Let SP=r, SA=a, <ASP=0; then the equation to the curve APS will be (14) A particle is projected with a given velocity from a point A (fig. 157) along a curve APO in which it is constrained to move, and is acted upon by a force always tending to O, and varying directly as the distance; to find the nature of this curve in order that the angular velocity of the radius vector OP may be invariable. = Let 40=a, OP=r, ‹ AOP= 0, μ the absolute force of attraction, w = the angular velocity of OP, B = the initial velocity of the particle; then the equation to the curve will be (15) A particle is acted on by an attractive force, tending to a centre and varying inversely as the square of the distance; to find the tautochrone. T If denote the time of the motion, and the notation remain the same as in problem (7), the differential equation to the tautochrone will be (16) To find the tautochrone when the central attractive force is constant. If ƒ denote the constant central force, the equation to the tautochrone will be (17) An infinite number of straight lines originate at a single point and lie in one plane; to determine the synchronous curve, gravity being the accelerating force. The given point being taken as the origin of co-ordinates, the axis of x extending vertically downwards, and that of y being horizontal; the synchronous curve will be a circle of which the equation is x2 + y2 = 1gk2x, where k denotes the common time of descent. Euler; Mém. de l'Acad. de St. Pétersb. 1819, 1820, p. 22. (18) There is an infinite number of cycloids, of which the bases all commence at the origin of co-ordinates, and coincide. with the axis of y, which is horizontal; to find the synchronous curve, gravity being the accelerating force, and the motion commencing from the origin. Let k denote the constant time of descent; then, the axis of x being vertical, the equation to the required curve depends upon the elimination of a between the two equations and will cut all the cycloids at right angles. John Bernoulli; Act. Erudit. Lips. 1697, Mai. p. 206. (19) A particle, acted on by a central attractive force, which varies as the distance, moves along a curve from one given point to another; to find the nature of the curve when it is brachistochronous. Let A (fig. 155) be the point at which the motion commences, and B the point at which the particle is to arrive in the shortest time possible. Let P be any point of the brachistochrone; let SP=r, p= the perpendicular from S, the centre of force, upon the tangent at P, SA=a. Then the equation to the curve between p and r will be p2 = A (r2 — a2), where A is a constant quantity, which is the equation to the hypocycloid. If from this equation we were to obtain by integration a relation between r and an angular co-ordinate 0, we should have another constant in the equation in addition to A. Both these constants would have to be determined by the conditions that the curve must pass through both A and B. Euler; Mechan. Tom. II. p. 191. SECT. 5. Inverse Problems on the Pressure of a Particle on Smooth Fixed Curves. (1) A particle descends down a curve line in a vertical plane by the action of gravity; to find the nature of the curve in order that the pressure may be invariable. Let OA (fig. 158) be the required curve; Ox, vertical, the axis of x, Oy, horizontal, the axis of y; P any point in the curve; let OM=x, PM=y, OP=s; let k be the constant pressure; the initial velocity of the particle, O being its initial position. Then, by formula (A) of Sect. (II.), we have where p denotes the magnitude of the radius of curvature at P. also, s being taken as the independent variable, where C is an arbitrary constant. Assume a to be the inclination of the curve to the vertical at the origin; then The relation between x and y may be obtained by a second integration, but the result is of little value in consequence of its complexity. For the investigation of the form of the curve which corresponds to the differential equation (2), the reader is referred to Whewell's Dynamics, part II. p. 95; or Earnshaw's Dynamics, p. 129. The problem of the Curve of Equal Pressure, in the case of gravity, was first proposed by John Bernoulli', and solved by L'Hôpital'. Various problems of a similar character were afterwards discussed by Varignon3. Commerc. Epistolic. Leibnitii et Bernoullii, Epist. VII. (2) A particle, acted on by gravity, descends from a point (fig. 158) down a curve 04, which it presses at each point of its descent with a force varying as the square of its distance below the horizontal line through 0; to find the nature of the curve OA, the initial velocity of the particle being zero. Let the axes Ox, Oy, be taken vertical and horizontal; let k be the pressure on the curve when x is equal to unity. Then, by the formula (A) of Sect. (II.), 1 Act. Erudit. Suppl. Tom. II. sect. vi. p. 291. 2 Mém. de l'Acad. des Sciences de Paris, 1700, p. 9. no constant being added because the curve passes through the which is the equation to the Elastic Curve of James Bernoulli'. Varignon; Mémoires de l'Académie des Sciences de Paris, 1710, p. 151. (3) A particle, acted upon by a force parallel to the axis of x, is constrained to move along a given curve OPA (fig. 158); to find the law of the force in order that the curve may experience an invariable pressure. Let k denote the constant pressure, ẞ the velocity of the particle at O, which we will take as the origin of co-ordinates, 1 Act. Erudit. Lips. 1694, p. 272; 1695, p. 538. |