713 in a given manner in that plane; to determine the motion of the particle. Let x, y, x, y be the coordinates, at time t, of the particle and point, a the length of the string, R the tension of the string, and m the mass of the particle. For the motion of the particle we have (1), with the condition (a, -#) 2 + (y-7) 2 = a 2 . Now 7 x, y are given functions of t. Take from both sides of the equations (1) the quantities in C -r^, m 7T^ res P ect i ve ly! ai!L ^ we have the equations of relative motion - x d 2 x These are precisely the equations we should have had if the point had been fixed, and in addition to the forces X, Y, and R acting on the particle, we had applied, reversed in direction, the accelerations of the point s motion with the mass as a factor. It is evident that the same theorem will hold in three dimensions. The acceler ations , c ~y are known as functions of t, and therefore the dt 2 dfl equations of relative motion are completely determined. Let there be no impressed forces, and suppose first that the point moves with constant velocity in a straight line. Here . are constant, and therefore no terms are intro- dt dt duced in the equations of motion. Again, suppose the point s motion to be rectilinear, but uniformly accelerated. The relative motion will evidently be that of a simple pendulum from side to side of the point s line of motion. In certain cases, when the angular velocity exceeds a certain limit, we shall have the string occasionally untended ; and this will give rise to an impact when it is again tended. While the string is untended the particle moves, of course, in a straight line. Suppose the point to move, with constant angular velocity u, in a circle whose radius is r and centre origin. Here, supposing the point to start from the axis of x, Hence the equations of motion are, since Whence or, in polar coordinates, for the relative motion, - J = - aParsm (6 - oit) , 1 J . dt dt Now - wt is the inclination of the string to the radius passing through the point ; call it <j>, and we have d?<p -T~ dt 2 r<a 2 the equation of motion of a simple pendulum whose length is The particle therefore moves, with reference to the uniformly revolving radius of the circle described by the point, just as a simple pendulum with reference to the vertical. A particle moves in a smooth straight tube which revolves with co>istant angular velocity round a vertical axis to which it is perpen dicular ; to determine the motion. Here, referring the particle to polar coordinates in the plane of motion of the tube, we have constant , P = (47), and thus for the acceleration along the tube r-ro> 2 = 0; whence rAg^ + Bg " . Suppose the motion to commence at time < = by the cutting of a string, length a, attaching the particle to the axis. The velocity of the particle at that instant along the tube is zero. Hence at t = l> r=0=A-B; so that A = B = ^, and r = ^(2 M + g~ w ). In fig. 51 let OM be the initial position of the tube and A that of the particle, and let OL and _ Q be the tube aud particle at time t. Then OA = a, arc AP = aiat, OQ = r, aud we have arcAP arc AP >A +g OA ) . From this we see that OQ and the arc AP are corresponding values of the ordinate and abscissa of a catenary whose parameter is OA. Here the vertical pressure on the tube is equal to the weight of the particle, while the- horizontal pressure is in d . t _ /. 7 ( "$) = - 2?ncor = - mu-a( S. - Z ) r df From this equation, combined with the value of r, we easily deduce for the horizontal pressure the value and it is therefore proportional at any instant to the tangent drawn from Q to the circle APN. Let the tube be in the form of a circle turning with constant angular velocity about a vertical diameter. Let AO (fig. 52) be the axis, P the position of the particle at any time. Let POA = 6 denote the particle s posi tion, and R the pressure on the tube in the direction of OP. We have o dt" - orrt sin = - R sin . Eliminating R, d*8 a-^-r- a<a~ sine cos Q - dt 2 Tli3 position of equilibrium will therefore be given by 9 sm0 = 0; or by 6 = 7, where " Integrating (1), -- i dt) =n 2 cos 2 0). (2). Suppose the particle to pass through the lowest point with velocity a<a l , we have ! = a)? - 2o> 2 cos 7(1 - cos 6) + or sin 2 ca 2 ) (1 -cos7) 2 +k -(cos0- COS7)- ^ , CO and can never vanish if ^f>4cos7, or ? > , that is, if the dt < n velocity at the lowest point be greater than that due to the level of the highest point. A rift If w 2 <--, the particle will oscillate; and, if -=- = 0, when a, a dt then ( ) = (cos - cos a) - 2 (cos 2 - cos 2 a) , dt J a^ = w 2 (cos - cos a)( ^ - cos a - cos
* t * /
= w 2 (cos - cos a)( 2 cos 7 - cos a - cos 0) ; and therefore, if 2cosy- cos>l, the particle will oscillate through the lowest point. If I>2cos7-cosa> -1, then, putting 2 cos 7 - cos a = cos /3 , 5?) =c,, 2 (co:,<?-cosaXcos-cos0), and the particle will oscillate on one side of the vertical diameter.
XV. 90
