MECHANICS. TRINITY COLLEGE, 1818. 1. DISTINGUISH between the real and apparent motions, of a person walking across the deck of a ship under sail; and also of a heavy body let fall from the mast-head :—and, for each case, prove their agreement geometrically. 2. If the angle, at which two given forces act, be diminished, the compound force is increased. 3. If a body be kept at rest by three forces, and lines be drawn, making each the same angle with the directions in which they act, and towards the same parts,—the sides of a triangle formed by these lines will represent the quantities of the respective forces. 4. Enumerate the simple mechanical powers; and give familiar instances of the lever (of both kinds), the wheel and axle, and the wedge. 5. Prove the general proposition of the screw: and find numerically the weight that could be sustained by a power of 1 lb. at the distance of three yards from the axis of the screw,-the distance of two contiguous threads being one inch. 6. If a lever be put in motion, the velocity of power velocity of weight weight : power. 7. A spherical body rests upon two planes, inclined to the horizon at the angles of 45° and 60° respectively. Compare the pressures. 8. Investigate the rule for finding the centre of gravity of any number of particles of matter A, B, C, D, &c.; and demonstrate geometrically, that the same point will be discovered, in whatever order the particles are taken, (e. g. whether we find E the centre of A and B, then the centre of C and A+B placed at E, and so on, or whether we begin with finding the centre of A, C, &c. &c.) [SUPP. P. II.] B 9. If a triangle, whose sides are in the ratio of 3, 4, 5, be suspended by the centre of the inscribed circle,—shew that it cannot remain at rest, unless the shorter side be in an horizontal position. 10. Find the centre of gravity of a quadrilateral pyramid. 11. Prove that the relative velocity before impact : relative velocity after force of compression: force of elasticity:—and that in all cases of imperfect elasticity, this ratio is greater than that of AaBb to Ap+ Bq°, (where a and b are the velocities of A, B before impact, p and q their velocities after.) 12. Shew that the velocity communicated by A to C through B, where B is of an intermediate magnitude, is greater than what would be immediately communicated from A to C in the case of perfect elasticity ;—also that this velocity is the greatest, when B is a mean proportional between A and B. 13. If a body strike upon a plane at an angle of incidence 0, and with velocity v-prove that the direction and velocity, after reflection, are found by the proportions tan.0' tan.8: force of compression : force of elasticity, and v': v :: sin.: sin.0'. What must be the angles of incidence and reflection-where the velocity before impact : velocity after :: √2: 1, and force of compression: force of elasticity :: √3: 1? And how do the general propositions apply to the case of perfectly hard bodies? 14. If v, v' be any two velocities of a body either falling or rolling down an inclined plane-what will be its velocity at the middle point of space between them ? 15. A body (P) weighing 1 lb. is thrown downwards from the top of a tower 270 feet high, with a velocity of 40 feet in a second; and at the same instant a body (Q) weighing 9 lbs. is thrown upwards in the same line from the foot of the tower with a velocity of 50 feet in a second. In what time and at what height will they meet? and after the impact (supposing them perfectly elastic) what will be their velocities and directions; and how long will it be before each of them reaches the ground? 16. Given a point without a circle; it is required to find the line of quickest descent to the circumference. |