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*41. In order to find approximately the area of a curved figure, a piece of ruled tracing paper (or celluloid) is laid over the figure as shown, and the sum of the ordinates y1, 2, y3. . . . is measured on a suitable area scale. Construct this scale, the readings on which shall give the area in square inches. What is the area of the figure? Construct an equivalent rectangle on the base BB. (50) *42. A cam mechanism is sketched. The cam C rotates uniformly about O, and actuates a slider S by means of the bent lever LL. Twelve successive positions of the slider, enlarged to full size at AB, numbered 0 to 11, are given, corresponding to twelve positions of the cam, at intervals of 30°, and similarly numbered as indicated. Set out the proper shape of the cam, working to the given dimensions and not copying any part of the diagram except the points on the line AB.

(50) *43. The figure shows a truss for the roof of an island platform of a railway station. The loads due to roof covering and wind pressure are given. Determine the forces in the

members of the frame.

The scale of the figure being inch to 1 foot, what are the bending moments on the supporting pillar at A and at B? (50) *44. The piece P, turning about O, drives the piece P1, turning about O, by the sliding contact of the curved faces SS and SS1, which are circular arcs with centres C and C1. The scale of the figure is 1 inch to 0.5 foot, and P rotates uniformly at 10 radians per second. For the position of the mechanism shown determine :

(a) The angular velocity of P1.

(b) The speed of relative sliding between the faces SS and SS1.

(50)

*45. A disc rotating about O at a speed of 1 radian per second has three masses my, my, mg of 4.3, 6.7 and 8.1 units (1 unit = 32-2 lbs.) fixed to it as shown, at radii r1, 72 r3 feet, and producing centrifugal forces on the axis O along the radii of magnitudes m1r1, m2r2 aud mr pounds. The scale of the figure being 1 inch to 1 foot, measure the radii and calculate the forces. Find and measure the resultant of the three centrifugal forces.

Determine G, the centre of mass, and find the centrifugal force of a single mass M = m1 + m2 + m3 situated at G. Find the position and give the magnitude of a single mass m, placed on the circumference of the disc, which shall balance the three masses mi, m, and m3.

(50)

#46. The triangle ABC is a diagram showing the intensity of earth pressure on a retaining wall, its area representing the total pressure P to the same scale that the area of the cross section of the wall represents the weight W of the wall. Determine the lines of action of P and W, the face AB of the wall being vertical. Find the centre of pressure on the base of the wall, this being the point where the resultant of P and Wintersects the base.

Also find the centres of pressure on the sections D, E and F.

(50) *47. A geometrical model is made by fixing a piece of bent wire ABC in a board as shown.

(a) What is the true angle between the wires at B?
(b) Show the projections of the conical surface that
is generated by BC when the wire AB is
turned about its axis.

(50)

*48. Bell buoy. Determine the shadow cast by the buoy on the horizontal plane, the direction of the parallel rays of light being given.

Indicate, both in plan and elevation, what portion of the surface of the buoy is in shade.

(50)

*49. Represent the buoy of Q. 48 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, to a scale of full size, and lines parallel to or 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 transferred from the diagram.

(50)

*50. Two projections of a prismatoid are given. Show the development, in a single piece, of one of the symmetrical halves of the sloping surface.

Draw the plan of the mid-section SS, determine its area a and the areas a1 and as of the ends of the solid. Measure the height and calculate the volume of the solid by using the prismoidal formula

volume =

height x

a1 + 4a + a2
6

(50)

51. The surface of a piece of ground is given by contours at vertical intervals of 5 feet, the linear scale of the plan being 1 inch to 50 feet. A road is to be cut at the given heights, the face of the cutting ou each side having a slope of 38° to the horizontal. Complete the plan of the finished earth work. (50) *52. You are given the curvatures of the projections of a tortuous curve SS at a point P. Determine the tangent line, the normal plane and the osculating plane at P.

(50)

53. A cone of revolution, vertical angle 64°, of indefinite length, lies with its curved surface on the ground; draw its plan. Determine a sphere 0·6" radius, which rests on the ground and touches the cone at a point 2 5" from its vertex. Show the indexed plan of the point of contact, and determine the common tangent plane at this point.

Honours.

(50)

NOTE. No candidate will be credited with a success in this examination who has not obtained a previous success in Stage 3, or in Honours, of the same subject.

A second sheet of drawing paper will be supplied, if necessary, after the first has been filled on both sides.

You must not attempt more than five questions in all, and of these No. 68 must be one; that is to say, you are allowed to take not more than four questions in addition to No. 68.

• ...

*61. In order to find approximately the area of a curved figure, a piece of ruled sheet celluloid is laid over the figure and adjusted as shown, and the sum of the ordinates Y1, y2 y3 is measured on a suitable area scale. Construct two such scales, the readings on which shall give areas in square inches and in square centimetres. Show that the scale is independent of the number of divisions over which the figure extends. What is the area of the given figure in square inches? Construct an equivalent rectangle on the base BB.

(80)

*62. A cam mechanism is shown. The cam C rotates uniformly about O and actuates the slider S by means of the bent lever LL. The slider has an intermittent motion as follows:

(a) A period of rest while the cam turns through 150°. (b) The upward half of a simple harmonic motion. from A to B while the cam turns through the next 120°.

(c) The downward half of another simple harmonic motion while the cam turns through 90°.

Set out the true shape of the cam profile, working to the given dimensions and not copying the diagram.

(85)

*63. The piece P, turning about O, drives the piece P1, turning about 01, by the sliding contact of the curved faces SS, S1S1, which are circular arcs with centres C, C1. The scale of the figure is 1 inch to 0.5 foot, and the piece P turns uniformly at 10 radians per second. For the position of the mechanism shown, determine :

(a) The angular velocity of P1.

:

(b) The speed of relative sliding between the faces

SS and S1S1.

(c) The angular acceleration of P1.

(85)

*64. The triangle ABC is a diagram showing the intensity of earth pressure on a retaining wall, its area representing the total pressure P to the same scale that the area of the cross section of the wall represents the weight W of the wall. Find the lines of action of P and W, the face AB of the wall being vertical. Determine the centre of pressure on the base of the wall, this being the point where the resultant of P and W intersects the base.

Also find the centres of pressure on the sections D, E, Fand G. (85)

*65. The curve shows the displacement y inches of the block in the link of a Stephenson valve gear, for any crank position degrees, during one revolution of the crank shaft. Express y approximately in terms of by the first three terms of the Fourier series

y = r1 sin (+ a1) + r2 sin (28 + α) + rg sin (36 + αs)

+ a constant.

Give the values of r1, 72, 73 and a1, a2, az.

(85)

*66. The figure shows a three-dimensional braced frame for a crane. Determine and measure the forces in the five bars which radiate from the joint numbered 3, the load being 5 tons.

(85)

*67. In a crank and connecting rod mechanism, see fig. (a), where OP = μ x CP, the inertia force due to the reciprocating mass m is equivalent to the inertia forces of the following revolving masses :—

m

(1) Primary forces. Masses at P and P', where

2

CP' is an imaginary crank equal to CP, turning in the opposite direction, the two cranks coming into coincidence along CA.

m

(2) Secondary forces. Masses approximately at Q

δμ

and Q, where CQ and CQ, each equal to CP, are imaginary cranks turning in opposite directions at twice the speed of CP, and coming into coincidence along CA.

(3) Forces of higher orders, very small, and here neglected.

Fig. (b) shows a symmetrical V engine, with two pistons A and B, equal reciprocating masses m, m, the connecting rods being coupled to one crank CP. For the given position of CP show the positions of the imaginary cranks CP', CQ, CQ' for both pistons. Suppose the mass at P to be balanced by the mass M, indicate by the help of a curve the nature of the resultant of the remaining inertia forces throughout one revolution of the crank. (80)

*68. A geometrical model is made by fixing a piece of bent wire ABCD in a board as shown.

(a) What is the angle between the lines AB and CD?

(b) Show the projections of the circular paths of the points C and D as the wire AB is turned about its axis.

(60)

*69. Two projections are given of a solid, the surface of which is developable. Show the development, in a single piece, of one of the symmetrical halves of the sloping surface.

Draw the plan of the mid-section SS and determine its area a, and the areas a, and a, of the ends of the solid. Measure the height and calculate the volume of the solid by using the prismoidal formula

a1 + 4a + α,

volume =

height x

6

(85)

*70. You are given the plan of the surface of a piece of ground, contoured in feet, the linear scale being 1 inch to 50 feet. The curve pp is the centre line of a road, 20 feet wide, which is to be made partly by cutting and partly by embankment, the former having a slope of 45° and the latter one of 38° to the horizontal. Complete the plan of the finished earthwork within the limits of the data. (85) *71. Determine the plan, and the elevations on xy and x'y', of the curve of interpenetration of the given cone of revolution and the given solid, the latter being formed by the rotation of a circle about a tangent.

(85) *72. You are given the curvatures of the projections of a tortuous curve SS at a point P. Determine the tangent line, the normal plane and the osculating plane at P. Find also the principal normal and the binormal at P.

(80)

*73. Draw the perspective projection of the given bell buoy, taking the vertical plane of projection as the picture plane, the point of sight S having the position shown in plan and elevation. You will observe that the object is in front of the picture plane.

(85)

SUBJECT II. MACHINE CONSTRUCTION

AND DRAWING.

INSTRUCTION S.

Read the General Instructions on page 3.

You are expected to prove your knowledge of machinery, as well as your capability of drawing neatly to scale. You are therefore to supply details omitted in the sketches, to fill in parts left incomplete, and to indicate, by diagonal lines, parts cut by section planes.

No credit will be given if the candidate shows that he is ignorant of projection. The centre lines should be clearly drawn.

Your answers should be clearly and cleanly drawn in pencil, except the tracing which must be done in ink.

In Stage 1 and Stage 2 the answers to the questions as well as the drawings should be made on the numbered papers supplied, comprising one sheet of drawing paper with tracing paper and

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