VII. PURE MATHEMATICS. (3).

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

I. Find from first principles the differential coefficients of sin x and €* respectively.

Do the same also from the expansions (assumed known) of sinx and e* in ascending powers of x.

Apply the result or any other method to find the nth differential coeffi cient of

5. Shew from the meanings of the forms oo, ∞°, 1°, that they are indeterminate, and that the first two are restricted in value and the third not so.

6. State fully the conditions that f(x) shall be a maximum or minimum for any given value of x.

If a parallelogram be inscribed in an ellipse the greatest possible value of its perimeter is equal to twice the diagonal of the rectangle described on the axes.

7. Find the equation of the tangent at the point x, y, of the curve y=f(x) and deduce that of the normal.

If 1 and p2 be the perpendiculars from the origin on the tangent and normal respectively, and if

8. If r and have their ordinary meanings in polar curves, prove that dr dp

the radius of curvature at any point is r

In the curve r=a sec 20 prove that the radius of curvature is

and prove that it cuts the curve at the angle whose tangent is 8.

IO.

II.

Trace the curve whose equation is given in the last question.

12. Find the length of the arc between the origin and any given point

on the curve ay1=x5.

VIII. STATICS.

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

1. State what are the axioms assumed in the Theory of Statics.

2. Shew that if three forces acting on a point keep it in equilibrium, each force is proportional to the sine of the angle between the directions of the other two.

3. A weight Q hanging freely over a pully supports P, which rests on an inclined plane. Determine the position of equilibrium and the pressure on the inclined plane.

4. Two forces act upon a rigid body in the same plane but not at the same point. Determine the magnitude, the point of application, and the direction of the resultant.

5. Explain what is meant by the arm, the moment, and the axis of a couple; and prove that the effect of a couple upon the equilibrium of a body is not altered if its arm be turned through any angle about one extremity in the plane of the couple.

6. Draw two diagrams representing the system of pullies in which W=2"P, and that in which W = (2′′ – 1) P, and state what are their respective advantages.

7. Weights in the proportion of 1, 2, 3 are placed at the three angles of a triangle; find by a geometrical construction the centre of gravity.

8. What laws have been established by experiment as regards the friction of plane surfaces? How is the friction estimated? Shew that the coefficient of friction between two given substances is equal to the tangent of the inclination of the plane of one of them when the body formed of the other is about to slide down.

9. The lengths of the arms of a false balance are (a) and (b), and the weight Wappears to balance P at the shorter arm (6), and Q at the longer arm (a). Shew that if the balance be of uniform density and thickness,

10. A uniform beam AB of given length and weight has its extremity A resting in a horizontal groove AC, and its extremity B in a vertical groove BC, and is kept in equilibrium by a string DC fixed at a given point D on the beam. Find the tension of the string, and the limits, as to the length and point of attachment of the string, under which equilibrium is possible.

IX. DYNAMICS.

[Great importance will be attached to accuracy of working.] (N.B.-When needed the measure of the force of gravity may be taken as 32 feet.)

I. If a particle moves uniformly in a circle, distinguish between the angular and linear velocities. What is the relative velocity of two particles moving uniformly in a straight line (1) in the same, (2) in opposite directions? What is the relative angular velocity of the hour and minute hands of a clock, and their relative linear velocity, if the minute hand be 9 inches and the hour hand 3 inches in length?

Two trains whose lengths respectively were 130 and 110 feet, moving in opposite directions on parallel rails, were observed to be 4 seconds in completely passing each other, the velocity of the longest train being double that of the other; find at what rate per hour each train is moving.

2. Enunciate the first law of motion. State briefly the evidence on which we accept the truth of the law. How is the velocity of a body affected when acted on by a uniformly accelerating force? If a body projected upwards with a velocity (u), ascend through a space (s), obtain the equation v2=u2 – 64s.

A tower is 288 feet high; at the same instant one body is dropped from the top of the tower and another projected vertically upwards from the bottom, and they meet half way; find the initial velocity of the projected body, and its velocity when it meets the descending body.

3. Prove that the times of descent down all chords in a vertical circle, whether drawn from the highest or the lowest points of the circle, are constant.

Two vertical circles whose radii are 10 and 6 feet touch each other at the highest point; a straight line is drawn from the point of contact to meet the outer circle; find the time of describing from rest the portion of this line intercepted between the two circles.

4. What is understood by the parallelogram of velocities? Find the resultant velocity of two uniform component velocities.

A particle moves in a straight line along a horizontal smooth plane with a velocity of 3 feet per second; after 2 seconds a velocity of 8 feet per second is imparted to it in a direction at right angles to its original motion; find the distance of the particle from its starting point after it has been in motion for 4 seconds.

5. Find the time of flight and greatest height of a projectile with reference to the horizontal plane passing through the point of projection.

If (t) be the time in which the projectile reaches a point P, and (ť) the time from Puntil it strikes the horizontal plane through the point of projection, prove that the height of P above that plane is gtt'. verify the expression for the greatest height.

Hence

6. State the third law of motion. If (W) be the weight of a body in pounds, what assumptions are made and what units are referred to in

A heavy body is placed on a smooth horizontal table, a pressure of 6 lbs. acts continuously upon it; at the end of three seconds the body is moving with a velocity of 48 feet in a second; find the weight of the body.

7. A weight (W) is drawn up a smooth inclined plane by means of a string, to the other end of which a weight (P) is attached that hangs freely over the top of the plane; find the accelerating force and the tension of the string.

If both (P) and (W) be 8 lbs., the inclination of the plane 30o, and the string be just on the point of breaking, find the greatest weight which the string would support if it were suspended from a fixed point vertically.

8. Define an impulsive force. How is such a force estimated? When one elastic ball impinges directly on another, describe briefly the action supposed to take place during their impact.

An elastic ball (m) moving with a given velocity impinges in direct impact on (m') at rest; find the velocity of (m') after impact, and determine the ratio of the relative velocity of the balls after impact to the original velocity of (m).

9. Point out briefly how a simple pendulum may be used to determine the force of gravity at the place where it swings.

A pendulum, whose length is L, makes (m) oscillations in a day its length is changed so that it makes (m+n) oscillations in a day; shew that

2n

m

L is the measure of this change nearly.

IO. In the theory of work what is meant by a foot-pound? Shew that the kinetic energy of a body in motion is equal to half its "vis viva."

A train is moving on a horizontal rail at the rate of 15 miles an hour; if the steam be suddenly turned off, how far will it run before it stops, the resistances being taken at 8 pounds per ton?