10. A uniform bar, revolving round a free axis intersecting it perpendicularly at its centre of gravity, snaps suddenly into two at any point; determine the subsequent motions of the fragments.

PRACTICAL MECHANICS.

1. A cylindrical pillar of dressed granite, sp. gr. 2.6, 20 ft. long, and 3 ft. in diameter, rests obliquely against a vertical wall, from whose lower edge its point of contact with the ground is distant 12 ft.; calculate in tons its pressure against the wall.

2. The section of a river wall of limestone, sp. gr. = 2.5, is a rectangular trapezium, whose upper and lower sides are 3 and 10 ft., respectively, and whose sloping face is turned towards the water; calculate in feet the limiting height to which it could be raised.

3. The several pairs of rafters of a common isosceles roof, inclined each 40° to the horizon, and sustaining each 2.5 cwts. of the entire weight of the roof, are braced by tie and collar beams trisecting their lengths; calculate in cwts. the strains common to each pair of braces.

4. The several supporting chains of an ordinary suspension bridge, 150 ft. in span, and 25 ft. in dip, support each an entire weight of 75 tons suspended from them vertically, and uniformly distributed horizontally; calculate in tons the central and terminal tensions common to each chain.

5. The upper flange of a Warren girder, divided into four bays by as many pairs of diagonals connecting it with the lower, and inclined at the usual angle of 60° to both, sustains a central load of 10 tons; calculate in tons the strains in the several bays of each flange.

6. For the same girder, perform the same calculations, on the supposition that the same load, in place of being concentrated as above at the middle point of the upper flange, were distributed instead uniformly over its entire length.

SCHOOL OF ENGINEERING.

EXAMINATION FOR ADMISSION TO THE MIDDLE CLASS.

THEORY OF THE STEAM ENGINE.

MR. GALBRAITH.

1. From the following data for a high-pressure non-condensing engine, find the horse-power :-Cylinder, 24 inches; stroke, 4 feet; revolutions, 25; evaporation, 1 cubic foot; steam not cut off during stroke.

2. If the pressure of the steam in the boiler be 50 lbs., to what extent may the horse-power be increased by altering the velocity?

3. If this engine be used for winding from a depth of 100 fathoms, find how many tons can be raised to the surface in these two cases in one hour, deducting 2 tons for winding gear.

4. If steam be cut off at 37 per cent of the stroke, and if the evaporation be reduced by 25 per cent.; find the horse-power corresponding to 25 revolutions.

5. Find also the least possible pressure of the steam in the boiler in order to produce this effect.

6. Find the diameter of the cylinder of a high-pressure non-condensing engine, which, with an evaporation of 1 cubic foot, and a piston-velocity of is to work without expansion at 60 horse-power.

200,

ELEMENTARY MECHANICS.

MR. TOWNSEND.

1. Given, on three concurrent lines, the directions of three forces in equilibrium, and the magnitude of one of them; determine the magnitudes of the other two.

2. Given, on three parallel lines, the directions of three forces in equilibrium, and the magnitude of one of them; determine the magnitudes of the other two.

3. Show that in both cases the sum of the moments of the three forces round every point in their plane = o.

4. State and prove the fundamental property which defines the centre of gravity of any body or system of bodies.

=

5. A weight w rests on a smooth inclined plane whose inclination i; determine in magnitude and direction the least force that will prevent its falling down.

6. Determine the ratio of the power to the resistance in the two species of Burton pulley, both in the direct and inverted positions of the systems.

7. In the motion of a body falling freely in vacuo under the action of gravity, prove the relations v=gt, s=gt2, v2 = 2gs.

8. In the motion of a body projected vertically upwards in vacuo, and acted on by gravity, determine the velocity of projection v requisite to attain a given height h, and the time t occupied in attaining it.

9. Determine in lbs. the moving force F which, acting freely for one second on the mass of a ton weight, would get up in it the velocity of one foot per second.

10. Determine in lbs. the weight W of the mass which, revolving uniformly in one second in a circle of one foot radius, would produce the centrifugal pressure of one ton.

11. State and prove the Principle of Archimedes respecting the action of water, or any other gravitating fluid, on a solid wholly or in part immersed in it.

12. Investigate the equation for the height to which water will rise in a pump at the first stroke of the piston.

3. Find four terms of the expansion of sin -1x by Maclaurin's theorem. 4. Find the integral of

7. If A, B, O are the angles of a plane triangle, prove that
I - cos 2A - cos 2B - cos 20-2 cos A cos B cos C = o.

8. Retaining the notation of the last question, prove that
sin 2A + sin 2B + sin 2 C = 4 sin A sin B sin C.

9. A base line AB of 1000 yards is measured; the angles BAC, BAD are, respectively, 31°46' 10" and 19° 0'56"; and the angles ABD, ABC are, respectively, 75° 49' 18" and 60° 42′ 48′′; find the distance CD.

10. Two sides of a triangle are 87.7 and 96.6, and the contained angle is 49° 34′ 55′′; calculate the remaining side and angles.

II. Find the projections of the line of intersection of two given planes, which are both parallel to the ground line.

12. Being given the traces of a plane, find the angles which it makes with the planes of projection.

DR. DOWNING.

1. Calculate the land required for the railway embankment with the following dimensions:-From the level of the ground at A where the depth is 0, to the point B the distance is 4 chains; from B to C 5 chains, from C to D 5 chains, and from D to E 7 chains. The depth at A = 0; at B = 12 ft.; at C = 31 ft.; at D = 31 ft.; at E on the level of the ground or zero. The breadth on the top of the embankment is 30 ft.; the slopes are 2 to 1, and the chain employed is 100 ft. in length. The area of each part and the total is to be brought out in acres, roods, and perches.

2. If in the part of the embankment from C to D the ground sloped transversely at the rate of 1 in 4, calculate the land required for this portion of the embankment, and prove the formula you employ.

3. Explain the principle of the contour lines as engraved on the Irish Ordnance Maps, and draw a portion of the lines passing round the sides of a valley down which a stream flows, having a rapid rate of fall, and explain the lines of equal soundings on the Admiralty charts of the coast.

4. On the contoured sheet given with this question, draw from the point A the centre line of a projected road, which shall have an inclination of 1 in 30, the vertical distance of the contours being 20 feet.

5. It is required to draw a scale of yards on an old map plotted to 7 Irish perches to an inch. State the linear and superficial ratio of the map to the actual dimensions on the ground.

6. Draw a scale of kilometers adapted to the Irish Townland Survey, of 6 inches to 1 mile.

7. Draw a careful sketch, showing how you would find 2.685 inches on the ordinary 12-inch Gunter's diagonal scale.

8. Calculate the contents of the following excavation :Distances, in statute chains, from

At A = 0, at B = 22 ft., at C = 29 ft.. at D= 9 ft., at E = o.

Base 30 ft., and slopes 3 to 1.

9. If between C and D the ground had sloped transversely at the rate of 1 in 4, compute the number of cube yards in that block, the other dimensions being as in the last question, and prove the formula by which you work out the result.

10. Give a full statement of the information required by the standing orders to be given on a Parliamentary section.

11. Give a full statement of the directions as to road bridges in the Railway Clauses Act.

12. Sketch carefully, and describe a vernier on a 5-inch Theodolite so as to read either single minutes or 20 seconds.

13. Explain the construction of the vernier of the mountain barometer, reading to thousandths of an inch.

2. Find three numbers such that their sum may be 33, the sum of their squares 467, and that the difference of the first and second may exceed the difference of the second and third by 6.

and show how the value of π may be calculated from it.

5. Prove the following formulæ :

sin (30°+ 0) = sin (30° - 0) + √3 sin

sin (60° + 0) = sin (60° – 0) + sin 0,

and point out their use in the computation of tables of sines.

6. Find the value of

tan A tan B + tan B tan C + tan C tan A,

when A + B + C = 90°.

7. Find the angles of the triangle whose three sides are 54.9, 70.3, and 85.8.

8. If two sides of a triangle are 29, 30, and the contained angle 63° 22′, find the remaining side and angles.

3. Being given the area of a right-angled triangle, construct it so that the sum of the sides containing the right angle may be a minimum. 4. Find the value of

5. One side AB of a quadrilateral is 1000 yards; the angles CAB, DAB are, respectively, 22° 56′ Ic" and 11° 27′ 50′′; and the angles ABD, ABC are, respectively, 117° 53' 8" and 103° 54' 50". Calculate the

side CD.

6. Find the angles of the triangle whose sides are 10, 11, 13.

7. Construct, by descriptive geometry, a right line passing through a given point, and meeting two given right lines.

8. Through a given point, draw a plane parallel to a given plane.