along p and S be the retarding force upon B arising from the inertia, we have du SXBAX dt A udu B de a222 p3 a*ß A udu F=0-S g3 B de a gede А :. udu = Fde = udu p3 B and udu = Ba'ßo de A+BP3 a BB 1 Х (see 436) B pe B X (1 A + B (A+B)a a”.(A+B). .(1) Aa' + A the equation to the curve, which is therefore one of Cotes' Spirals. Again, in all curves 2 2 Pe ✓ Code + de) 673. Let w be the given weight of the cylinder; then the accelerating force down the plane is R being the distance of the centre of gyration from the axis of rotation, and 0 the inclination of the plane. 22 2 :. F= sin. A sin. 0. 3 78 + 2 574. Let W be the weight of the cylinder, " the radius of its base, then its inertia is w x re, and the accelerating 2 :. 12. (2P + W) = Pg P x g - - 24). 12 12 575. Supposing m' not equal to m, and putting cm' = x, ac =cb = a, then since the efficacy of m in opposing the motion of m', is measured by тх m', v (a? + x2) the whole moving force is 2mi V (a? + x2) and the mass moved is 2m + m' 1 2m + m But at the time the velocity is a maximum, the accelerating force = 0. 2mx ma a? + 2? am And .. x = (4m", - m')' the distance required. = { } 577. Let , be the radius of either globe, M its mass, k, k' the distances of their centres of gravity from the axis of rotation 1, r the distances of their centres of oscillation from the same, when the bodies are unconnected ; then since (Creswell's Translat. of Venturoli, p. 141), the length of the compound pendulum is Mkl + Mk'l kl+k'T Mk + MK ktk and by the question k = 3r, k' = 5r. Therefore 31 + 51 L = 578. Let a be the length of the rod; then supposing generally the distance of the point of suspension from its extre mity, the length of the pendulum is (see Venturoli) a 3 (-9) + 3n3 3 a 579. Since the centre of Initial Rotation is distant from that point of impact by the same interval as the centre of oscillation, considering that point as the point of suspension, the distance required is (Venturoli) S Aa2 + B62 Aa! + Bbe Aa - B6 Aa Bb A + B A, B being the masses, and a, b the lengths of the arms of the lever. 580. Generally, if P denote the power moving the system, whose weight is w, acting at the distance r from the axis of rotation, then the force which accelerates P is Pr2 Prs + W.R? s=gFt gFt :.0 = 360° x 2πη or the body moves at the rate of gFt revolutions in a second. 271 Hence, by the question, gFt 277 Pro = 10 gF 32 = 6'. 32". 6173 nearly. |