IV. STATICS AND DYNAMICS. I. Enunciate and prove the "Triangle of Forces." A small heavy ring, which can slide freely upon a smooth thin rod, is attached to the end of the rod by a fine string. If the rod be held in any position inclined to the vertical, draw a triangle representing the forces acting upon the ring. 2. A rectangular box, containing a ball of weight W, stands on a horizontal table, and is tilted about one of its lower edges through an angle of 30°. Find the pressures between the ball and the box. 3. State the conditions of equilibrium of three parallel forces acting upon a rigid body. A rod ABC, 16 inches long, rests in a horizontal position upon two supports at A and B one foot apart, and it is found that the least upward and downward forces applied at C which would move the rod are 4 oz. and 5 oz. respectively. Find the weight of the rod and the position of its centre of gravity. 4. Show how to find the centre of gravity of a body composed of two parts whose weights and centres of gravity are known. A solid figure is formed of an upright triangular prism surmounted by a pyramid; if the length of every edge of this figure be a feet, find the height of its centre of gravity above the base. 5. Show that the algebraical sum of the moments of two forces (whose lines of action intersect) about any point in the plane containing the forces is equal to the moment of their resultant. OA, OB are chords, 4 and 5 inches in length, of a circular disk OACB, whose diameter OC is 6 inches. If forces of 3 and 4 lbs. act from O along these chords respectively, find how the disk will begin to move, the point C being fixed. 6. Describe, and show how to graduate, the Danish Steelyard. 7. Explain how the measure of the velocity of a moving point depends upon the units of space and time. If the unit of time be half a minute and the unit of length be 2 yards, what will be the measure of the velocity of a body which describes, at a uniform rate, 14 miles in 3 hours? 8. Investigate the formula s = ft2 for the space described from rest by a particle subject to uniform acceleration f, and hence deduce a corresponding expression in the case where the particle has an initial velocity u. A particle is observed to describe 7 feet in 3 seconds, and 13 feet in the next 3 seconds; find its acceleration. 9. The side BC of a triangle ABC is vertical; show that, if the times of falling down the two sides BA, AC be equal, the triangle must be isosceles or right angled. IO. Distinguish, with examples, between the volume, mass, and weight of a body, and find the relation between the units of mass and weight in order that W may be equal to Mg. If the measures of the mass and weight be the same, and the unit of length be 2 feet, find the unit of time. II. If a force of 15 poundals act upon a mass of 13 pounds, what velocity will it generate in 8 seconds? 12. A particle is projected in any manner in a vertical plane, show how to find its position at the end of a given time. 7. A stone is thrown from the top of a tower with a velocity of 50 feet a second in a direction making an angle of 30° with the horizon. Find the distance of the stone from the point of projection at the end of 5 seconds. FURTHER EXAMINATION. V. MATHEMATICS (1). [Full marks may be obtained for about three-fourths of this paper. importance is attached to accuracy.] Great I. Define a "conic section," and show that a straight line generally cuts a conic section in two points. Enumerate carefully all the cases of exception. 2. Prove that the subtangent at any point of a parabola is double of the abscissa, and the subnormal is half of the latus rectum. If one side of an equilateral triangle be a focal radius SP of a parabola, and another side lie along the axis, show that the third side will be either the tangent or the normal at P, and find in each case the length of SP. 3. Prove that, in any central conic, the square on the ordinate PN bears a constant ratio to the rectangle under the abscissae AN, A'N. PNP' is a double ordinate of an equilateral hyperbola whose transverse axis is AA'. Prove that the directions of AP, A'P' are perpendicular to one another. 4. If from any point O two straight lines be drawn, OPQ, OP'Q', intersecting an ellipse in P, Q; P', Q', show that the rectangles OP. OQ, OP'. OQ' are to one another as the squares of the diameters parallel to PQ, P'Q' respectively. If an ellipse, inscribed in a triangle, touch one side in its middle point, prove that the straight line joining that point with the opposite angle of the triangle will pass through the centre of the ellipse. 5. If from any point P on a hyperbola a straight line, drawn in a given direction, meet the two asymptotes in R, R' respectively, prove that the rectangle RP. PR' is constant. Hence show that a straight line touching the hyperbola, and terminated by the asymptotes, is bisected at the point of contact. 6. If a right cone be cut by a plane, determine by geometrical construction the positions of the foci and directrices of the curve of section. Show that sections made by parallel planes have the same eccentricity. 7. Given the coordinates of two points, P, Q, find the coordinates of the point in which the straight line PQ is divided in a given ratio. Apply your results to show that the straight line joining the middle points of the diagonals of a quadrilateral, and the straight line joining the middle points of either pair of opposite sides, bisect each other. 8. Find the angle between the straight lines whose equations, referred to rectangular axes, are lx+my+c= 0, l'x+my+c' = 0. OACB is a parallelogram, whose sides OA, OB are of lengths a, b, and make angles a, ẞ respectively with the axis Ox. Write down the equations of the two diagonals OC, AB, and show that the tangent of the angle between them is 2ab sin (B-a) 9. Prove that the equation of any tangent to the circle x2+12 = c2 may be written in the form x cos a + y sin a = c. Prove also that the locus of the foot of the perpendicular let fall upon the tangent, from the point in which the circle cuts the positive direction of the axis of y, is the curve whose equation is (x2+y2 − yc)2 = c2 {x2 + ( y −c)2}. IO. Show that the equation of the normal to the parabola y2 = 4ax, which is inclined at an angle ℗ to the axis, may be written in the form y = (x -a- a sec20) tan 0. If a chord of a parabola, whose inclination to the axis is tan-12, be normal to the curve at one end, show that it will subtend a right angle at the vertex. II. Investigate the equation of the polar of any point with respect to a given ellipse or hyperbola. An ellipse and a hyperbola have the same principal axes. Show that the polar of a point on either curve with respect to the other touches the first curve. 12. Write down the equations of the latera recta of the conic whose polar equation is VI. = I-e cos 0. Also show that if r1, r2, rg be the lengths of three focal radii, each of which is equally inclined to the other two, I I I 3 MATHEMATICS (2). [Full marks may be obtained for about three-fourths of this paper. Great importance is attached to accuracy.] I. If a3+b3+c3 = 3abc, where a, b, c are finite, show that any one of the three quantities a2bc, b2- ca, 2-ab, is a mean proportional between the other two. 2. Find the coefficient of x in the expansion of (1+x2+x1) (2 − 3x − 2x2) − 1 -1 in a series of powers of x, and show that 87597365 nearly. 3. A bag contains 3 balls, which are equally likely to be either white, black, or red. A white ball is drawn and replaced, and then a black ball is drawn and replaced. Find the probability that the next drawing will give a red ball. |