2 2 2 fat C, r and varies nearly inversely as the square of her distance from 15 0.0 – 2B + a)}. (dr)? do, 2μ + dt dR 1, dt dr investigate the equation for the perturbation in longitude. What terms in the resulting equation are most important ? 246. Apply the equations of motion to shew that a body acted upon by a central force will describe a curve lying in one plane, and that the areas described about the centre of force are proportional to the times. 247. A body describes an ellipse round a centre of force in the centre, find the periodic time and also the time of describing a given angle after leaving an apse. 248. Explain Newton's method of finding the angle between the apsides in orbits nearly circular. Ex. Force of (dist.)". 249. The velocities of a body at different points of a curve, 1836 described about a centre of force, are inversely as the perpendiculars from the centre upon the tangents at those points. 250. Find the law of force and the periodic time in an ellipse about the focus. Hence, assuming the law of gravitation, deduce Kepler's third law. 251. Explain Neuton's method of finding the angle between the apsides in an orbit nearly circular. Ex. Force varying partly as the distance and partly inversely of the distance. as 252. When the sun is moving from apogee to perigee, shew that, in consequence of his disturbing force, the moon is always in advance of her mean place. 253. The radius vector of a planet's orbit is affected with a small periodical inequality; shew that its effect may be represented by continued and periodical alterations of the eccentricity and longitude of the perihelion ; the period of either being P.T where P is the period of the planet, and T that of the P - T inequality. 254. A body describes an ellipse about a centre of force in one of its foci, and its velocity is slightly increased in the direction of its motion; shew that the eccentricity will be thereby diminished or increased, as the distance of the body from the focus is greater or less than its mean distance ; and the apse will advance or recede, as the body is moving from the lower to the higher apse, or from the higher to the lower. 255. Find the horary motion of the moon's nodes in a circular orbit; and shew that the mean horary motion of the nodes is half the horary motion when the moon is in syzygy. 256. If a body attracted to a centre of force be projected with a given velocity from a given point, its velocity, when at a given distance from the centre, is independent of the direction of projection. 257. In the expression for the moon's longitude there occurs K2 the term sin (2gpt – 2y); shew that it is nearly the dif 4 ference between her longitudes measured on her orbit and on the ecliptic. 3m Explain the effect of the term ko sin{(2—2m-g)0--28+y} in the expression for the tangent of the moon's latitude. What do the several letters in this term represent? 258. The equation for determining the perturbation of the radius vector of a planet being d? (ror) dR dR 0 (9) rồr + 2n so dt tr dt2 de + n2 drs explain the method of solving it by approximation. Shew that a force, which goes through all its values nearly in the time of a revolution, will produce a considerable inequality in the radius vector. 259. The apparent orbit of P to a spectator at S in motion may be described round S fixed by the action of the same force, if P be projected with a proper velocity and in a proper direction. (Newton, Prop. 58.) SECTION XVIII. QUESTIONS IN HYDROSTATICS AND THE THEORY OF SOUND. 1821 = ax. 1. A clepsydra is constructed to mark equal portions of time, in the form of a paraboloid having its vertex downwards, the equation to the generating curve being yo How must the scale on the axis be graduated ? 2. Explain the construction of the steam engine ; and having given the weight upon the piston, the quantity of steam admitted, and the content of the cylinder, find the velocity of the piston at any point, and the time of describing the cylinder. 3. Compare the pressure on the surface of a sphere filled with water, with the weight of a sphere of mercury of the same magnitude. 4. Find the centre of pressure of a trapezoidal plane surface immersed vertically in a fluid, two of whose sides are parallel to each other, and to the surface of the fluid. 5. A cylinder of given length is pressed down in a vertical position into a fluid, so that its upper end is on a level with the surface, the specific gravity of the cylinder being one half that of the fluid: the pressure being removed, to find the greatest height to which the upper end of the cylinder will rise above the surface of the fluid. 6. Distinguish between the centre of gravity and centre of pressure, and shew that the former is always nearer to the surface of the fluid than the latter. 7. Graduate a thermometer according to Fahrenheit's scale. 8. A small aperture (a) is made in the vertical side of a cylindrical vessel filled with a fluid ; the area of its horizontal section being A. Compare the latus rectum of the parabola first described by the spouting fluid with the length of a pendulum vibrating once while the surface of the fluid descends to the orifice. 9. A vessel is kept filled with a fluid ; and an aperture is made in its perpendicular side in the form of a parabola, the vertex of which coincides with the surface of the fluid. Find the depth of a horizontal section such that if the whole fluid issued with its velocity, the quantity discharged in a given time would be the same as when each horizontal section flows with its own velocity. 10. An isosceles triangle is immersed perpendicularly in a 1822 fluid with its vertex coincident with the surface and its base parallel to it. How must it be divided by a line parallel to the base, so that the pressure upon the upper and lower parts respectively may be in the ratio of 1:7? 11. How may the phenomena of the trade winds be explained ? 12. A vessel of given altitude empties itself through an orifice of given dimensions in its lowest point, and the upper surface descends with a given uniform velocity ; find the content of the vessel. 13. Find the time of emptying a sphere filled with fluid through an orifice in its lowest point. 14. An upright cylindrical vessel empties itself through an orifice in the base; compare the pressures upon the concave surface at first, and when half the time of emptying has elapsed. 15. The orifices in the equal bases of two upright prismatic vessels are in the ratio of 2:1, and the vessels are emptied in equal times; compare their altitudes. 16. A life-boat contains 100 cubic feet of wood, specific gravity .8; and 50 feet of air, specific gravity .0012. When filled with fresh water, what weight of iron ballast, specific gravity 7.645, must be thrown in before it will begin to sink? |