspended from a wheel and axle, is not that of the state bordering upon motion.*

Let S, be the space through which the weight P, moves whilst its velocity passes from V, to V,: observing that U'=P,S,, and that U=P,S,=P,, substituting in equa

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tion (126), and solving it in respect to V,, we have

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