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1. A body moves under the force of gravity in the arc of a cycloid: find the centrifugal force, and the normal pressure from gravity at each point.
2. Two inclined planes are placed back to back; the inclination of one is 1 in 7, and of the other I in 10; a weight of 20 lbs. lies on the first, and is connected by a string with a weight of 30 lbs. which lies on the second. Find the direction in which motion will take place, the dynamical force of descent, and the tension of the string.
3. The direction of elevation which corresponds to the maximum range on an inclined plane bisects the angle between the plane and the vertical line passing through the point of projection?
4. Deduce the six equations of equilibrium of a solid body.
5. Deduce the equation of the curve which a homogeneous cord assumes under the force of gravity when suspended from two points.
6. A series of rough inclined planes pass through a given horizontal line; find the locus of the point to which bodies which fall down them arrive at a given time, if they start together from the given line.
7. For any system of forces parallel to a line, the sum of the moments is equal to the moment of the resultant round any perpendicular to the
8. For any system of forces parallel to a plane, the sum of the moments is equal to the moment of the resultant round any perpendicular to the plane?
9. Given the co-ordinates and the masses of any system of particles, prove the formulæ for the co-ordinates of their centre of gravity.
10. Required in pounds the moving force which, acting freely for a second on the mass of a ton weight, will get up in it a velocity of 20 miles an hour.
11. Required in pounds the centrifugal force of a mass a ton weight which moves uniformly at a velocity of 20 miles an hour in a circle of a mile radius.
12. A mass m, projected on a horizontal plane with a velocity v, is retarded in its motion by the friction of the plane; if the coefficient of friction =μ, required the space it will describe, and the time it will take to describe it.
13. G is the centre of gravity of a triangle ABC; prove that forces in the direction of and proportional to GA, GB, GC, will keep it at rest.
14. A ring R is attached to the end of a thread AR, which is fastened at A; BRW is another thread passing through the ring, and supporting a weight W. Find the position of equilibrium, A and B being in the same horizontal line.
15. A ladder rests in a given position against a smooth wall, with its foot upon a rough pavement; determine the weight which must be placed at its foot to prevent it from sliding, the coefficient of friction for the ladder and weight being both given.
16. Two weights move on inclined planes whose directions are at right angles to each other, and are connected by a string passing over their intersection; if the tension of the string be a maximum, what is the inclination of either plane to the horizon?
17. A particle runs down the interior of a vertical semicircle; find the pressure at the lowest point.
18. If a pendulum, which beats seconds at the base of a mountain, lose 10 oscillations in the day when taken to the summit, what is the height of the mountain ?—the Earth being supposed a sphere whose radius is 4000 miles.
1. A projectile is discharged from a point on a horizontal plane at a distance a from the foot of a wall whose height is b; calculate the velocity of projection, and the tangent of the angle of elevation, so that it should touch the top of the wall and reach the ground at a distance c beyond it.
2. Two bodies, P and Q, move on two rough inclined planes which are placed back to back, and whose inclinations are a and ß, and are connected by a string which passes over a pulley at the top; show that the accelerating force of P is given by the formula
Psin (a)-Q sin (ẞ + ø)
where tan = the coefficient of friction; and hence calculate the velocity with which P will reach the bottom of the plane.
3. A uniform beam rests with one end on a given inclined plane, the other end being suspended by a string from a fixed point above the plane; determine the position of equilibrium, the tension of the string, and the pressure on the plane.
4. A uniform bar attracts a point by a force varying as the inverse square of the distance; calculate the magnitude and direction of the resultant attraction, and show that it is the same as that of a circular arc whose centre is the given point, and radius the perpendicular from the point on the line, bounded by lines drawn from the given point to the ends of the given line.
5. A polygonal frame is held in equilibrium by a system of forces acting at its vertices, and intersecting at a common point; given the point and vertices, exhibit by a figure the ratios of the forces, and of the stresses they produce on its sides.
6. A weight w, resting on a rough inclined plane, is acted on by a horizontal force f tending to push it up the plane; show that equilibrium will exist if ƒ lie between the limits w.tan (i− p) and w.tan (i+ø), where i is the inclination of the plane, and the angle of friction.
7. A mass m, falling down the inner surface of a smooth hemispherical bowl, strikes against another mass m' in equilibrium at the lowest point of the bowl; if both bodies be perfectly elastic, required the heights to which they will respectively rise after collision.
8. Two bodies setting out from rest attract each other with a force of mutual action varying directly as their distance asunder; given their masses, absolute force of attraction, and initial interval of separation, required the entire time they will take to come together.
9. Find the position of equilibrium of a beam in a smooth hemispherical bowl-(a) when it lies completely inside the bowl; (b) when part of the beam projects beyond the rim.
10. A semicircular plate moves freely about a pivot at one end of its diameter; a cord attached to the other end passes vertically over a fixed pulley, and sustains a weight. Find the inclination of the diameter to the cord, the position being one of equilibrium.
11. An elastic ball strikes another which is at rest obliquely; prove that, if the ratio of their weights be equal to the modulus of elasticity, the balls will move after impact in directions at right angles to each other, whatever be the direction of the impact.
12. A particle runs down the exterior surface of a vertical circle; find the point at which it will leave the curve, and the magnitude of the parabola described.
Experimental and Natural Science.
1. A portable barometer, whose neutral points of pressure and temperature are 29.65 and 56°, and whose correction for capacity is, is observed to stand at 30.34, the temperature of the quicksilver being 50° Fahrenheit; what is the exact pressure indicated by this observation reduced to 32°?
JUNIOR SOPHISTERS—EXPERIMENTAL AND NATURAL SCIENCE. 21
2. A square sheet of iron, whose side is 20 inches, and coefficient of linear expansion for 180° is, is heated to 300°; what augmentation of surface does it experience ?`
3. What is the theoretical ascensional force of a balloon inflated by N cubic inches of coal gas, having a density equal to half that of atmospheric air, the atmosphere at the time being at the temperature t, and pressure p?
4. How many pounds of water at 200° should be added to 3 pounds of ice at 18°, so that the liquefaction of the latter should be just accomplished?
5. If a volume of air at 32° be heated until its bulk is doubled, and be then compressed into its original volume, what will be its temperature?
1. If a power be in equilibrium with a weight, by means of a number of toothed wheels, prove that the Power is to the Weight as the continued products of the diameters of the pinions is to the continued product of the diameters of the wheels.
2. A rectangular mass of cast iron rests upon an inclined plane of oak, and is upon the point of slipping down it, and also upon the point of overturning. Its base is two feet square, what is its height?
3. Prove that the work done in walking up an inclined plane is equal to the work done in walking along the base of the plane, plus the work done in lifting the weight of the body through the height of the plane.
4. The prisoners confined in the military prison of Dublin perform their punishment shot-drill under the following conditions :-Each man lifts a 32 lb. shot to his breast (3 ft.) from a tressel, carries it through 9 ft. (4 paces by drill), and lays it down on a similar support; he then returns unloaded, and takes up another shot, and so repeats the double journey; of course, after a certain time, all the 32 lb. shots are transferred from one side to the other of the working gang, and they must then reverse the order of proceedings, and carry back the shots; six of the double journeys occupy one minute. If we assume that the same work is done in laying down as in lifting up the shot, prove the following expression for the work done per minute:
w = weight of man in lbs.
a = distance in feet to which the 32 lb. shot is carried.
h = height in feet to which it is lifted.
n = number of double journeys per minute.
Assuming the average weight of the prisoners at 141 lbs., and that they are employed for 3 hours, find the work done.
5. How much sweat would be evaporated by the preceding work, supposing it to vaporize as freely as water?
1. What is meant by absolute temperature and absolute zero?
2. If an air thermometer contain eleven grains of air, the absolute temperature will be equal to the number of cubic inches multiplied by the pressure expressed in pounds per square inch?
3. What was Despretz's modification of Dr. Hope's experiment to prove that water has a point of maximum density?
4. Prove the following formula for expressing the density of any vapour as referred to its own liquid :
in which s is the specific gravity of the vapour, d the specific gravity of the liquid, p the pressure in inches of mercury, and T the absolute temperature.
5. What is the French Commissioners' formula for expressing pressure in terms of temperature, as adapted to the Fahrenheit scale, and pounds on the square inch? Deduce from it a Table of Temperatures for 10 lbs., 20 lbs., 30 lbs., 40 lbs., and 50 lbs., on the square inch.
History and English Literature.
1. Give an account of (a) the Anglo-Saxon Chronicle; (b) the Chronicle of Ingulphus; (c) the writings of William Fitzstephen.
2. Write a short essay on the feudal system; explaining in particular the origin of feuds, the successive steps through which the tenure of fiefs passed, the duties of the vassal, the feudal incidents, the practice of sub-infeudation, and the leading causes of the decline of the system.
3. Give some account of (a) Anselm; (b) Simon de Burley; (c) Robert of Jumieges.
4. M. Thierry remarks that the progress, settlement, and direct results of the Norman Conquest form several distinctly marked epochs ?
5. Give an account of the Statute of Winton. In what king's reign was it passed? What important changes in the administration of justice were gradually introduced after its enactment?
6. What committee is known in English history by the name of the "Lords Ordainers?" Give a short account of their proceedings.
7. Write short notices of the Continental transactions which, in the years 1237 and 1257, led to disputes between the King of England and his Parliament.
8. Give a sketch of the history of England, from the death of John Duke of Bedford, till the expulsion of the English from France.