where is the coefficient of elasticity of the string, and c its natural length. 61. A uniform circular disc moving in any way is placed gently upon a rough horizontal plane. Assuming that the friction between any element of the disc and the plane varies as the relative velocity and is in a direction opposite to it, find the motion of the disc, and shew that if u and a be the velocity of the centre and the angular velocity about it at any instant, uwo uw, where u, and w, are the initial values of u and w. 62. A homogeneous straight rod AB is constrained to move in a vertical plane with its middle point in a horizontal groove, and its upper extremity against a smooth curve; find the nature of the curve when the rod descends from one given position to another in the least time possible, the initial angular velocity being given. 3 63. A number (n) of equal uniform rods A ̧С1В1, A‚Ç‚B2, ...are placed on a smooth horizontal plane so that the end A of the 2nd is in contact with the middle point C, of the first, the end A, in contact with C... and the angles CA¿B1, C‚„... are each equal to 0, so that the figure ACCC...Cn is a portion of a regular polygon. At the end A, an impulse P is applied inwards in the direction making an angle 7/2-0 with AC1. Prove that the impulse between the 7th and r+1th, supposing them smooth and rigid, π where a and ẞ are the roots of the equation 22-(2 sec 0 + 3 cos 0) z + 1 = 0. 64. A homogeneous inelastic hemisphere of radius a and mass m is let fall with its base vertical on a smooth inelastic horizontal plane. Prove that its pressure on the plane when the base is horizontal is equal to where v is the velocity with which it strikes the plane. Shew that the hemisphere will leave the plane immediately upon its base becoming vertical if 15v 16 Jag, and that, if 675v2/1024Tag is an integer, the hemisphere will again strike the plane with its base vertical. 65. A solid body of mass M rests with its flat base on a smooth horizontal plane, on which it is free to slide. Two points inside the solid, both lying in a vertical plane through its centre of gravity, are connected by a fine smooth hollow tube, down which a particle of mass m slides from the highest point to the lowest. If the tube is the brachistochrone, prove that its intrinsic equation is 66. A solid hemisphere of mass M rests on a perfectly rough horizontal plane, its upper surface, which is a perfectly smooth plane, being horizontal. Prove that if a particle of mass m is gently placed on it at a distance c from the centre, the initial radius of curvature of the path described by it will be equal to 3mc/Mk2, where k is the radius of gyration of the hemisphere about a tangent at its lowest point in the undisturbed position. CHAPTER XV. MOTION IN THREE DIMENSIONS. 266. WE now proceed to consider the motion of a system referred to three rectangular axes, either fixed, or moving in a given manner. As in Art. (34) we employ 01, 0, and 0, to represent the angular velocities of the system of axes. Taking w1, w, and w, as the angular velocities, at any instant, of a rigid body about the axes, it follows as in Art. (34) that the angular accelerations are respectively From the definition of the linear momenta and the angular momenta of a system it follows that these quantities are vectors and are subject to the parallelogrammic law. Let P1, P2, P3 represent the linear momenta of a system in the directions of the axes, and h1, h2, h, the angular momenta of the system about those axes. Then it follows, as in Art. 34, that, if we take OL, OM, and ON to represent either the quantities P1, P2, P3 or the quantities h1, ha, ha, the rates of change of these quantities are, on the same scale, the velocities of the point of which OL, OM, ON are co-ordinates, and are therefore respectively The equations of motion of the system are obtained by equating these expressions to the components of the acting forces and of the acting couples. The equations of motion, in the forms thus obtained, were first given by Mr R. B. Hayward, F.R.S., of St John's College, Cambridge. They are contained in a paper, published in 1856, in Part I., Vol. X., of the Cambridge Philosophical Transactions. 267. If x, y, z be the co-ordinates of a particle m of the system, and if u, v, w be the component velocities of the particle, h1 p1 = Σmu, p1 =Σmv, p = Σmw; = Em (wy - vz), h2 = Σm (uz — wx), h2 = Σm (vx-uy). The total motion of the system at the instant in question is thus represented by three linear momenta in the directions of the axes and three angular momenta about those axes. These are equivalent to a single linear momentum and a single angular momentum. 268. If the origin is not a fixed point, the expressions for the rates of change of the linear momenta are unaffected, but the expressions for the rates of change of angular momenta will require modification. Let a, ẞ, y be the component velocities of the origin, and suppose the axes to have no rotation. the angular momentum, at the time t + St, about the axis Ox, fixed in space = Σm {(w + Sw) (y + dy + B&t) − (v + dv) (z + dz + ydt)} = = h1 + Sh1 + (p3ß — P2Y) St, and, subtracting h, and dividing by St, we obtain the additional term so that the complete expression for the time-flux, about the instantaneous position of the axis, of the angular momentum If the origin be the centre of gravity of the system, the expressions for angular momenta and their rates of change are those of Art. (266). It will be seen that the terms pß-py of the previous article disappear in this case, for P2 = MB, and p2 = My. 269. Motion of a rigid body about a fixed point. In this case u = zw2- yW3, V = XW3 zw1, w = yw1 — XW2, and therefore h ̧ = Σm (y2 + 22) w1 − Σ (mxy) w2 – Σ (mxy) w3, and, if we represent the three moments of inertia by A, B, C and the three products of inertia by D, E, F we have If the expressions D, E, F all vanish the axes are said to be principal axes; if two vanish, the corresponding axis is a principal axis. In the case of a sphere, or a solid bounded by any regular polyhedron, when the centre is the origin, D, E, F all |