If 2a be the length of the rod, b that of each string, and a the inclination of the strings to the horizon in the position of equilibrium, the time of oscillation is equal to (17) A thin uniform rod, one end of which is attached to a smooth hinge, is allowed to fall from a horizontal position: to find the vertical strain on the hinge when the horizontal strain on it is the greatest. If W be the weight of the rod, the required vertical strain is equal to 11 꿍ᅲ. (18) A given square board, two edges of which are horizontal, is supported by two vertical strings attached to its higher edge at given points: supposing either of the two strings to be cut, to find the initial tension of the other. If W be the weight of the board, a the length of an edge, and the distance of the point of attachment of the uncut string from the middle of the higher edge, the required tension is equal to a2 W a2 + 66* (19) An equilateral triangle is suspended from a point by three strings, each equal to one of the sides, attached to its angular points: if one of the strings be cut, to find the instantaneous change in the tensions of the other two. Let T, T', be the tensions of either of the two strings before and just after the third string is cut then T': T: 36: 43. (20) A uniform circular table is supported by three equal and equidistant props placed at the circumference: if one prop be suddenly removed, to find the alteration in the pressure on each of the other props in the first instant. The pressure on each of the remaining props is instantaneously diminished by one-twelfth of the weight of the table. (21) An elliptic lamina is supported, with its plane vertical and transverse axis horizontal, by two weightless pins passing through its foci: if one of the pins be released, to determine the eccentricity of the ellipse in order that the pressure on the other may be initially unaltered. The required value of the eccentricity is 2\ (22) A uniform rod is suspended by two strings of equal lengths, attached to its extremities and to two fixed points in the same horizontal plane, the distance between which is equal to the length of the rod. An angular velocity of such magnitude is communicated to the rod about a vertical line through its centre that it just rises to the level of the fixed points: to find the tension of either string the instant after the communication of the angular velocity. The instantaneous tension of either string is seven times as great as it was before motion commenced. (23) A heavy rod is suspended from a fixed point by two inextensible strings without weight, the strings and the rod forming an equilateral triangle: supposing either of the strings to be cut, to determine the initial tension of the other. If W be the weight of the rod, the required tension is equal to (24) A uniform sphere, moveable about a fixed point in its surface, rests against an inclined plane: supposing the diameter which passes through the fixed point to be horizontal, to determine whether, if the plane be suddenly removed, the pressure on the fixed point will be increased or diminished. The pressure will be increased or diminished accordingly as the inclination of the plane is less or greater than tan1. (25) A hemisphere oscillates about a horizontal axis which coincides with a diameter of the base: to compare the maxi mum pressure on the axis with the weight of the hemisphere, the base of the hemisphere at the commencement of the motion being inclined to the horizon at an angle of 60o. The greatest pressure = 128 × weight of hemisphere. (26) A cone, moveable about the horizontal diameter of its base, which is fixed, is supported, its axis being horizontal, by a vertical string fastened to its vertex: supposing the string to be cut, to compare the initial pressure on the fixed diameter with the pressure in the former case. If a be the vertical angle of the cone, P the pressure on the horizontal diameter before the string is cut, and P the pressure after it is cut, then (27) An angular velocity having been impressed upon a heterogeneous sphere about an axis, perpendicular to the vertical plane which contains its centre of gravity G and its geometrical centre C, and passing through G (fig. 192), it is then placed upon a smooth horizontal plane: to determine the magnitude of the impressed angular velocity in order that G may rise to a point in the vertical line SCK through C, and there rest; the initial magnitude of the angle between CG and the vertical radius CS being given. = Let CGc, k the radius of gyration about G, a = the initial value of the angle GCS, and the required angular velocity; then w will be determined by the equation = (k2 + c2 sin2 a) w2 = 2cg (1 + cos a). Euler; Nova Acta Acad. Petrop. 1783; p. 119. (28) A uniform rod, not acted on by any forces, is in motion, its ends being constrained to slide along two fixed rods at right angles to each other in one plane: to find the wrenching force at any point. Let AB be the rod, C any point in it, O the intersection of the two fixed rods; let CH, CK, be perpendiculars from C = upon OA, OB, respectively; let m the mass of AB. Then the angular velocity w of AB will be invariable, and the wrenching force at C will be equal to mw2. CH. CK. Mackenzie and Walton; Solutions of the Cambridge SECTION 2. Single Body. Axis of Rotation rotating. The problems in this section, the solutions of which are worked out in full, or which are proposed for the exercise of the student, require a knowledge of theorems the demonstrations of which are given in ordinary treatises on Rigid Dynamics. Every student is of course expected to be familiar with the notation and theorems in the excellent work by Mr Routh, On the Dynamics of a System of Rigid Bodies. (1) A plane lamina, not acted on by any forces, of uniform density and thickness, the boundary of which is a curve represented in polar co-ordinates by the equation r = a + b sin2 20, moves about its pole as a fixed point: to determine the nature of the cone described in space by its instantaneous axis. The moments of inertia of the lamina about the prime radius vector and a line through the pole perpendicular, in the plane of the lamina, to this radius vector, which are principal axes at the pole, are equal: hence, by Euler's equations, we see that, w1, w,, w,, denoting the angular velocities about these two axes and the principal axis normal to the lamina at their intersection, where a and ẞ are constant quantities. Thus, o denoting the angular velocity about the instantaneous axis, Letr be the radius vector of Poinsot's momental ellipsoid, which coincides with the instantaneous axis, and p the perpendicular from the centre on the tangent plane to the ellipsoid at the extremity of this radius vector: then where, A, A, C, denoting the moments of inertia about the principal axes, and Hence T= A (w,2 + w‚3) + Cw ̧2 = A (a3 + ß3) + Cy3, 1 G = {42 (w," + w2") + C2∞ ̧2}* = (A2a2 + C2y2,1. 1 But the position of the perpendicular p is absolutely fixed in space, and the position of the radius vector r coincides with the instantaneous axis1: hence the instantaneous axis describes a right cone in space. (2) A solid of revolution, not acted on by any forces, is revolving about a fixed point at its centre of gravity: supposing the instantaneous axis and the axis of figure to be equally inclined to the invariable line, to find the magnitude of this inclination. Let C denote the moment of inertia about the axis of figure, and A that about a perpendicular line through the centre of gravity. Then, the principal axes at the fixed point being taken as axes of co-ordinates, the equations to the instantaneous axis are Since the inclinations of the invariable line to the instantaneous axis and to the axis of figure are equal, we have, supposing 1 Routh, Dynamics of a System of Rigid Bodies, Second Edition, p. 328. |