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10. Explain the method of integration by parts, and integrate with regard to x the expressions

ex cos x and x log (x+1).

II. Find the limiting value of the sum of the series

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[Full marks may be gained by correctly answering three-quarters of this paper. Great importance is attached to accuracy.]

I.

Three forces acting at a point make equilibrium. If they make angles of 120° with each other, prove that they are equal. If the angles are 60°, 150°, 150°, in what proportions are the forces?

2. Two given forces meet at a point. Find in what direction a third force of given magnitude must act at the point, if the resultant of the three is the greatest possible.

3. State the conditions necessary for equilibrium for any system of forces acting in one plane on a rigid body.

If the rigid body have one point fixed, what is necessary for equilibrium? What, if it have two points fixed?

4. The door of a room is open: determine (as far as possible) the forces which it exerts upon its two hinges; given all the dimensions, and its weight.

5. Two given forces act at two given points: if they are turned round those points in the same direction through any two equal angles, show that their resultant will always pass through a fixed point.

Extend this theorem to any number of forces.

6. A uniform bar AB 10 feet long and weighing 50 lbs. rests on the ground. If a weight of 100 lbs. be laid on it at a point distant 3 feet from B, find what vertical force applied at the end A will just begin to lift that end.

7. Prove in any way that a couple cannot be balanced by a single force. If it be balanced by two forces, what is necessary with regard to them?

8. A heavy horizontal circular ring rests on three supports at the points A, B, C of its circumference. Given its weight, and the sides and angles of the triangle ABC, find the pressure on the supports.

9. Two inclined planes have a common vertex and equal slopes; one is smooth and the other rough. A given weight Wis supported on one by a string passing over the vertex and attached to another weight resting on the other plane; this weight being just sufficient to prevent W descending. Prove that the weight required is less when W is on the rough plane, than when it is on the smooth one.

IO. Four equal heavy bars are jointed so as to form a rhombus ABCD: A and C are joined by a string. The whole is suspended from the angle A; find the tension of the string, by means of the principle of Virtual Velocities (or otherwise).

II.

If the resultant of any systems of parallel forces in one plane acts outside the two extreme forces of the system, show that some must act in an opposite sense to others.

A heavy bar lies on a rough floor: a horizontal force, too small to move it, is applied at one end, at right angles to the bar; describe generally the forces which have been called into operation to prevent motion ensuing.

12.

A circle is pushed over an equal circle till it completely covers it. Find the centre of gravity of the crescent-shaped space left uncovered just before they coincide.

13. A heavy rectangular block ABCD rests with AB on the ground; a rope is attached to the corner C and pulled round a fixed pulley P vertically over D till motion ensues. Give a geometrical construction determining the least height for the pulley P, if the block is to begin to revolve round A without slipping along the ground (given all particulars).

IX. DYNAMICS..

[Great importance will be attached to accuracy.]

[If needed, the measure of the force of gravity may be taken as 32 feet.]

1. A body moves in any direction in a vertical plane: given the horizontal and vertical velocities at any instant, show how to find the actual velocity of the body, and the direction in which it is moving.

A body describes uniformly a vertical circle, whose radius is 12 inches, in 6 seconds, and the centre of the circle moves uniformly in a horizontal line with the same velocity: show how to find the true velocity of the body whenever its direction is inclined at 45° to the horizon.

2. State the first law of motion, and the evidence on which it is accepted.

In an observed case of a body in uniform motion, are we to infer that no force acts on the body? Illustrate this by example.

3. How is uniform force measured? If a body be acted upon by a uniform force, and be initially projected in the direction of that force with a velocity (u), and if it acquire a velocity (v) in (1) seconds, prove that (s) the space described may be determined by the equation s=t u +

A train at the foot of an incline is moving at the rate of 60 miles an hour; the steam is suddenly turned off, and it then runs on the horizontal plane for 3 miles before it stops; if friction be the constant retarding force, find its measure. Determine also the space the train describes in three minutes from the instant the steam is turned off.

4.

tion is g

A body slides down a rough inclined plane, show that the accelera

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If the height of the inclined plane be 12 feet, the base 16 feet, find how far a body will move on the horizontal plane after sliding down from rest the length of the inclined plane, supposing it to pass from one plane to the other without loss of velocity, the coefficient for both planes being .

5. In the case of a projectile in vacuo, find the direction of the projectile with respect to the horizon at any point of its course.

If (0) be the angle of elevation at which a projectile strikes a given mark, (a) the angle of projection, and (45°) the angle which the given mark subtends at point of projection, prove tan a +tan 0=2.

6. State the law of motion that connects the statical and dynamical measures of force. Obtain the equation from which that connexion is shown, and explain the units of reference in that equation.

A body weighing 36 lbs. is moved by a constant pressure, which generates in it a velocity of 8 feet per second, find the statical measure of that pressure.

7. A ball is projected vertically upwards with a velocity of 160 feet per second, when it has reached its greatest height it is met in direct impact by an equal ball which has fallen through 64 feet; find the times from the instant of impact in which the balls reach the ground, the elasticity between them being.

8. Assuming the expression for the time of a small oscillation of a pendulum in a circular arc, calculate to two decimal places the length of a second's pendulum in inches.

A simple pendulum beating seconds is lengthened by of an inch; find the number of seconds it will lose in 24 hours.

9. A string 4 feet long can just support a weight of 9 lbs. without breaking, a weight of 8 lbs. fixed to one end of the string describes a circle uniformly round the other end, which is fixed on a smooth horizontal table; determine the greatest number of revolutions the revolving weight can make in a minute so as just not to break the string.

IO.

II.

Determine the time of a small oscillation in a cycloid.

Show how to find the work accumulated in a moving body.

A bullet with an initial velocity of 1500 feet strikes a target 1200 yards distant with a velocity of 900 feet in a second, the range of the bullet being assumed to be horizontal; compare the mean resistance of the air with the weight of the bullet.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JULY, 1886.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. AND VI.)

[Ordinary abbreviations may be employed; but the method of proof must be geometrical. Great importance will be attached to accuracy.]

I. Any two sides of a triangle are together greater than the third.

Prove that the shortest line, which can be drawn with its ends upon the circumferences of two concentric circles, will, when produced, pass through the centre.

2. Prove one case of the following proposition :

If two triangles have two angles of the one equal to two angles of the other, each to each; and one side equal to one side, viz., either the sides adjacent to the equal angles in each, or sides which are opposite to equal angles in each; then shall the other sides be equal, each to each; and also the third angle of the one equal to the third angle of the other.

The vertical angle A of the isosceles triangle ABC is half a right angle, and the perpendiculars AD, BE let fall from A, B upon the opposite sides intersect in F. Show that FE is equal to EC.

3. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given angle.

W. P.

I

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