8. In the system of pulleys in which each pulley hangs by a separate string, all the strings being parallel and the power acting upwards, find the relation between the power and the weight when the system is in equilibrium.

If there are three pulleys it is found that a certain weight can be supported by a power of 7 lbs., but if there are four pulleys the same weight can be supported by a power of 4 lbs. ; find the weight supported, and the weight of the pulleys: the pulleys being of equal weight.

9. A square table stands on four legs, placed respectively at the middle points of its sides; find the greatest weight that can be placed at one of the corners without upsetting the table.

10. State the laws of Friction.

A body is supported on a rough inclined plane by a force acting along the plane. If the greatest magnitude of the force when the plane is inclined at the angle a to the horizon is equal to the least value of the force when the plane is inclined at the angle a' to the horizon, show that

DYNAMICS.

1. Explain how velocity is measured (1) when constant, (2) when variable.

A body, moving uniformly, passes over a mile in fifteen minutes; if 64 be the measure of the velocity, and eleven feet be the unit of length, find the unit of time.

2. State and prove the parallelogram of velocities.

A man is riding, at the rate of 12 miles an hour, along a straight road at right angles to a railway, on which a train is passing at the rate of 16 miles an hour; find the velocity of the man relative to the train.

3. Explain how acceleration is measured.

What is the measure of the acceleration due to gravity (1) when a foot and half a second are units of length and time, (2) when the units are a mile and eleven seconds?

4. State the Second Law of Motion.

A ball, which is at rest on a smooth horizontal table, receives, at the same instant, two impulses in given directions employ the Second Law of Motion and the parallelogram of velocities to determine the motion produced, the velocities due to each impulse being known.

5. A ball is projected vertically upwards with a velocity of 128 feet per second; find after what time it will have attained a height of 192 feet above the point of projection, and explain the double result.

Find also the greatest height attained.

6. Two weights, P and Q, are connected by an inextensible string which passes over a fixed smooth pulley; determine the acceleration of the system when in motion under the action of gravity only, and find the tension of the string.

=

If P 2 Q, and if the system have no motion initially, find the velocity at the end of three seconds, and determine the instantaneous effect and the subsequent change of motion produced by suddenly attaching at that time a weight Q to the ascending body.

7. A body is projected with a velocity u in a direction inclined at the angle a to the vertical; if at the time t the direction of its motion be inclined at the angle e to the vertical, prove that

At what time will the body be moving at right angles to the direction of projection?

8. Find the range of a projectile on the horizontal plane through the point of projection.

If the velocity of projection be given, find the greatest range, and show that, for any range short of the greatest, there are two directions of projection, which are equally inclined to the direction of projection giving the greatest range.

9. Determine the change in the motion of a smooth inelastic ball produced by its impact on, 1st a fixed surface, 2nd a fixed edge.

An inelastic ball, sliding along a smooth horizontal plane with the velocity of 16 feet per second, impinges upon a smooth horizontal rail at right angles to the direction of its motion; if the height of the rail above the plane be half the radius of the ball, prove that the latus rectum of the parabola subsequently described is one foot in length.

10. An elastic ball impinges in a given direction and with a given velocity upon a fixed smooth plane; the coefficient of elasticity being given, determine the velocity of the ball immediately after the impact.

1 2

A ball of elasticity falls from a height of 64 feet upon a

horizontal plane; find the height to which it will rise at the first rebound, and the time at which the rebounding will

cease.

11. A shot weighing 700 lbs. is fired with a velocity of 1700 feet per second from a gun weighing 38 tons. Find the velocity with which the gun recoils, neglecting the weight of the powder. If the recoil be resisted by a constant pressure equal to the weight of 17 tons, through what space in feet will the gun recoil?

12. If a point move with uniform velocity v in a circle of radius r, prove that its acceleration is in the direction of the centre of

Explain the difficulty of walking rapidly in a curve on smooth ice, and show how it is possible for a man, in skating, to describe a circle on one foot.

13. Write down the expression for the time of oscillation of a simple pendulum of length 1, and find, approximately, the length of a pendulum which oscillates in one second.

K

SET IX.

EUCLID.

1. Define a plane superficies, an obtuse-angled triangle, a segment of a circle, and reciprocal rectilineal figures; and show how the latter differ from similar figures.

2. The angles at the base of an isosceles triangle are equal; and if the equal sides be produced, the angles at the other side of the base are equal.

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

ABCD is a parallelogram, C being the corner opposite to A. If P, Q are the middle points of CB and CD, show that the triangle APQ is equal to the sum of the triangles BPQ, CPQ, and DPQ.

4. In a right-angled triangle the square on the side subtending the right angle is equal to the sum of the squares described on the other two sides.

If a quadrilateral be such that its diagonals are at right angles to one another, the sums of the squares described on the opposite sides are equal.

5. If a straight line be bisected, and produced to any point, the square on the whole line thus produced, and the square on the part produced, are together double of the square on half the line bisected, and of the square made up of the half and the part produced.

6. A segment of a circle being given, describe the circle of which it is a segment.

7. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them shall be equal to the rectangle contained by the segments of the other. Prove this for the case when one straight line only passes through the centre of the circle.

ACDB is a semicircle, whose diameter is AB; AD, CB are any two chords meeting one another in P. Prove that

AB2 = PA. AD + PB. BC.

8. When is a triangle said to be inscribed in a circle? In a given circle show how to inscribe a triangle equiangular to a given triangle.

Show that the perpendiculars drawn through the middle points of the sides of the triangle pass through the centre of the circumscribing circle.

9. Inscribe an equilateral and equiangular pentagon in a given

circle.

If ABCDE be the inscribed pentagon, prove that

BE2 AB. BE+ AB2.

10. The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or the consequents of the ratios.

O is the centre and OA the radius of a given circle, and V is the middle point of OA. P and Q are two points on the circumference on opposite sides of, and equally distant from A. QV is produced to meet the circle in L. Show that, whatever be the length of the arc PQ, LP always meets OA produced in a fixed point.

11. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to the sum of the rectangles contained by the opposite sides.