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of empty trucks, on another part whose inclination is B. Find the weight of a truck.

8.

Determine the relation between the power and the weight in that system of pulleys in which each of the strings, all of which are supposed parallel, is attached to the weight, taking into account the weight of the pulleys.

If the pulleys be equal, and the power equal to the weight of one of them, and the number of pulleys be five, prove that the ratio of the weight to the power is 57: I.

9. Determine the relation between the power applied and the weight supported in the case of the screw.

Find the mechanical advantage of a screw whose diameter is 6·78 inches, and the distance between whose successive threads is 71 inch

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10.

What are the three different forms of levers?

may be reduced to one general case.

Shew how they

Two equal uniform beams AB, BC are freely jointed at B, and A is fixed to a hinge at a point in a wall about which AB can revolve in a vertical plane. At what point in BC must you apply a vertical force to keep the two beams in one horizontal line? Find the value of the force.

II. If a force moves its point of application through a certain space, find how much work is done. What is the difference between work done by a force and against a force, and how is it expressed mathematically?

A capstan, whose diameter is 20 inches, is worked by a lever which measures 5 feet from the axis of the capstan. Find in foot-pounds the amount of work done in drawing up by a rope one ton over 35 feet of the surface of a smooth plane inclined to the horizon at an angle cos-1. The rope may be supposed to be always parallel to the surface of the plane. Find also the force applied to the end of the lever, and the distance through which its point of application moves.

(Note.

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When needed the measure of gravity may be taken as 32 feet.)

1. Obtain a formula that expresses the relation between space, time, and velocity when the motion is uniform.

A train moving uniformly describes 88 yards in 3 seconds; find its rate per hour in miles. In what time will it travel 600 miles with a stoppage of 5 minutes after every 100 miles?

Supposing the circumference of the earth at the equator to be 25,000 miles, and the time of the earth's rotation to be 24 hours, find the velocity

in space, in miles per hour, of a cannon ball when it is fired at the equator with a velocity of 1,650 feet per second (1) in the direction of the earth's rotation, (2) in the opposite direction.

2. If a body's velocity is observed to increase or decrease, what must be the condition of increase or decrease when the force acting on the body is uniform? A body is set in motion in a right line; at the end of 3 seconds it is moving at the rate of 224 feet per second, at the end of 4 seconds at the rate of 256 feet per second, at the end of 5 seconds at the rate of 288 feet per second; what is the measure of the force acting on the body?

3. If a body acted on by gravity is projected upwards with a velocity V, find its velocity at the end of (t) seconds, and shew that it will just V2 2g

ascend to a height h, when h=·

A body so projected from the ground descends to the ground again and rebounds continually, losing at each rebound half the velocity with which it strikes the ground; what is the whole space described by the body?

4. Shew that velocities may be resolved and compounded analogously to the resolution and composition of forces. If two sides AC, CB of a triangle represent in magnitude and direction the component velocities of a particle, prove that the third side AB will represent the actual velocity of the particle, in magnitude and direction.

The velocity of a ship in a straight course on an even keel is 83 miles an hour; a ball is bowled across the deck, perpendicular to the ship's length, with a uniform velocity of 3 yards in a second; describe the true path of the ball in space, and shew that it will pass over 45 feet in 3 seconds nearly.

5. Assuming that the path of a projectile in vacuo is a parabola, shew that the velocity at any point is that which would be acquired in falling freely from the directrix to that point. If y be the vertical height of the projectile above the horizontal plane when it is moving with a velocity (~), and be the velocity of projection, prove v2= V2-2gy.

6. When a projectile strikes a given mark, find the direction in which it is moving at the instant of impact.

If (A) be the point of projection, (a) the angle of projection, PN and P11 equal perpendiculars from the curve to the horizontal plane passing through A, (ß) and (81) the angles which PN and P11 respectively subtend at A, prove tan ẞ+tan ẞ1 = tan a.

7. A weight (W) is drawn up a smooth inclined plane by a weight P hanging over the top of the plane and attached to (W) by an inextensible string; find the accelerating force and the tension of the string.

Three weights (w) are fastened to a string whose length (7) is equal to that of an inclined plane; one weight is attached to each end, and the other weight to the middle of the string; when one weight hangs over the top of the plane the weights are in equilibrium; if the second weight also is just made to hang vertically, find the velocity with which the third weight reaches the top of the plane.

8. Prove that the times of descent down all the chords drawn from the highest point of a given vertical circle are the same; express the time of descent for such a circle in an invariable form.

A particle P descends from the highest point down the chord which is the side of a regular hexagon inscribed in the circle, and Q down the vertical diameter; if P=2Q, shew that their common centre of gravity will descend along the chord which is the side of an equilateral triangle inscribed in the circle, assuming that the path of the centre of gravity is a straight line.

9. If a smooth sphere impinge directly upon another sphere, explain the mechanical action that takes place during the impact.

When an elastic sphere (m) impinges on another elastic sphere (m1) at rest, find generally the velocity of each after impact; if, after the impact, (m) remain at rest and (m1) move on with one eighth of the velocity with which it is struck, find the elasticity and the ratio of the radii of the two spheres, supposed of the same material.

10. How is the resistance of an inclined plane modified by friction? If (4) be the limiting angle of resistance on a rough inclined plane whose inclination is (a), shew that the accelerating force down the plane is

g sin (a – 4) ̧

cos

II. When a given mass which is made fast to the end of an inextensible string revolves uniformly round the other end of the string, which is fixed, find the tension of the string.

If the weight of the body is given in pounds, how is the mass expressed ? A string 5 feet long can just sustain a weight of 20 pounds; if the revolving weight be 5 pounds, determine the greatest number of complete revolutions that can be made in one minute by the string without breaking.

12. Find the time of an oscillation of a heavy particle moving down the arc of a cycloid. Derive from this the time of an oscillation of a pendulum in a small circular arc.

A pendulum whose length is makes (m) oscillations in 24 hours; when its length is slightly changed it makes (m+n) oscillations in 24 hours; shew that the pendulum has been diminished in length by a part equal to / nearly.

2n

m

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JUNE, 188L

PRELIMINARY EXAMINATION.

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

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I. Define a plane angle, a scalene triangle, a rectangle, the angle of a segment of a circle. What is meant by a parallelogram being "applied to a straight line? Give an instance from Euclid.

2. When is one straight line said to be at right angles to another straight line? Draw a straight line perpendicular to a given straight line of unlimited length from a given point without it. Why is the given straight line described as of unlimited length?

3. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel.

Prove that a quadrilateral, which has two opposite sides and two opposite obtuse angles equal, is a parallelogram.

4. In any right-angled triangle the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.

If BC is the hypotenuse of a right-angled triangle ABC, and BE is the square described on BC, and AK on AC, shew that AE and BK intersect at right angles to one another.

W. P.

I

3

5. If a straight line be divided into two equal and also two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal to the square on half the line.

If of the two unequal parts one is three times the other, shew by means of the above proposition that the rectangle contained by the unequal parts is three-fourths of the square on half the line.

6. Prove that the straight line drawn at right angles to the diameter of a circle from the extremity of it falls without the circle.

If the centre of a circle be joined to the vertices of a circumscribing quadrilateral, then any one of the four angles at the centre is supplementary to that one which is not adjacent to it.

7. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If from any point on the circumference of a circumscribing circle perpendiculars be drawn to the sides of the inscribed triangle (produced if necessary), prove that their feet will lie in the same straight line.

8. About a given circle describe a triangle equiangular to a given triangle.

Having given the vertical angle of a triangle, and the segments into which the base is divided by the inscribed circle, construct the triangle.

9. Describe an isosceles triangle, having each of the angles at the base double of the third angle.

Shew how it is possible to describe within the smaller circle used in the construction a triangle equal in all respects to the required triangle.

IO. Define similar rectilineal figures; and shew the necessity for each part of the definition.

If two similar triangles ABC, A'B'C' are so placed that their homologous sides are parallel, shew that AA', BB', CC' will meet in a point, unless the triangles are equal, in which case these lines will be parallel.

II. If the vertical angle of a triangle be bisected by a straight line, which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

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