6. Define the centre of gravity: and find it for any number of points in the same plane. 7. Let a uniform beam be supported on two given inclined planes: find the position in which it will rest. 8. Shew under what condition a system of forces, acting any how in space, can have a single resultant: and find the magnitude and direction of that resultant. 9. Find the differential expression for the centre of gravity of a solid of revolution: and apply it to a parabolic frustum. 10. Find the equations to the common catenary. 11. In the direct impact of imperfectly elastic bodies, find the velocities lost by A and gained by B in the direction of A's motion. 12. Find the direction in which a perfectly elastic ball must move, from a given point, so that it may return to the same point after impinging successively on the four sides of a rectangle and prove that in its course it describes a parallelogram. 13. Let a body be projected with a given velocity, and accelerated or retarded in the line of its motion by an uniform force: find the time in which it will describe a given space. 14. Let two bodies P and Q be connected by a string passing over a fixed pulley, and let P descend: after P has described a given space let a weight w be removed from P, leaving the remainder (P — w) lighter than Q. Trace the subsequent motion. 15. Prove that a projectile describes a parabola: and find the directrix, focus, and latus rectum. 16. Two bodies are projected from the same point with the same velocity, so as to fall again on the horizontal plane passing through the point of projection, at the same point; and the times of flight are to each other as m : n. Find the directions of projection. 17. Explain the construction of the cycloid: and make a body Oscillate in it. 18. The length of the second's pendulum, in vacuo, is 39-1386 inches: hence find the actual force of gravity. 19. Let two elastic balls oscillate in the same cycloid, one on each side the vertical line drawn through the common point of suspension: explain their motions, supposing the descents to begin (1). At the same time from different altitudes; (2). At different times from the same altitude. 20. Define friction; and shew how it may be estimated by experiment. HYDROSTATICS. TRINITY COLLEGE, 1820. 1. THE pressure of a fluid against any surface in a direction perpendicular to it, varies as the area of the surface multiplied into the depth of its centre of gravity below the surface of the fluid. 2. A hollow cone without a bottom stands on a horizontal plane, and water is poured in at the vertex. The weight of the cone being given, how far may it be filled so as not to run out below? 3. What must be the magnitude and point of application of a single force that will support a sluice-gate in the shape of an inverted parabola? 4. Find the specific gravity of a body which is lighter than the fluid in which it is weighed. 5. If the specific gravity of air be called m, that of water being 1, and if W be the weight of any body in air, and W' its weight in water, its weight in vacuo will be 6. Three globes of the same diameter and of given specific gravities, are placed in the same straight line. How must they be disposed that they may balance on the same point of the line in vacuo and in water? 7. If a homogeneous hemisphere, floating in a fluid, be slightly inclined from the position of equilibrium; shew that the moment of the fluid to restore it to that position, is not affected by placing any additional weight at its centre. 8. A regular tetrahedron moves with its vertex forwards, in a direction perpendicular to its base: compare the resistance on the ob lique planes with that on the base. 9. If the particles of an elastic fluid repel each other with forces varying inversely as the fourth power of their distances, the compressive force on any portion varies as (density). 10. Explain the method of measuring altitudes by means of the barometer and thermometer. 11. Two barometers, whose tubes are each 7 inches long, being imperfectly filled with mercury, are observed to stand at the heights h and h', on one day, and k and k' on another. Find the quantity of air left in each, reducing it to the density when the mercury is at the standard altitude of 30 inches, and supposing the temperature invariable. 12. Construct a common forcing pump; and shew what is the force requisite to force the piston down. 13. In the common sucking pump, given the lowest point to which the piston descends, find the length of the stroke that the pump may work. 14. A cylinder which floats upright in a fluid, is pressed down below the position of equilibrium: when it is left to itself, find the time of its vertical oscillations, neglecting the resistance. 15. A vessel generated by the revolution of a portion of a semihyperbola round its conjugate axis, is emptied by an orifice at the centre of the hyperbola: find the time. 16. A close vessel is filled with air n times the density of atmo◄ spheric air. A small orifice being made, through which the air rushes into a vacuum, find the time elapsed when the density is diminished one half. 17. A tube of uniform diameter consists of two vertical legs connected by a horizontal branch. When it is made to revolve with a given velocity round one of its vertical legs as an axis, find the height to which the water will rise in the other. 18. Let a spherical body descend in a fluid from rest; having given the diameter of the sphere and its specific gravity relatively to that of the fluid, it is required to assign the time in which the sphere describes any given space. 19. If the density of a medium vary inversely as the distance from a centre, and the centripetal force inversely as any power of the distance from the same centre, a body may describe a logarithmic spiral about that point. 20. If the resistance on a body which oscillates small ares in a fluid vary as the nth power of the velocity, the difference of the arcs of descent and ascent will vary as the nth power of the whole arc. TRINITY COLLEGE, 1821. 1. A GIVEN cylindrical vessel full of water, is taken to such a depth beneath the earth's surface that the water sinks - th of an inch 1 n round the edge, find the distance of the surface of the water from the earth's centre. 2. Prove that the centres of pressure and of percussion coincide; and find the centre of pressure of a trapezium, two of whose sides are parallel to the surface of the fluid. 3. If s be the specific gravity of a body, ascertained by weighing it in air and water, and m be the specific gravity of the air at the time when the experiment was made; the correct specific gravity, or that which would have been found if the body had been weighed in a vacuum instead of air, is s + m(1 − s). 4. A square is immersed in a fluid whose specific gravity is to that of the square as 1 to r; shew that when one angle of the square is immersed there will be 12 different positions of equilibrium if r lie 8 32 9 32 between and ; and when three angles are immersed, that there 23 will be 12 different positions when r lies between and 24 32 5. In a floating body, whose transverse section is the same from one end to the other, find the distance of the Metacentre from the centre of gravity of the part immersed; and shew that the equilibrium is stable, or unstable, according as the metacentre lies above or below the centre of gravity of the body. |