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Hence

These are the accelerations parallel to x and y. by § 14, the acceleration along the radius vector is

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16. When a point is in motion in any curve, to find its accelerations along, and perpendicular to, the tangent, at any

instant.

Let x, y, z be the co-ordinates of the point at the end of the time t, s the length of the arc described during that interval. Then, since by the equations to the curve x, y and z are functions of s,

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To find the acceleration along the tangent, we must mul

tiply these component accelerations by

dx dy dz

ds' ds' ds'

spectively, and add. Thus the tangential acceleration is

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re

as we have already seen. Also in the normal, towards the center of curvature, we have the acceleration

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We have assumed, in the above, the following equations from Analytical Geometry,

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2

2

(diz)* + (17)* + (13)* or (resultant acceleration)*

dt2

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Now is the acceleration along the tangent, and the

d's
dt

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form of the equation shews, and consequently is the acceleration perpendicular to the tangent.

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and thus the acceleration perpendicular to the osculating

plane vanishes. The acceleration

ρ

must therefore be along a normal to the path drawn in the osculating plane; that is, along the radius of absolute curvature.

18. We are therefore led to expand the definition given in § 12 thus:-Acceleration is the rate of change of velocity whether that change take place in the direction of motion or not.

What is meant by change of velocity is evident from § 10. For if a velocity ОA become OC, its change is AC, or OB.

Hence, just as the direction of motion of a point is the tangent to its path-so the direction of acceleration of a moving point is to be found by the following construction.

From any point O draw lines OP, OQ, etc., representing in magnitude and direction the velocity of the moving point at every instant. The points, P, Q, etc., form in all cases of motion of a material particle a continuous curve, for an infinitely great force is requisite to change the velocity of a particle abruptly either in direction or magnitude. Now if be a point near to P, OP and OQ represent two successive values of the velocity. Hence PQ is the whole change of velocity during the interval. As the interval becomes smaller, the direction PQ more and more nearly becomes the tangent at P. Hence the direction of acceleration is that of the tangent to the curve thus described, called by its inventor, Sir W. R. Hamilton, the Hodograph.

The amount of acceleration is the rate of change of velocity, and is therefore measured by the velocity of P in the curve PQ.

19. The Moment of a velocity about any point is the rectangle under its magnitude and the perpendicular from the point upon its direction. The moment of the resultant velocity of a point about any point in the plane of the components is equal to the algebraic sum of the moments of the components, the proper sign of each moment depending on the direction of motion about the point. The same is true of moments of acceleration, and of momentum as defined later.

Consider two component velocities, AB and AC, and let AD be their resultant (§ 10). Their half moments round

the point are respectively the areas OAB, OCA. Now OCA, together with half the area of the parallelogram CABD, is equal to OBD. Hence the sum of the two half moments

together with half the area of the parallelogram is equal to AOB together with BOD, that is to say, to the area of the whole figure OABD. But ABD, a part of this figure, is equal to half the area of the parallelogram; and therefore the remainder, OAD, is equal to the sum of the two half moments. And OAD is half the moment of the resultant velocity round the point O. Hence the moment of the resultant is equal to the sum of the moments of the two components. By attending to the signs of the moments, we see that the proposition holds when O is within the angle CAB.

20. Now if one of the components always passes through the point O, its moment vanishes. This is the case of a motion in which the acceleration is directed to a fixed point, and we thus prove the theorem that in the case of acceleration always directed to a fixed point the path is plane and the areas described by the radius-vector are proportional to the times; for the moment of velocity, which in this case is constant, is evidently double the rate at which the area is traced out by the radius-vector.

the

21. Hence in this case the velocity at any point is inversely as the perpendicular from the fixed point upon tangent to the path, the momentary direction of motion.

For evidently the product of this perpendicular and the velocity at any instant gives double the area described in one second about the fixed point, which hast just been shewn to be a constant quantity.

22. The results of the last three sections may be easily obtained analytically, thus. Let the plane of motion be taken as that of x, y; and let the origin be the point about which moments are taken. Then if x, y be the position of the moving point at time t, the perpendicular from the origin on the tangent to its path is

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