If p, q, r, be the perpendicular distances of the molecule from BC, CA, AB, respectively, then P_L x = = = = If λ=μ=v, then p=q=r, or the molecule will rest at the centre of the inscribed circle, a theorem proved by Ferdinand Joachimsthal, in the Cambridge and Dublin Mathematical Journal, Vol. I. p. 93. (13) Two equal straight rods, the particles of which attract each other with a force varying inversely as the square of the distance, are parallel to each other and perpendicular to the lines joining their ends, and are held asunder by strings attached to their middle points: to determine the tension of the strings when the rods are at a given distance from each other. = If a the distance between the rods and b=the length of either, the required tension is equal to (14) Each particle of two rods of infinite length, which coincide in direction with two conjugate diameters of an elliptic wire, attracts with a force varying inversely as the square of the distance: to find the position of equilibrium of a bead moveable along the wire. Let a, b, be the semi-axes of the ellipse, o the acute angle between the two conjugate diameters: then the particle will rest at the intersections of the ellipse with a concentric circle the square of the radius of which is equal to (15) Each particle of two infinite rods, at right angles to one another, attracts with a force varying inversely as the nth power of the distance: to find the form of the rigid curve on which a particle, subject to the attraction of the rods, will rest in all positions. The rods being taken as axes of co-ordinates, the equation to the curve is unless n = 2, in which case it is xy= = c2. (16) To find the resultant attraction of a homogeneous globe on an external particle, the law of attraction being that of the inverse cube. If a be the radius of the globe, c the distance of the attracted particle from the centre, and μ the attraction of a unit of mass at a unit of distance, the required attraction is equal to (17) A quantity of homogeneous matter of uniform density is in the form of a portion of a paraboloid of revolution bounded by a plane perpendicular to the axis: to find its attraction on a particle of unit mass at the vertex. If P be the density, c the length of the axis of the frustum, and the length of the latus rectum, the attraction is equal to πpl log (w.ew-1), where = cot 4, 4 being defined by the equation (18) A portion of a spherical surface, intercepted between two parallel planes, attracts a particle placed in a normal to the two planes through the centre of the sphere: to find the resultant attraction on the particle, the law of attraction being that of the inverse square. Let r', r', be the distances of the particle from the two boundaries of the portion of the surface, c its distance from the centre of the sphere, a the radius of the sphere: then the required attraction is equal to where μ is the attraction of a unit of area of the spherical surface condensed at a unit of distance from the particle. (19) A portion of a thin spherical shell, the projections of which upon three planes through the centre at right angles to each other are given, attracts a particle at the centre: supposing the law of attraction to be that of any function of the distance, to find the direction of the resultant attraction on the particle. If A, B, C, be the given projections and the three planes be co-ordinate planes, the equations to the required direction are Ferrers and Jackson: Solutions of the Cambridge Problems, 1848 to 1851, p. 373. (20) A brittle rod AB, attached to smooth hinges at A and B, is attracted towards a centre of force C according to the law of nature. Supposing the absolute force to be indefinitely augmented, to determine where the rod will eventually snap. If E be the point of snapping, then, a, ß, denoting the angles BAC, ABC, respectively, CHAPTER VIII. MISCELLANEOUS PROBLEMS. (1) STRINGS are fastened to any number of points A, B, C, .........., in space, and are pulled towards a point P with forces proportional to PA, PB, PC, ......: shew that, wherever the point P be situated, the resultant of these forces will always pass through a fixed point. Let a, b, c, be the co-ordinates of P referred to three rectangular axes: then, x, y, z, being the co-ordinates of any one of the points A, B, C,......, the components of the resultant will be equal to μ (na - Σx), μ (nb - Zy), μ (nc - Σz), which will therefore be proportional to the direction-cosines of the resultant. The equations to the resultant will therefore be multiplying each of these fractions by n and adding unity to each we get Hence we see that the resultant always passes through a point of which the co-ordinates are (2) Find the amount of work done in drawing up a common Venetian blind. How must the same problem be solved for a curtain ? = Let W the weight of each bar of the blind; a= the distance between two consecutive bars; n = their number. Then the work done will be equal to W (a + 2a+3a+......+na) = n(n+1) Wa. Let P-the sum of the weights and 7=the height of the window: then P=nW and l=na, and the work done is equal to Let n = ∞ then the Venetian blind is mechanically the same as a curtain, the number of its bars being infinite and the weight of each indefinitely small. Thus, P being the weight and the length of the curtain, the work done is equal to Pl. The work done in raising the curtain may also be estimated by integration. The weight of a length dx of the curtain is P da: hence the τ (3) The frustum of a paraboloid of revolution, the density of its circular sections varying as their areas, stands upon its vertex on a horizontal plane: to find the length of its axis when the equilibrium is indifferent. If the vertex of a solid of revolution, of which the axis extends vertically upwards, be placed upon the summit of another solid of revolution the axis of which extends vertically downwards, then, as is proved in most works on Statics, the equilibrium will be stable, unstable, or indifferent, accordingly as the altitude of the centre of gravity of the upper body above the point of contact is less than, greater than, or equal to r, r', being the radii of curvature of the two surfaces at the point of contact. |