7. (a) If f) is holomorphic within and on a simple contour C, show 'hat Sf(z)dz = с (b) If f(z) is a rational integral function and if C encloses all the poles of f(z), evaluate f" (a) = (b) Evaluate 8. (a) If f(z) is holomorphic within and on C and a is within C, show that 0. Sf(z)dz. с n!S f(z) 2πi3 (z − a)n+1 COS X (x2 + 1)(x2+4) I 22-32+2 dz. dx. 9. (a) If f(z) is holomorphic in a ring about the point a, develop an expansion for ƒ(z) at any point within the ring. Show that the series converges and state its character as a is (1) a multiple pole of order n, (2) an essential singularity. (b) Expand in the three regions (1) |≈|<1, (2) 1<|≈|<2, (3) |≈|>2. 10. (a) What is meant by Newton's Parallelogram? (b) Given 0=w5+w3z+2w2x2+wz3+2°+3w*x+3w3x2+2w2z3 +higher degree terms, find the forms of the expansions for w in terms of and compute the first two terms of each expansion. PRELIMINARY HONOUR PHYSICS. Mechanics. 1. (a) Find the vector equation of a straight line passing through the termini of two given vectors. 2. (b) Prove by vector methods that the diagonals of a parallelogram meet in a point which bisects them. 2. A stream whose velocity varies as the distance from the nearer shore is crossed in a boat by a man whose velocity is 5 miles per hour. If width of stream is a, find the equation of the path he takes between the shore and mid-stream. 3. (a) Prove that a force may be measured by the space rate at which it changes the kinetic energy of a body. (b) A particle moves in a straight line from a distance a towards a centre to which it is attracted with a force varying as the cube of the distance. Find the time it takes the particle to reach the centre. 4. (a) A particle is subjected to two simple harmonic motions in directions perpendicular to each other, of unequal amplitude, with periods in ratio 1 : 2, and phase difference zero. Find the equation of the resulting path. (b) Obtain the equation of motion for a particle subject to a damped S. H. M. State (without solving) the two solutions, and give an illustration of each from any branch of Physics. 5. (a) What is your test for a conservative system of forces? (b) You are given the value of the potential at a point due to any configuration of bodies. Derive the mathematical conditions of equilibrium of a particle placed at this point, and show how to distinguish between the various kinds of equilibrium. 6. (a) Derive from fundamental considerations the general equations of motion when a body is acted on by impulsive forces. State each of these in words. (b) Apply these equations to find the direction of a blow struck by a body capable of rotation about a fixed axis, so that no pressure results on the axis. Find also the relation between the magnitude of the blow and the resulting angular velocity. 7. A cone of height h and radius R with apex pointing downwards is capable of rotation about a horizontal axis coinciding with a diameter of its base. Find (a) The length of the equivalent simple pendulum. (b) The angular velocity if it falls from the position of unstable equilibrium with reference to this axis, (1) using the principle of energy, (2) using the fundamental equation of motion. FACULTY OF ARTS. PRELIMINARY HONOUR PHYSICS. Laboratory Paper. 1. Write a report on the "Height by Aneroid and Sextant" exercise, using your own observations as recorded in the "data book" pages supplied you. 2. (a) Explain the graphical method of deducing the focal length of a lens from a series of "object and image" observations. (b) What advantage has the graphical method over that of averaging the values obtained from the individual pairs of readings? 3. (a) Explain the difference in slope of the two curves on the plot for the determination of g by Kater's pendulum. (b) If the knife edges are symmetrically placed with regard to the centre of gravity the pendulum will obviously swing about either with the same period. Under what conditions do these points become the points required in the Kater's pendulum experiment? (c) What relation must obtain as to relative percentage error in the observation of length and of time so that they have equal influence on the error of the result? 4. (a) You are provided with a galvanometer, two 10,000 ohm standard resistance boxes, a straight uniform wire attached to a graduated scale 100 cm. long, a standard Weston cell, a storage battery and a high resistance. Describe all the steps you would take to calibrate a thermocouple over a given range of temperature. (b) If you were also provided with a standard resistance capable of carrying currents as high as 5 amperes, how would you use the above apparatus to calibrate an ammeter? 5. A spectrometer was used to take the following set of readings dealing with the dispersion of two glass prisms and the first order spectrum of a transmission grating: Prism I Prism II Grating 47° 45' 38° 41' 22° 18' 48° 11' 38° 56' 19° 54' 49° 20' 39° 25' 16° 21' 50° 16' 39° 53' 14° 32' 47° 20' 38° 20' 16° 20' Wave-length .0000667 .0000589 .0000495 .0000441 Unknown (a) Record these results graphically, to the most appropriate scale, and explain any differences obtained in the resulting curves. (b) Obtain graphically the best value for the unknown wave-length. (c) If you were anxious to have spectral lines as intense as possible, would you use a prism or a transmission grating? Give reasons. 6. A millivoltmeter has a resistance of 0.800 volts and scale reads to 10 millivolts. Find, (a) The resistance of the shunt box to go along with it to enable you to use it as a milliammeter with a range up to 1 ampere. (b) The arrangement necessary to enable you to use. it as a voltmeter reading up to 100 volts. |