One of the scales of a common balance having been slightly displaced from its position of rest, in a vertical plane at right angles to the beam; to investigate the nature of the oscillatory motions of the two scales and of the beam.

Let AB (fig. 237) be the original position of the beam, PQ its position at any time t; p, q, the projections, on the directions AP, BQ, respectively, of the positions of the scales, considered as material points, at the same time. Let AC=a=BC, AP = 2 = BQ, Pp = a, Qq=y, Ma = the moment of inertia of the beam round C, m = the mass of each scale, 7 = the length of the string by which each scale is suspended. If we put, for simplicity,

we shall have, for the complete expression of the motions, the

initial value of x being c, while those of y,

dx dy

are all

dt' dt'

Investigations of the last two problems are given in a paper on the Sympathy of Pendulums, in the Cambridge Mathematical Journal, Vol. II. p. 120, by D. F. Gregory and A. Smith.

CHAPTER XII.

IMPULSIVE FORCES.

IF two rigid bodies impinge against each other, their motions both of translation and of rotation will generally experience modification, the determination of the nature of which, in the case of bodies of which the positions and motions are assigned at the instant before impact, constitutes the general problem of collision. The process of collision may be divided into two stages of indefinitely small duration: in the former stage, by the force of compression, which we will denote by R, the two points at which the bodies touch each other are constrained to assume equal resolved velocities in the direction of the common normal to their surfaces; in the latter stage, by the force of restitution, if the bodies be not inelastic, an additional reaction. eR takes place between them, where e denotes their common elasticity. Let w,, w,, w,, denote the angular velocities of one of the bodies about its principal axes and v1, v, v, the components of the velocity of its centre of gravity, at the conclusion of the former stage of the collision; let w, w, w,, vị, vý, vg, denote the analogous quantities in relation to the other body. Then, for the expression of the motion of the former body, as modified by the force of compression, we shall have six equations involving, together with known quantities, the symbols w,, w, w1, V1, V2, V3, R; and in like manner for the latter body we shall have six equations involving w', w,', w,', vý, vý, vý, R. Thus we shall have in all twelve equations involving thirteen variables. Another equation is supplied by the condition that the points of the two bodies at which their contact takes place shall have an equal resolved velocity in the direction of the common normal. Thus we shall be able to determine completely the modification of the motions of the two bodies due to the force of compression as well as the magnitude 37

W. S.

39

ვა

of this force. An additional modification must be applied, in the case of elastic bodies, in consequence of the force of restitution eR, which, from the investigation for the former stage of the collision, has become a known force. If one of the bodies be immoveable, the simplification of the method of investigation which we have described is obvious, the thirteen equations of which we made mention being reduced in this case to seven, and the common normal velocity of the two points of contact being zero. For ample information on this subject the student is referred to Poisson's Traité de Mécanique, Tom. II. p. 254, seconde édition.

SECT. 1. Single Body. Smooth Surfaces. Axes of Rotation, before and after impulsive action, parallel to each other.

(1) A beam of imperfect elasticity, moving anyhow in a vertical plane, impinges upon a smooth horizontal plane: to determine the initial motion of the beam after impact.

We will commence with supposing the beam to be inelastic; in this case the extremity of the beam which strikes the horizontal plane will continue after impact to slide along it without detaching itself. Let PQ (fig. 238) represent the beam at any time after impact; KL being the section of the horizontal plane made by the vertical plane through PQ; G the centre of gravity of PQ; draw GH at right angles to KL. Let GH =y, QG=a, GQH = 0, k = the radius of gyration about G, m = the mass of the beam; w, w', the angular velocities of the beam about G estimated in the direction of the arrows in the figure, just before and just after impact; u, v, the vertical velocities of G estimated downwards just before and just after impact; B the blow of impact.

w'

Then, ww being the angular velocity communicated by the blow, we shall have, if ẞ be the value of 0 at the instant of impact,

and, uv being the velocity of G which is destroyed by the

blow,

and therefore, t denoting the interval between the instant of impact and the arrival of the beam at the position represented in the figure,

dy

do

= a cos 0

dt

dt

hence, — v, — w', being the values of dy de

at the instant after

dt' dt

(3).

B),

a2

B) =

= a cos Bw+

k2

B= mk2

w' = ∞ + a cos B

Ba cos B
mkTM

=u-aw cos B,

aw cos B

a2 cos2 B+k2

aw cos B

a2 cos* B+ k2

aw cos B

a* cos* B+k2

= a cos B

acos B+k

Next let us suppose the beam to be imperfectly elastic, its elasticity being denoted by e; in this case the value of B given in (4) must be increased in the ratio of 1+e to 1; and therefore, instead of the equation (4), we have

which determines the magnitude of the blow of impact: substi tuting this value of B in (1), we get

The velocity of G parallel to the plane KL will be the same before and after impact. The end B of the beam will evidently after impact detach itself from the horizontal plane, since v is less and a greater when e has a finite value than when it is equal to zero.

(2) The edge BC, (fig. 239), of a vertical lamina is placed on a line Oy of greatest slope on an inclined plane: after sliding a given distance along the plane, it impinges against a small obstacle at C: to determine the impulsive reaction of the obstacle and the motion of the lamina immediately after impact.

Let G be the centre of gravity of the lamina; draw GH at right angles to Oy; Ox parallel to HG. Let GH= a, CH=b, m = the mass of the lamina, k = the radius of gyration about G; c=the velocity of G immediately before impact. We will commence with supposing the lamina to be perfectly inelastic; in this case the point C of the lamina will remain during impact in contact with the obstacle, the lamina rotating about this point. Let R, S, denote the impulsive reactions of the obstacle parallel to Ox, y0; and let u, v, denote the velocities of G parallel to Ox, Oy, on the completion of the impact; also let w represent the angular velocity of rotation about G at the same instant.

Then we have, for the motion of translation,