in the wall, if the angle of inclination of the beam to the wall be given, show how to find, by geometrical construction, the length of the string CD and the height of D above A.

Prove that for the problem to be possible the given angle BAD must be acute or obtuse according as AC is less or greater than AB.

9. Find the relation between the power and the weight in the system of pulleys where each hangs by a separate string, and prove that the principle of virtual velocities holds good in this

case.

Prove also that this principle holds in the case of a boat propelled by oars; the power being the pull of the rower's arm and the weight the resistance of the water to the boat which may be assumed to be a rectangular trough.

10. State how "work" is measured. A block weighing one ton is in the form of a rectangular parallelopiped 8 feet high on a square base, whose side is 6 feet. It is placed upon a rough board with the sides of its base parallel to the length and breadth of the board, and the centre of the base 6 feet from one extremity of the board.

The board is now tilted round this extremity until the block topples over without sliding; find the work done.

DYNAMICS.

1. Define momentum, force, acceleration.

If an accelerating force be represented by 10, when feet and seconds are taken as units, what must the unit of time be in order that the same force may be represented by 67 when a yard is the unit of length? 2. State the second law of motion; and show that if a body move with velocity in a given direction its velocity in any direction making angle a with the given direction is v cos a.

Three equal particles are placed at the angles of the triangle ABC, and move respectively from A to B, B to C, and C to A, the three motions being accomplished with uniform velocity and in the same time; prove that their centre of gravity remains

at rest.

3. A body moves with uniform velocity in a circle whose centre is O. Find the velocity and acceleration perpendicular to a given diameter YOY', and show that the motion of the particle relative to YOY' is the same as that of a particle moving in a diameter at right angles to YOY' under the action of a force to O, varying as the distance.

4. A body moves from rest under the action of a constant force. Find the time of describing a given space, and explain the double result.

If O be a fixed point, C a point in a vertical plane through O, find the locus of C when the time of sliding down CO varies as the length of CO.

OX is a horizontal line, C a fixed point in it, OY is drawn in any direction in vertical plane through OC. Find by geometrical construction the point in OY from which the time of sliding down to C in a straight line is the least.

5. Prove that the velocity acquired by a body in sliding down a smooth curve in a vertical plane is the same as that acquired in falling perpendicularly through the same vertical height.

ABC is a circular arc whose middle point is B. A particle is projected from A so as to slide along the curve, and is acted on by a constant force in the plane of the curve. If the velocities at A, B, C respectively be known, show how to find the magnitude and direction of the force.

6. Two particles are simultaneously projected from the same point with given velocities and inclinations to the horizon in different vertical planes; prove that their centre of gravity describes a parabola.

If one of the vertical planes remain fixed while the other assumes all possible positions, find the locus of the foci of all such parabolas.

7. If a particle be attracted to a centre of force varying as the distance, prove that the time of falling from rest to the centre is independent of the initial distance.

8. Define a cycloid, and prove that the length of the arc measured from the vertex is twice the corresponding chord of the generating circle.

A pendulum of length l has one end of the string fastened to a peg on a smooth plane inclined to the horizon at angle a. With the string and the weight on the plane its time of oscillation is two seconds. Find a, having given that a pendulum oscillates in one second when suspended ver

of length

tically.

2/2

9. Find the central force under which a particle will move with uniform velocity in a circle.

A particle is projected horizontally with velocity from the highest point of a smooth vertical circle; find when it will leave the curve.

If a parabola with vertical axis be substituted for the circle, prove that the particle will never leave it unless it leave it at the vertex.

2

10. Two elastic spheres of masses m,, m,, impinge obliquely; V1, V2 are their velocities at any time. It being given that m, v +m, v, is the same after impact as before, prove that the velocity of separation is equal to that of approach.

Two unequal perfectly elastic particles capable of moving freely in a smooth circular tube are projected in opposite directions with unequal velocities along the tube from the same point in it. Determine the angular distances from the point of projection of the points of successive impacts.

SET IV.

EUCLID.

1. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.

A is the greatest angle of a triangle ABC; show how to divide A into two parts whose difference is equal to B.

2. Parallelograms on the same base and between the same parallels are equal to one another.

On the side of the base BC remote from the vertex A of an acute-angled triangle a rectangle BDEC is described. If AD, AE meet BC at F and G, and AH is drawn parallel to BD, and meeting DE at H, show that the quadrilateral AFHG is equal to the triangle ABC.

3. In any right-angled triangle, the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle.

Give a geometrical construction for a square three times as great as a given square.

4. If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section.

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

Prove that the converse proposition is also true.

If in any quadrilateral the diagonals are equal to one another, show that the straight lines joining the middle points of opposite sides are at right angles.

6. Show how to draw a straight line touching two given circles.

Prove that the difference of the squares on the common tangents to two equal circles is equal to the square on a

diameter.

If two equal circles cut at right angles, show that the common tangent is equal to the common chord.

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

8. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular to one another.

9. Similar triangles are to one another in the duplicate ratio of their homologous sides.

From an isosceles triangle cut off a part bearing to the whole triangle the ratio of the base to a side.

10. If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.

11. Prove that the circle described through the middle points of the sides of a triangle passes also through the feet of the perpendiculars from the angular points on the opposite sides. Find the centre of this circle, and show that its radius is half that of the circle circumscribing the triangle.