First Stage or Elementary Examination.

INSTRUCTIONS.

You are not permitted to attempt more than SEVEN QUESTIONS. You may select them from any part of the paper.

1. State how to find the position of the centre of gravity in the following cases :-(a) a square lamina, (b) a triangular lamina, (c) a cone; each body being of uniform density.

If a body is held suspended by a thread fastened to a point of it, in what position does it come to rest? (12.) 2. Forces of 10, 13. and 16 units act at a point and are in equilibrium; show by a diagram how they act, and note the angles between their directions.

(12.) 3. A rod (AB) can turn freely round a hinge at A, and rests in an inclined position with its end B against a smooth vertical wall; show in a diagram how the forces act which keep the rod at rest, and name them.

(16.). 4. Define the moment of a force with respect to an assigned point.

A uniform rod rests in a horizontal position on two supports 8 ft. apart, one under each end; it weighs 6 lbs.; a weight of 24 lbs. is hung to it from a point distant 3 ft. from one of the points of support; find the pressure on each point of support.

(14.)

5. State what is meant by the sensibility of a balance, and explain why the sensibility is great when the arms are long, and also when the centre of gravity of the beam is near the point of support.

6. Define a foot-pound of work.

(14.)

A man weighing 140 lbs. puts a load of 100 lbs. on his back and carries it up a ladder to a height of 50 ft.; how many foot-pounds of work does he do altogether, and what part of his work is done usefully?

(10.)

7. If the velocity of a body is increased uniformly in each second by 32 ft. a second, by how many feet a second is its velocity increased in one minute?

If a velocity is 1920 ft. a second, what is it in yards a minute?

(10.)

8. Write down the formula for the distance described in a given time, by a body whose velocity is uniformly accelerated.

A body thrown upwards against gravity reaches a greatest height of 121 ft.; find the velocity with which it is thrown up, and the number of seconds that will elapse before it returns to the point of projection. (g = 32).

9. A body whose mass is 10 lbs. is moving at the rate of 50 ft. a second; what is the numerical value of its kinetic energy at that instant? If from that instant it moves against a constant resistance equal to one-twentieth of its weight, how far does it go before being brought to rest? (g = 32.) (14.) 10. When a body is wholly or partly immersed in a liquid, what is the magnitude of the resultant pressure of the liquid on the body?

A body, whose specific gravity is 14 and volume 3 cubic ft., is placed in a vessel in which there is water enough to cover it; what pressure does the body produce on the points of the bottom at which it is supported ? (12.) 11. Explain briefly the method of finding the specific gravity of an insoluble body by means of the balance.

If the body weighs 732 grains in vacuo, and 252 grains in water, what is its specific gravity?

(10.)

12. A tube filled with water is inverted with its open end in water, no air having got in; the top of the tube is 20 ft. above the surface of the external water. If the waterbarometer stands at 34 ft., what is the pressure, in pounds per square foot, at a point of the inside of the top of the tube? (1 cubic ft. of water weighs 1000 oz.)

What would be the consequence of making a hole through the top of the tube, and why would the consequence follow?

(14.)

Second Stage or Advanced Examination.

INSTRUCTIONS.

Read the General Instructions on the first page.

You are not permitted to attempt more than eight questions. You may select them from any part of the paper.

21. State and prove the rule for finding the resultant of two parallel forces acting towards the same part on a rigid body.

Parallel forces of 10 and 20 units act towards the same part at A and B; a force of 15 units acts from A to B; find the resultant of the three forces, and show in a diagram how it acts.

(20.)

22. Show that two couples, whose moments are equal and of opposite signs, are in equilibrium when they act in the same plane on a rigid body.

(15.)

23. ABCD is a quadrilateral figure, P and Q are the middle points of the opposite sides AB and CD; O is the middle point of PQ; show that four forces, represented by OA, OB, OC, OD respectively, are in equilibrium. (20.) 24. ABC is a rigid equilateral triangle (whose weight is put out of the question); the vertex B is fastened by a hinge to a wall, while the vertex C rests against the wall, under B; if a given weight is hung from A, find the reactions at B and C.

What are the magnitude and direction of the forces exerted by the weight on the wall at B and C ? (30.)

25. Define the coefficient of friction.

A weight of 500 lbs. is placed on a table, and is just not made to slide by a horizontal pull of 155 lbs.; find the coefficient of friction, and the number of degrees in the angle of friction by drawing it to scale; or, if you have no instruments, explain how to calculate the number of degrees. (15.) 26. Find the relation between the power and the weight in the screw press, taking into account the friction between the threads of the screw and of the companion screw. (15.) 27. Define a foot-pound of work, and a horse-power.

A steam-crane, working with 3 horse-powers, is found to raise a weight of 10 tons to a height of 50 ft. in 20 minutes; what part of the work is done against friction? If the crane is kept at similar work for 8 hours, how many footpounds of work are wasted on friction?

(15.) 28. Find the position of a body at the end of a given time from the instant at which it is thrown with a given velocity in a given direction, the motion being supposed to take place in

vacuo.

(30.)

A body is thrown in a direction making an angle of 30° with the horizon, and passes through a point whose horizontal distance from the point of projection 400 √3 ft., and vertical height above the point of projection 76 ft.; find the velocity of projection (g = 32.) 29. A particle, whose mass is 10, moving with a velocity 5, meets and impinges directly on another particle whose mass is 20 and velocity 3; the coefficient of restitution is 0.125; find from first principles the velocities of the particles at the end of the impact.

State the dynamical principles employed in answering this question, and define the coefficient of restitution.

(25.) 30. A flywheel weighs 10,000 lbs., and is of such a size that the matter composing it may be treated as if concentrated on the circumference of a circle 12 feet in radius; what is its kinetic energy when moving at the rate of 15 revolutions a

minute? How many turns would it make before coming to rest, if the steam were cut off and it moved against a friction of 400 lbs. exerted on the circumference of an axle 1 ft. in diameter ? (g = 32.)

(30.) 31. Define the centre of pressure of a fluid on a plane area, and find its position in the case of a rectangular area, one edge being on the surface of the fluid.

Find where the centre of pressure of the rectangle would be, if its plane were vertical and its upper edge (which is horizontal) below the surface of the water at a distance equal to the height of the rectangle. (25.) 32. Given the specific gravities of two liquids, show how to calculate the specific gravity of a mixture of given volumes of the two liquids, assuming that the mixture takes place without change of volume.

Three pints of a liquid, whose specific gravity is 0.8, are mixed with five pints of another liquid, whose specific gravity is 104; find the specific gravity of the mixture, (a) if there is no contraction, (b) if on mixture there is a contraction of 5 per cent. of the joint volumes. (20.)

Honours Examination.

INSTRUCTIONS.

Read the General Instructions on the first page.

You are not permitted to attempt more than eight questions. You may select them from any part of the paper.

41. Find the position of the centre of gravity of a very fine piece of wire, bent into the form of a circular arc of given radius.

Calculate (by Guldin's rule) the area of the surface of the solid formed by the revolution of the arc of a quadrant of a circle round its chord. (40.) 42. A given force acts at right angles to the plane of a given couple; show that the three forces cannot be reduced to fewer than two, and that the reduction can be effected in an infinite number of ways.

Let ABCD, ABEF be two faces of a cube; a given force acts along the edge BC, and an equal force along the diagonal AE; show how to resolve them into a couple and a force at right angles to the plane of the couple. (60.)

43. A and B are two points in a horizontal line; the ends of a thread, whose length is less than 3 AB, are fastened to A and B; let C and D be points dividing the thread into three equal parts; if equal weights are fastened at C and D, find the tensions of the parts of the thread. (40.) 44. A rough cylinder is placed endwise on a rough horizontal plane, against which it is pressed by a given weight; find the moment of the couple which will just turn it against the friction of the plane. (40.) 45. Find an expression for the radius of curvature at any point of the neutral axis of a prismatic beam slightly bent by forces, acting in a plane containing the axis of the beam, at right angles to the axis in its undeflected state.

A, B, C, D are four points taken in order along the axis of a beam of uniform cross section; the beam is kept at rest by forces acting at these points; those at A and B form a couple, and so do those at C and D; the weight of the beam is neglected; show that the radius of curvature of the axis has the same value at all points between B and C. (50.)

46. Define the virtual moment or virtual work of a force.

If

X and Y are rectangular components of a force P, show that the virtual work of P equals the sum of the virtual works of X and Y.

Apply the principle of virtual velocities to find the relation between the power and the weight on a rough inclined plane, when the power is about to make the weight slide up the plane.

(50.) 47. If x and y are the co-ordinates of a moving point, explain briefly why its velocities parallel to the axes of x and y are dx dy and

dt

dt

A rod moves with its ends A and B in two fixed rectangular axes, Ox and Oy, respectively; given, at any instant the velocity of A along Or, find that of B along Oy, and the velocity and direction of the motion of the middle point of the rod.

(50.)

48. Find the moment of inertia of a square about one edge, and deduce from it that of a right angled isosceles triangle about the hypotenuse.

(40.) 49. Define the centres of suspension and oscillation of a compound pendulum; state exactly what is meant by their convertibility and prove it. (40.) 50. A perfectly flexible chain hangs over a smooth fixed point, and is allowed to move under the action of gravity; when it is in any given position find;-(a) the acceleration of its velocity, (b) the tension at a given point of its length.