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Q. 61. The figure shows a skew pantograph. Four pieces of sheet material are hinged together at A, B, C, D, these points forming the corners of a jointed parallelogram. If one of the pieces is pinned or hinged to the drawing board at P, and if any point Q in an adjacent piece is moved along a curve, there is some point R in the opposite piece which traces a similar curve, but turned through some angle.

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Determine P and R so that the curve traced by R shall be a half size (linear) copy of that traced by Q, and be turned through a clockwise angle of 75°.

Only three attempts. Two of these only plotted the five points. The third made a further construction, but did not solve the problem.

Q. 62. The figure represents one half of a symmetrical arch. You are required to divide the curve LLL into eight parts, so that the length of each part shall be approximately proportional to the mean square of the depth of the part. That is is to be the

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same for all the segments. Give the value of this common ratio. Attempted once only and with no success. The answers to Questions 61 and 62 were disappointing. Honours candidates should display more resource in attacking unfamiliar problems.

Q. 63. The slide valve of a steam engine is actuated by a Joy gear. The figure gives eight positions 0, 1, 2 7 of the valve,

corresponding to the eight crank positions of

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Measuring from the point A, the displacement of the valve for any crank position is given approximately by the Fourier equation

x = c + α1 cos y + a2 cos 20 + a3 cos 30+ a, cos 40

+ b1 sin + b, sin 20 + b, sin 30.

Determine the eight constants in this equation.

If the speed of the crank shaft is 10 radians per second, what is the velocity of the valve when () = 0?

Harmonic analysis is now becoming well known, and the answers were good. The best results were obtained by the vector or radial method. The accuracy of this graphical process is shown by the enclosed table in which the answers used by the Examiner are for comparison placed above those obtained by candidate No. 3053.

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Q. 64. 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

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O being a fixed point of reference, and angles being measured anti-clockwise from the eastward direction. Plot the points O, A, B to a scale of " to 0'1'. What is the position of R relatively to A?

At the instant under consideration the velocity of A is 515 feet per second, and its acceleration 4560 feet per second per second, the angular speed of the body being 10 radians per second, and the angular acceleration 60 radians per second per second.

Draw vector triangles of velocity and acceleration, or velocity and acceleration images, from which the velocity and acceleration of any point of the body can be found.

Measure the velocity and the acceleration of B relatively to A. Show the point Z in the body the instantaneous acceleration of which is zero.

There were some

Attempted by about 35 per cent. of the candidates. really good answers, but the majority were only partially successful. This is a fundamental problem in plane motion, and is capable of very useful extension to a series of connected links constituting a machine, and to the force and couple actions due to inertia.

Q. 65. A card is pinned to a board at three points A, B, C; linear scale 1.

A number of forces (not shown) are then applied to the card in the plane of the board. The pins B and C being now removed, it is found that the moment of the system of forces about A measures 16.8 lb. inches. In a similar manner it is found that the moments of the system about B and C measure 4'6 and -79 lb. inches respectively.

Find the force which will balance the system, and show it acting on the card, so that the latter will not move if all three pins are removed.

It

Of the twenty attempts, ten were worthless, and seven were good. was generally overlooked that the line of the resultant must pass through the outer or inner centres of similitude of circles with centres at A, B, C, and radii proportional to the given moments, and could thus be very readily found.

Q. 66. The figure indicates how the position of a point P in space may be defined by a vector OP. Here O is an origin, OZ a vertical axis, ZX a reference plane, and ZN a plane containing OP. Then the vector

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If m, m2, m, are the masses of three bodies in space, and 41, 42, and A, 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

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Draw the plans of G1, G2, G3 to a scale of 1" to 1' when

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Determine G and measure r,

Nineteen attempts, all fairly good except one;

but even amongst Honours men the mistake is sometimes made of taking the magnitude of 1- (a + bß)

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Q. 67. You are given a bowstring truss under the action of wind forces, and supported at the ends A and B. Determine the supporting forces at the ends, that at A acting in a vertical line. Also draw the force diagram or reciprocal figure for the truss, and measure the forces in the bars a, b, c and d, distinguishing pulls from thrusts.

The most popular optional question. The answers were very satisfactory.

Q. 68. 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 draw the elevation of the wire and plate on xy.

(b) Obtain the projections of a line from B which meets the wire AD at an angle of 70°.

(c) Draw the plan and elevation of the projection of the wire
AD on the plane through BC and D. What angle does
the wire make with this plane?

The answers to all the parts of this question were as a rule very good.
Q. 69. The figure represents a hanging lamp shade and a tilted mirror.
Draw the two elevations and the plan showing the image of
the conical shade in the mirror.

Also draw a perspective view of the shade, the mirror, and so much of the image as is visible in the mirror, the point of sight being on the same horizontal level as the vertex of the conical shade, and at a distance from the picture plane equal to three times MM.

Thirteen candidates selected this question. There were only three bad attempts, but there were few full and complete solutions. As in the other stages, the projection on the mirror was sometimes given instead of the image

Q. 70. The figure represents a pipe made from sheet metal. Draw a development of the pipe showing the shape of the sheet from. which it is bent.

Draw the plan of the section of the pipe made by the horizontal plane SS. Find the area of this section, and the area of either end of the pipe, the linear scale of the figure being.

There were five good attempts and nine worthless answers. See the remarks under Q. 50.

Q. 71. The line LL rotates about the axis AA. Draw the plan and elevation of figure generated.

Determine the shadow cast on the horizontal plane by parallel rays of light, one of which R is given.

Attempted eighteen times and with very creditable results. There was a pleasing variety in the methods adopted."

DAY EXAMINATION.

There were more candidates in Stage 1 than last year, but fewer in Stage 2, the numbers for 1905 and 1906 being respectively 120 and 128 in Stage 1, and 144 and 89 in Stage 2.

STAGE 1.

Results 1st Class, 68; 2nd Class, 28; Failed, 32; Total, 128.

In Stage 1 the work was very satisfactory, and the results, as regards the proportion of highly marked papers, compare favourably with those of the evening, or of any previous examination. The general observations in paragraphs (a), (b), (c), and (d) of the report on the Evening Examination

apply to this examination, in further illustration of which see the remarks under Questions 3, 8, 10, 27, 28 and 29. Attention may be called to the value of tracing paper in connection with such problems as the determination of the lengths of curves, the plotting of loci, the drawing of rolling curves, the copying of figures in new positions, etc. Questions 4, 5, 14, 23, 26 and 32 afforded opportunities for the application of this expeditious, accurate and pleasing method, of which the candidates did not take sufficient advantage.

Draw

Q. 1. In this example employ a decimal scale of inch to 1 unit. a right-angled triangle ABC, base BC= 766 units, hypothenuse BA 10 units. Measure the altitude CA and the base angle B, and calculate the sine, cosine and tangent of B.

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Give the correct values of the angle, sine, and tangent taken directly from the examination tables supplied.

Attempted by all the candidates except five. As a rule the figure was accurately drawn and measured, and the calculated results fell within allowable limits. Many candidates failed to observe that as the lengths of the base and hypothenuse were given the cosine of the angle B was known, and hence the correct value of B could be obtained directly from the tables. Hence also the correct values of sin B and tan B appeared in the tables. Accordingly, in answer to the last part of the question, many gave, as the correct value of the angle B, the value which they themselves had obtained by measurement with the protractor, using the tables to find (by interpolation when necessary) the correct values of the sine and tangent of this same angle.

Q. 2. The lines AA, BB, if produced, would meet in a point out of reach. What is the distance of this point from the point K? Measure the angle between the lines in degrees and in radians. Attempted by about 40 per cent. of the candidates, nearly every one of whom knew how to determine the required angle. A fair proportion also knew how, by means of proportion, to calculate the required distance, but only a comparatively small number were able to draw and measure with that degree of accuracy necessary to determine the numerical result correct within allowable limits.

Q. 3. One link of a chain is shown. Draw the figure to the given dimensions, which are in millimetres. Show the construction for determining the centre C, and measure the radius r. joins of the tangential arcs.

Mark the

N.B.-A mere copy of the diagram will receive no credit. Although attempted by 41 per cent., probably less than 4 per cent. were able to determine by construction the required centre, and only quite a few knew that the joining points of the arcs were on the lines joining the centres of those arcs. On the whole the answers were distinctly poor.

Q. 4. There is certain point O about which the triangle ABC could be turned, so as bring A to A', and AB along A'L, as, for example, by the employment of tracing paper, with a pricker inserted at 0. Determine the point O, and draw the triangle in its new position.

Attempted by about one-third of the candidates; the answers were fairly satisfactory. The majority first drew the triangle in its new position and then determined by construction the position of the centre 0; while others, by pricking through, first copied the given triangle on tracing paper and then determined, by trial, the centre O, about which the tracing paper should be turned in order to bring the given triangle into its new position.

Q. 5. A piece of sheet material has a motion determined by the condition that two pins a and b fixed in it slide in two grooves AA and BB cut in a fixed plate K. Determine the path traced on K by the point P carried by H.

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More than 50 per cent. attempted this question. It was observed that although it was intended to test the candidates' acquaintance with one of the many uses to which tracing paper may be put, probably one-half determined the locus, or, more generally, only a part of it, by means of intersecting arcs, described with the compass. The solutions obtained by this more clumsy method were not only less complete, but more inaccurate than those obtained by the more ready method of using tracing paper.

Q. 6. When a person looks into the angle formed by two perpendicular plane mirrors, he sees himself by a double reflection, and without the customary lateral inversion. Verify this by the following construction:

4 is the object. The image b in the mirror OM of a point a in it is found by drawing a m b perpendicular to OM, and setting off mb equal to ma. Find the images of the other points a, a, and complete the image B. In like manner find the image C of the object A in the mirror ON. Now find the image of B in ON, and the image of C in OM.

About 25 per cent. attempted this. As a rule only three images were drawn, viz., B and C, and either the image of B in ON or the image of C in OM, but few demonstrated the coincidence of the two last images.

Q. 7. A road is formed along the side of a hill, as shown in cross section at RR, by cutting away material 4, and depositing it as an embankment B. The scale of the figure being 1 cm. to 1 yard, determine :

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(a) The number of cubic yards of material removed at 4 per yard length of road.

(b) The point Freached by the foot of the embankment, assuming the material removed at A to expand th

in bulk.

N.B.-The point

and is not to be copied.

on the diagram is purposely misplaced,

Attempted by 30 per cent. As a rule the candidates adopted a correct method of procedure, and there was a good proportion of fairly correct

answers.

Q. 8. A ship sails for two hours with a velocity of 150 miles per hour (directions being measured anti-clockwise from the East). The vessel then slows down and changes her course, and goes for 14 hours at 120 miles per hour. Finally, she sails for hour at 8125 miles an hour.

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Plot the course of the ship to a scale of 1 inch to 10 miles, and find the magnitude and direction of the straight line from start to finish.

Over 90 per cent. attempted this, but although a quite straightforward problem, there was a large number of incorrect answers. The mistakes were due not so much to ignorance of the rule for vector addition, as to the candidates' ignorance of the directions of the cardinal points of the compass. Q. 9. Three light ropes are knotted together at a point 0. Three boys A, B, C pull at the ropes, and in one position of equilibrium the angles between the ropes are measured, and found to be

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105°, BOC

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If the pull exerted by the boy A is 95 lbs., what are the pulls of B and C ?

Use a scale of inch to 10 lbs.

Attempted by 70 per cent., the solutions being correct in nearly every case.

Q. 10. You are given the projections of a piece of wire AB fixed in the horizontal plane at B.

(a) Find the true length of the wire, and the angle the wire

makes with the horizontal plane.

(b) From the plan project a new elevation of the wire on x'y',

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