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7.

State without proof the method of determining maxima and minima values of given functions of x.

If A be the least of the three angles of a triangle whose sides are a, b, and c, prove that the length of the shortest line which bisects the area is

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8. If x be the angle between the radius vector and tangent of a curve and p the perpendicular upon the latter from the pole, prove that

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9. Shew how to find the envelope of a system of curves whose equation contains one arbitrary parameter.

Prove that the polar equation between p and r of the envelope of the lines

is

x cos 20+ sin 20 = 2a cos 0,

p2= (a2 – r2).

IO. Find the co-ordinates of the centre of curvature of the curve y=f(x) at the point x, y.

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13. Find the area of the loop of the curve r=a0 log,0.

I.

VIII. STATICS.

[Great importance will be attached to accuracy.]

Enunciate the "Polygon of Forces" and deduce it from the "Parallelogram of forces."

Forces of 3, 4, 5 and 6 lbs. act in directions respectively north, east, south, and west; find the magnitude and direction of their resultant.

2. Find the magnitude and direction of the resultant of two parallel forces acting in opposite directions on a rigid body.

Explain the meaning of your results in the case where the two forces are equal.

3. What is meant by the Resolved part of a given force in a given direction?

Prove that, if any number of forces meet in a point, the sum of their resolved parts in any direction is the same as the resolved part of their resultant in that direction.

4. What is meant by the Moment of a force about a straight line?

Prove that the sum of the moments of two forces acting in a plane about a point in that plane is equal to the moment of their resultant about that point.

5. Prove that, if the sum of the moments of a number of forces acting in one plane about each of three points in that plane not in the same straight line is zero, the forces are in equilibrium.

The magnitude of a force is known, and also its moments about two given points A and B. Find by a geometrical construction its line of action.

6. Find the centre of gravity of a number of heavy particles in a plane.

At the centres of the three circles escribed to a triangle particles are placed of masses inversely as the radii of those circles. Find the centre of gravity of the system.

7. Find the position of equilibrium of a balance when the weights in each scale pan, the length of the arms, the weights of the beam and of the scale pans and the position of the centre of gravity are given.

8. Explain what is meant by the principle of virtual velocities.

A pulley is suspended by a vertical loop of string to a wheel and axle, and supports a weight, one end of the string being wound round the axle, the other in a contrary direction round the wheel. Find by virtual velocities the power which, acting at right angles to one end of an arm a so as to turn the axle, will support the weight.

9. A string without weight is suspended from two pegs not in the same horizontal line and passes through a small smooth heavy ring free to slide along the string. Find the position of equilibrium of the ring.

If the ring, instead of being free to slide, be tied to a given point in the string, find equations to give the ratio of the tensions of the two portions of the string.

IO. A continuous string, without weight, length 1, hung over two smooth pegs in the same horizontal line, distant a apart, hangs in two loops, on each of which is placed a small smooth heavy ring, one of weight W, the other of weight W'. Find an equation to determine the tension of the string.

IX. DYNAMICS.

[Great importance will be attached to accuracy.]

[N.B.-When needed the measure of the accelerating force of gravity may be taken as 32.]

I. If a body start with a velocity V under the influence of a uniform acceleration ƒ in the direction of motion, find its velocity after a time / and the distance over which it will have gone in that time.

2.

A ball is dropped from a height 1⁄2 under the influence of gravity, and another ball is discharged from the point, where the first would strike the ground, with velocity sufficient to carry it to the height h. At what moment ought it to be discharged so as to meet the other half way?

3. What is meant by the work done by a given force?

Find how many foot-pounds of work are required to change the velocity of a body of Wlbs. weight from v to v' feet per second.

4. Describe any means by which the acceleration due to gravity may be ascertained.

What is meant when we say that gravity is an acceleration of 32? If an hour were the unit of time, and 4000 miles were the unit of length, what number would represent the acceleration due to gravity?

5. Prove that a body moving freely under the influence of gravity moves in a parabola.

Given the velocity and direction of its projection, find the velocity and direction of its motion at any subsequent time t.

6. A body is projected from a given point with velocity v. Find the direction of its projection so that it may pass through another given point, distant / horizontally and k vertically, from the point of projection.

7. Two heavy particles of equal weight are connected by a string, one lies on a rough table, and the other hangs over the edge; find the distance over which they will have moved after a given time t from rest.

8. Two imperfectly elastic balls impinge on one another in a direct line; given their velocities before impact, find their velocities after impact. Prove that the motion of the centre of gravity of the bodies is the same after impact as before it.

9. Prove that where a body descends from rest down a smooth curve in a vertical plane its velocity depends only on the vertical height through which it has descended.

An elliptical wire of eccentricity is placed in a vertical plane with its major axis inclined to the horizontal at an angle 60°. If a small ring is allowed to slide along the wire from rest at the higher extremity of the major axis, shew that its velocities at the two ends of the minor axis are as √2: 1.

IO.

A body moves uniformly in a circle: find the acceleration towards the centre.

With what number of turns per minute must a weight of 10 lbs. revolve horizontally at the end of a string 15 inches long, in order to cause the same tension on the string as if a 1 lb. weight were hanging at rest held by the string vertically?

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JULY, 1884.

PRELIMINARY EXAMINATION.

I.

I. EUCLID (BOOKS I.-IV. AND VI.). (Obligatory.)

[Great importance will be attached to accuracy in working.]

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them of the other.

Two equal straight lines AB and CD are joined towards opposite parts by the equal straight lines AD and CB intersecting in O. Prove that both the triangles OAC and OBD are isosceles.

2. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

No straight line can be placed within a parallelogram greater than the greater diameter.

3. Triangles upon equal bases, and between the same parallels, are equal to one another.

The line AD is drawn from the vertex of the isosceles triangle ABC perpendicular to the base, and is bisected in E. If BE produced meet AC in F, prove that the triangle BCF is double of the triangle BAF.

W. P.

I

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