1. SIDNEY SUSSEX COLLEGE, MAY 1830. HAVING given the time in which a body would fall through a given space at the Earth's surface, the radius of the Earth and the Moon's distance, find the periodic time. 2. Find the time of a body's describing any portion of a parabolic orbit. 3. Investigate Newton's construction for the velocity and time when a body moves from rest directly to or from a centre of force 1 varying according to any given law. Also if force o and the d2 body descend from an infinite distance, trace the curves and draw their asymptotes. 4. If a body be projected, in a direction inclined at any angle to its distance, from a centre of force which find the equation to the orbit described; shew that it possesses five species, and point out which of them have apses and asymptotes. 5. Apply Prop. 44. to determine the whole force at p when the fixed orbit is an ellipse and the force in the focus. 6. If a body oscillate in a hypocycloid, the force being as the distance and tending to the centre of the globe, time of an oscill. time to centre: √(SA): √(SC). (Prop. 52). 7. The force ∞ : 1 d2 ;; shew that the axis major of the ellipse de scribed by P round s in motion: the axis major of that described by P rounds at rest in the same periodic time: 3/(S+P): 3/S. (Prop. 60.) 8. Investigate fully the effects produced on the inclination of P's orbit to the ecliptic by the ablatitious force. 9. If all particles of matter attract each other with forces which 1 a corpuscle placed within a spherical shell is equally attracted in all directions. 10. If a body move in a plane acted on by a central force, and if this force = P at a distance d2u d02 P (), the equation to the trajectory described is +u- =0. (Whewell's Dynamics.) h2u2 11. If a body move through one or more spaces bounded by parallel planes, and be acted upon by a force which is perpendicular to those planes and which is at the same distance from them, the angle of incidence is to the angle of emergence in a given ratio. (Prop. 94.) 12. S and H are two equal centres of force, S attractive and H repulsive. If SH be bisected in C and BC drawn perpendicular to it, shew that a body placed any where in BC will describe a semi-ellipse about S, of which BC is the axis minor, and S, H the foci. 13. The Moon is retained in her orbit by the force of gravity. ST. PETER'S COLLEGE, MAY 1831. 1. IF a body be acted on by a central force, it's velocity at a given distance will be independent of the curve described. Shew what condition must be satisfied in order that this may be the case, the body being acted on by any forces. Will it be satisfied if a system of bodies move round a common centre of force, and be acted on by their mutual attractions ? 2. If a body be acted on by a central force which varies shew that the differential equation to the curve is 1 (dist.)3 c being the distance, and ẞ the angle of projection; the velocity of projection being e times that which would be acquired from infinity by the action of the same force. Find the integral equation to the orbit, when the velocity is greater than that in a circle at equal distance, and the area described less. [SUPP. P. II.] 3. Explain fully what is meant by the fixed, and moveable orbits, and the orbit traced out in fixed space, in Newton's 9th Section. Find the expression for the force in the orbit traced out in fixed space. What is the object of this Section? To what kind of orbits does it apply; and how is the expression, mentioned in the last question, rendered identical with that for the law of force in any such proposed orbit? 4. If a body move in a cycloid on the surface of an inclined plane, find the time of one oscillation when it descends from a given point, taking account of friction. 5. Two bodies attracting each other with forces which ∞ 1 (dist.) being placed together, one of them is projected with a given velocity; determine their motions, and where they will again be together. 6. Explain clearly what is meant by the plane of the Moon's orbit, and how it passes from any position to the consecutive one when it's inclination is varying. Investigate the motion of it's nodes. (Newton, Prop, 66.) 7. What is the evection? In what corollary does Newton point out the cause of it; and in which does he indicate that of the annual equation? 8. State clearly the nature of the elliptical orbit. (Newton, Vol. III.) 9. Deduce the expression for the horary motion of the node in a circular orbit, and thence shew that if N be the longitude of the node, where is the longitude of the Moon, and m the ratio of the periodic times of the Sun and Moon. Give the interpretation of the terms in this expression, 10. Enunciate D'Alembert's principle, as applicable to the case of the action of continuous forces, and also to that of impulsive ones. 11. The centres of oscillation and suspension are reciprocal. What must be the position of the axes about which the body oscillates? 12. If a rotatory motion, and a motion of translation be co-existent in any system of bodies, shew that in the determination of the value of 2. mv2, we may consider them as independent of each other. ST. JOHN'S COLLEGE, JUNE 1832. 1. THE orbit which the Sun appears to describe about a planet is an ellipse; prove this and determine the periodic time, 2. Two bodies start at the same time from the farther apse in an ellipse, one to describe the ellipse, the other the circle on the axis major, the farther focus S being the centre of force in both cases; compare the absolute forces in the two orbits, that the periodic times may be equal. 3. If yP and Pz be the external and internal spaces due to the velocity at the point P in an ellipse, force in focus; shew that Sy, Sp, Sz are in harmonical progression. 4. A weight P suspended by a string (c) is drawn 0° from the vertical by the action of a force placed in the same horizontal plane with the original position of P, and at a distance (a) from it, shew 5. Investigate the equation to the isochronous curve, when the force is constant, and acts in parallel lines. 6. The perpendiculars drawn from a point in a triangle upon the sides are a, b, c, and the angles which the sides subtend at the point are a, ß, y; shew that a particle placed at that point and acted on by the attraction of the sides (which a) will be kept at rest, 7. In a catenary attracted to a fixed centre, if P be the force of attraction and p the perpendicular on the tangent from the centre at any point, then the force by which the body would revolve in the P curve formed by the catenary o 8. If x, y be the co-ordinates of the highest point of the curve described by a body acted on by gravity and projected in vacuo, at an angle of inclination (a) to the horizon, their decrements, when the body is projected in a rare medium, in which Rk. velocity, 9. A sphere, when acted on separately by three forces, revolves round three diameters inclined at the same angle to each other and with the same angular velocity, determine the angular velocity and the new axis of rotation, when the three forces are applied at the same instant. 10. Construct for the inclination of the lunar orbit to the plane of the ecliptic at a given time. (Newton, Vol. III. Prop. 35.) 11. The equation for determining the projection of the Moon's orbit on the ecliptic is P and T being the whole forces on the Moon, parallel and perpendicular to the projection on the ecliptic of the Moon's distance from the Earth. 12. A body P is projected with a given velocity (a) in a direction perpendicular to its distance SA from a centre of force S, (which a distance) and which itself moves uniformly with velocity (V) in the direction AS produced. Determine the equation to the orbit described and shew that the motions of P and S are parallel when the co-ordinates of P measured from the original position of S are (a) and (−1)V. |