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(1) Teachers might, with advantage to themselves and to their students, occasionally

ask for written answers to questions on certain parts of the subject which are of fundamental importance.

In this way, misconceptions and difficulties could be detected and corrected, and,

at the same time, useful practice would be provided for the students. Q. 1. In working this question employ a decimal scale of 1 inch to

1 unit.

Draw a circular arc, radius 10 units, centre 0. Mark a chord AB of this arc, 3:47 units long, and draw the radii OA, OB. Measure the angle AOB in degrees.

From B draw a perpendicular BM on OA, and at A draw a tangent to meet OB produced in N. Measure carefully BM and AN (on the above unit scale), and calculate the sine and the tangent of the angle AOB.

Give the correct values of the angle, the sine and the tangent,

taken directly from the examination tables supplied. The first two parts were well answ

swered, indicating a widespread knowledge of trigonometrical ratios, but a rather common error was to make the chord 3:47 inches long instead of 3 47 half-inch units. With regard to the last part, comparatively few discovered from the tables that the angle whose chord is 0-347 is exactly 20°, and therefore that the correct values of the other functions were known. Q. 2. The figure is the plan of a building site to a scale of 1 inch to 10

yards. Determine the area of the site in square yards. Of the optional questions this was the favourite, and the answers were fairly satisfactory. The most usual though not the best method was division into triangles and calculation of their separate areas by measurements of bases and heights. Errors in scaling and measurements in vulgar fractions were few, and the arithmetic was not so bad as in some previous voars. Some, in attempting to reduce the figure to an equivalent triangle, went wrong over the re-entrant angle. Q. 3. Draw the electric lamp to the dimensions given, which are in

millimetres. Mark carefully the points of junction of the several arcs.

N.B.- A mere copy of the diagram will receive no credit. Attempted by about 60 per cent. of the candidates. Very few obtained correctly the points of junctions of the circular arcs, or determined properly the centres of the upper and lower arcs. Students seem to have à very scant knowledge of elementary tangential properties of circles. Teachers should give attention to this matter. Q. 4. In arranging some elementary experiments in statics, cyclist

trouser clips are used as spring balances. In order to be able to measure pulls, a series of weights are hung on a clip, and the corresponding openings AA are measured. The results being plotted, give the curve shown. For instance, when the pull is OM ounces, the opening is ON inches. On SS construct a decimal scale of ounces, which, being applied to the spring at

A A, shall measure any pull up to 32 ounces. Rarely attempted, not more than 10 per cent. of the candidates taking the question. There were a few full and complete answers, but the great majority got no marks. A scale of ounces in equal parts was generally shown, and even this was seldom sub-divided decimally. Q. 5. The figure is the plan of a corner of a landing. S is a stone, on which

a stove is to rest. Show the corner of the landing carpet folded over along the line LL. On this fold draw the hole which must be cut in the carpet, so that when the latter is turned back

into position the hole shall just fit the stone S. This question was, perhaps, too easy. It was frequently attempted and very well answered.

Q. 6. Two pieces of sheet material are hinged together at A One of

them is pinned or hinged to the drawing board at B, and the
point P on the other piece is moved in a straight line from
to M. Find the paths described by the points and R of the

second piece. Neither a favourite question nor well answered. The results were very inaccurate, and few discovered the geometrical character of the loci. It might, perhaps, have been better to have stated explicitly in the question that AB, AP, AQ, and AR were equal, and that P, A, Q, were in a straight Jine. Q. 7. A and B are two points in a body which is moving in a plane. At

a certain instant the positions of A and B are given by the vectors.

OA=22 OB=5'5'

75

33°

O being a fixed point of reference, and angles being measured anti-clockwise from a line drawn along the ter square to the right (eastwards).

Plot the points 0, A, and B to a scale of 1" to 1'. Measure the magnitude and direction of the vector AB, that is, the position

of B relatively to A. This was usually well done, though some gave the vector sum 0 A +OB instead of the vector difference OB-04. Also occasionally West was substituted for East, and clockwise rotation for counter-clockwise. Q. 8. The vertices A, B, C of a triangle, referred to a point 0 as in Q. 7,

have the positions defined by the vectors

OA=3" OB=4" OC=2"

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1166

(3"

, say.

16°

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116°

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Determine the centre of area, G, of the triangle, having given that

1 OG=

+ 4" + 2" )

3 Measure a and a. Seldom attempted and rarely well done. Very few obtained G in any way. Some made the vector addition 0 A +OB+ OC but did not plot Ğ from the result. The vector equation given was in fact seldom understood. Q. 9. A piece of card is pinned temporarily to a drawing board, and

forces P and Q are applied to the card as shown. It is desired to balance these two forces by a force of 30 ounces acting as A, so that the card shall remain at rest on removal of the pins. What must be the line of this third force, and what the original

magnitudes of Pand Q? Not very frequently attempted. Candidates often failed to see that the line of the required force must pass through the intersection of P and Q. But having assumed an arbitrary line through A, they were generally able to draw a corresponding triangle of forces, and to measure results consistent with their initial error. Q. 10. A thin metal plate ABC, resting on the ground, is shown in plan

A piece of thin wire AD, 2.5" long, with one end soldered to the plate at A, is also shown in plan at Ad.

(a) What is the distance of the end D of the wire from the

plate ?

(b) What angle does AD make with the plate ?

(c) Draw the elevation of the plate and wire on xy. There were many good answers, but far too many worthless ones, and, on the whole, the results were disappointing. A frequent mistake was to draw the elevation of AD on ay 2" long, and to give in answer to (b) the angle between this elevation and xy. Sometimes (a) was correctly answered, and the angle just mentioned given in answer to (b). The given xy was often incorrectly transferred ; and the elevations a', 1', c' were often shown above the xy line. Q. 11. The roof of a house is rectangular in plan, and two adjacent

surfaces are each inclined at 32° to the horizontal. Wbat is the inclination to the horizontal of the “hip,” or line where the

two surfaces meet? Not very frequently attempted. The question was not well understood. Q. 12. The hopper shown in the diagram is required to be lined on its

inner surface with sheet metal bent out of one piece. Find the

shape of this piece, the joint being along AA. Many good answers. With the poorer candidates, a common mistake was to give the shape of the elevation as the shape of the development. There was some want of accuracy in extending the development, after having obtained the true shape of one face. Q. 13. Two elevations of a kitchen salt-box are given. Copy the

elevation A, and from it project a plan. From the plan project an elevation on x'y'.

N.B.—The view B need not be drawn. A favourite question and satisfactorily answered. The principal defect was inaccuracy in transferring dimensions. Q. 14. The figure shows a portion of a timber joint. Represent this in

pictorial projection, in a manner similar to that used for the cube, where lines parallel to oy and oz are drawn horizontally and vertically, and to a scale of full size, and lines parallel to ox are drawn by using the 45° set-square, and to a scale of half size.

N.B.- The figure need not be copied, dimensions being taken

directly from the diagram. Fairly often attempted, and with much success. This kind of projection is attractive to students, and they are able to apply it well.

STAGE 2

Results : 1st Class, 253 ; 2nd Class, 436; Failed, 218; Total, 907. In this stage there was an increase of 8 per cent., the numbers of candidates for 1905 and 190€ being respectively 843 and 907. The work was very satisfactory and the proportion of failures unusually small. The previous remarks under (b), (c) and (d) apply in this stage, though not quite to the same extent. Further, more care should be given to the accurate determination of the lengths of curves, as detailed in the special remarks on Questions 21 and 32. And in some schools the use of blunt soft pencils is still permitted. Q. 21. In working this question employ a decimal scale of } inch to

1 unit.

Draw a circular arc, radius 10 units, centre O. Mark a chord AB of this arc, 3:47 units long, and draw the radii OA, OB. Measure the angle AOB in degrees.

From B draw a perpendicular BM on OA, and at A draw a tangent to meet OB produced in N. Measure carefully BM, AN, and the arc AB (on the above unit scale) and calculate thé sine and tangent of the angle, and the angle in radians.

Give the correct answers for the degrees, sine, tangent, and radians, the numbers being taken directly from the examination tables supplied.

Very well answered in regard to the first two parts. The remarks under Q: 1 apply to the last part of the question. The tracing paper method of finding the length of the arc was often successfully used, and with due care gave the most accurate result. Q. 22. The figure is the plan of a field in which there is a pond P;

scale inch to 10 yards. Find the total area of the field

(including the pond), and the area of the pond in square yards. A favourite question and very well answered. The measurements were usually made in decimals. A good proportion showed a fair knowledge of some method of obtaining the area of the curved figure. Q. 23. In arranging some elementary experiments in statics, cyclists'

trouser clips are used as spring balances. In order to be able to measure pulls, a series of weights are hung on a clip, and the corresponding openings AA are measured. The results are as tabulated.

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Plot a curve showing the relation between P and x, the scale for P being !" to 1 ounce. Use this curve to graduate a decimal scale of ounces, which being applied to the spring at AA shall

measure any pull up to 32 ounces. A fair number plotted the curve and did so well, but very few understood how to obtain the required scale. Q. 24. The figure is the plan of a corner of a landing, scale 1" to 1'. S is

a stone on which a stove is to rest. Show the corner of the landing carpet folded over along the line LL. On this fold draw the hole which must be cut in the carpet, allowing a margin of 1}" all round for turning under, so that when the carpet is turned

back into position, the hole shall just fit the stone S. The candidates were well able to answer this question, but sometimes the drawing was a little inaccurate. Q. 25. Four pieces of sheet material are hinged together at A, B, C, D,

these points forming the corners of a jointed parallelogram. One of the pieces is pinned or hinged to the drawing board at P, and the hinge point B is moved in a straight line from L to M. Find the locus of Q, that is the path that would be traced by a

pencil moving with Q. Frequently attempted and fairly well done. Owing to slight inaccuracies in working, a large number failed to obtain a straight line locus, and very few appeared to recognize the special properties of the mechanism. Q. 26. A and B are two points in a body having plane motion. At a certain instant the positions of A and B are given by the vectors

0A = 0-44769 OB = 1'1'33', O being a fixed point of reference, and angles being measured anti-clockwise from a line drawn along the tee-square to the right (eastwards). Plot the points 0, A, B, to a scale of " to 01'. What is the position of 'B relatively to A ?

At the instant under consideration the velocity of A is 5160 feet per second, and the angular speed of the body 10 radians per second. To a scale of 4 inch to 1 foot per second, draw a vector triangle, or velocity image, showing the velocity of A, the velocity of B relatively to A, and the velocity of B. Read off the

velocity of B. Seldom attempted, and good answers were very rare indeed,

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Q. 27. If mi, M.2, m.z are the masses of three bodies in a plane, and

A1, A2, A3, three vectors defining the positions of their centres
of mass G1, G2, G3, then the position of the centre of mass G of
the three bodies is given by the vector equation-
OG = A = da

m 4. + m 4, + myd,

mi + m2 + mz
Plot G1, G2, G3 to a linear scale cf one-half, when
8".

7"47", A3
mi
105,

1'0, Determine G and measure a and a. Rarely attempted. The plotting of the triangle was fairly satisfactory, but the vector equation was not well understood. A few obtained the answers by the vector method suggested, but the majority employed other and less direct methods. Q. 28. A cardboard lever is pinned or hinged to a board at the point o

Forces P and Q are applied to the lever as shown.

The linear scale of the drawing being ), and the magnitude of P being 1:85 lbs., what is the moment of Pabout O Find the magnitude of Q if the lever is balanced.

Show a force which, being applied to the lever near 0, shall relieve the pin from all pressure, and allow of its being removed

without disturbing the equilibrium. A fair number of candidates gave good answers. Some failed to express the moment of P in proper units. In other cases an incorrect balancing force was shown which did not pass through 0, or did not pass through the inter-section of P and Q. Q. 29. A thin metal plate ABC resting on the ground is shown in plan.

A piece of thin wire AD, 2.5" long, with one end soldered to the plate at A is also shown in plan at Ad.

(a) Find the distance of the end D of the wire from the plate, and the angle between wire and plate.

(b) Draw the elevation of the wire and plate on xy.

(c) From the elevation project a new plan on Xyy1. Generally well done. A considerable number, however, were unable to project the new plan properly. Q. 30. The roof of a house is rectangular in plan, and two adjacent sur

faces are inclined to the horizontal at 30° and 40° respectively. Find the inclination to the horizontal of the “hip," or line in which the two surfaces meet.

What is the magnitude of the dihedral angle between the

surfaces ? Answered by about 35 per cent. of the candidates, with, for the most part, satisfactory results. Q. 31. The figure represents a hanging lamp shade and a tilted mirror.

Draw the two elevations of the image of the conical shade in the mirror.

N.B.- If P is any point in space, and Pits image, then the

plane of the mirror bisects PP' at right angles. Not very frequently attempted. A common mistake was to give the projection of the shade on the mirror, instead of the image as specified in the question. Q. 32. The figure shows a funnel made of sheet metal. Draw the de

velopments of the cylindrical and conical portions, showing the shapes of the plates from which the funnel is bent. Omit all allowances for overlap at the joints.

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