For the curve x=a cos p, y=b sin ø, tan(z/c)=y/x, shew that the cosine of the angle between the osculating planes at points in the yz and ze planes is ab √ (a2+c2) √ (b2+c2)` ANALYTICAL STATICS AND DYNAMICS. HONOURS. 1. Two forces, acting along the sides OA, OB of a triangle OAB are represented in magnitude by mОA, nOB; prove that their resultant will pass through a point G in AB, such that m AG= GB, and will be represented by (m+n) OG. Three uniform rods of the same material and thickness are joined at their extremities so as to form a triangle ABC (sides a, b, c), which is then suspended from a point O by three strings OA, OB, OC; shew that their tensions are proportional to b+c, c+a, a+b respectively. 2. A number of weights are placed at given points whose distances form a given plane are known; find the distance of the centre of gravity from the same plane. When a given weights are placed at given points their centre of gravity is G. When the weight at every point is broken up into n-1 equal parts, and one of these parts is placed at each of the remaining -1 points, G' is the centre of gravity. Shew that G', O, G are in the same straight line, O being the centre of gravity of equal weights placed at the given points; and that GG'=nOG'. 3. Find the general conditions of equilibrium of a rigid body subjected to forces (X1, Y1, Z1), (X, Y, Z1⁄2), . . . acting at the points (X1, Y1, 1), (2, Y2, Za), 4. State the principle of virtual work, and sketch the line of argument you would use to establish it. Three equal weightless rods AB, AC, AD are hinged at A and connected by strings BC, CD, DB of equal length. B, C, D are placed on a smooth horizontal table so that the system forms a regular tetrahedron, and a given weight is suspended from A. Find the tension of the strings. 5. Find the equation to the common catenary. 6. Find expressions for the accelerations of a particle referred to rectangular axes in one plane, rotating with given uniform angular velocity round the origin. A particle is placed on a horizontal table, which is rotating with constant angular velocity w, round a vertical axis, and is subject to a retardation in the direction of the relative velocity, and proportional to it. Shew that its equations of motion, referred to axes fixed in the table are (D2+kD-w2)x-2wDy=0, (D2+kD—w2)y+2wDx=0; where D denotes differentiation with regard to the time. 7. Establish the equation referring to central orbits, is the angle that the tangent to a central orbit makes with the radius vector r, and the orbit is given by the equation ƒ (r, ø)=0. Prove that Apply this result to finding the law under which an ellipse is described about a centre of force at a focus. 8. Discuss the large oscillations of a simple pendulum, ¿.e. (i.) find the equation of motion; (ii.) effect a first integration. 9. A uniform rod swings as a conical pendulum, the point of support dividing the rod in the ratio of 1 to 3. Discuss the motion, and find the action at the point of support. 10. Explain D'Alembert's principle. In the case of a uniform rod sliding in a vertical plane in contact with a smooth vertical wall and a smooth horizontal plane, obtain expressions for the effective forces, and find the equation of motion. SPHERICAL TRIGONOMETRY AND ASTRONOMY. HONOURS. 1. Define polar triangles, and prove that if ABC is the polar triangle of A'B'C', then A'B'C' is the polar triangle of ABC. Prove also that AA', BB' and CC' meet in a point. 3. Prove geometrically that, if C is a right angle, (i.) sin a sin e sin A. (iii.) cos c=cos a cos b. Shew that the difference of the sides of a right-angled spherical triangle of given hypothenuse is least, when their sum is a quadrant. 4. Find the radius of the small circle circumscribing a spherical triangle. If O is the pole of this circle, prove that 5. Find the area of a spherical triangle in terms of its angles. π In a spherical triangle a=b and c, prove that the 6. Describe the principal errors of adjustment of the transit circle, and explain how a small level error existing alone may be measured and corrected. In latitude 45° the observed time of transit of a star in the equator is unaffected by the combined errors of level and deviation; prove that these errors must be nearly equal to each other. 7. Two stars, whose north polar distances are ô, and 2, and difference of right ascension 6 hours have the same azimuth 7 at a place whose co-latitude is p, shew that cot2+cos2=sin2p(cot2è,+cot2ĉ). k 8. Define the equation of time, and shew that it vanishes four times a year. If is the sun's longitude, and a the equation of time due to the obliquity w of the ecliptic alone, shew that cotx-cot 27-cot2cosec 21. 9. Explain how you would proceed to find the sidereal time, having given the mean time at any place. In finding the time of the sun's meridian passage from the 15 sint tant, where t is half the interval between the observations, A the sun's horary change in declination, the latitude of the place, and the declination of the sun at either observation. 10. Shew that the effect of aberration will be to make the stars, referred to the celestial sphere, describe small ellipses about their true places. Find expressions for the aberration of a star in latitude and in longitude, and shew that when they are equal sin 2-2 cot(0-1), where A is the latitude of the star and 7, O the longitudes of the star and the sun. 11. Shew how the hour lines are determined in a vertical east and west dial. INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS. HONOURS. Question 1 (i.), (ii.) and (iii.) are not to be done by Third Year Students. 4. Find the intrinsic equation to the semi-cubical parabola 6. Find the mean value of ordinates of an ellipse drawn at equal distances along the major axis between the centre and one extremity. FOR THIRD YEAR STUDENTS ONLY. 7. Shew how to integrate a non-homogeneous equation of the first degree in x, y. Solve (2x+3y+4) dx+(2x+3y+5)dy=0. 8. Having given that y=Ae+ Be" is the solution of -2y=0, shew how, by suitable choice of A and B |