10. Find a formula for the area of a curve referred to polar coordinates. + 11. Find a formula for the volume of a solid of revolution. Find the volume generated by the revolution about the axis of x of any portion of the ellipse 22 1. 62 = + a? PURE MATHEMATICS.-PART III. The Board of Examiners. Write essays on three of the following subjects : (i) Change of variables. (ii) Tangents and normals to plane curves. (iii) Definite integrals. (iv) Straight lines in space. (v) Generating lines of a conicoid. (vi) Ordinary differential equations of the first order but not of the first degree. (vii) Linear partial differential equations of the first order. MIXED MATHEMATICS.--PART I. The Board of Examiners. 1. Define power, and express a horsepower in c.9.5. units, taking a kilogramme as 2.2 pounds, and a foot as 30.5 centimetres. 2. Two particles of masses my, my are connected by an inextensible string of length l. Being placed close together they are acted on by forces F1, F: in opposite directions. Find their velocity immediately after the string becomes tight, neglecting gravity. 3. Investigate the acceleration of a particle moving uniformly in a circle. 4. Two equal particles of mass m are attached to two points B, C of an endless string of length 31 forming an equilateral triangle ABC in a vertical plane. The whole rotates under gravity about a vertical axis through A, with BC horizontal, making n revolutions a second. Find the tensions of the string. 5. Prove that two couples on a rigid body in the same plane and of equal and opposite moments, will balance. 6. A uniform stick of weight W is supported partly by a string attached at one end, and partly by a rough plane, inclined at an angle a, with which the other end is in contact, the string and stick lying in a vertical plane through a line of greatest slope with the string making an angle o with the inclined plane, and with the stick. Find the tension of the string and the least possible coefficient of friction, 7. Investigate the formula for finding the distance of the centre of mass of a system of particles from a plane. 8. A simple pulley sustains a weight W, and the rope which supports the pulley leaves the wheel at an angle a with the vertical. Find the tension of the rope, and examine separately the equilibrium of the wheel and the case. 9. Assuming the normality of fluid pressure, deduce its equality in all directions at the same point. 10. Find the least force required to open outwards a rectangular door on a vertical hinge to an exhausted vessel, when the top of the door is at a depth h below the surface of water, its breadth being b, and height l. 11. Volumes vi, v, of gas at pressures P1, P2, and equal temperatures are introduced into a vessel of volume V. Find the pressure of the mixture. 12. The top of a barometer tube contains air, so that when the atmospheric pressure is 30 inches it reads 29 inches, and 4 inches is unoccupied by the mercury: Find the reading when the pressure is 29.5 inches, assuming the level of the mercury in the cistern and also the temperature constant. MIXED MATHEMATICS.-PART II. The Board of Examiners. 1. Two particles are projected at the same instant from the same level and in the same vertical plane with velocities v, v and elevations a, a', the points of projection being at distance d. The resistance of the air being neglected, find the relations between these quantities that the particles may collide, and where the collision takes place. 2. Investigate the time of a small oscillation of a simple pendulum. 3. A particle of mass m hangs at one end of a light elastic string suspended from a fixed point, the string being double its unstretched length l. The string being now stretched to a length 41, the particle is let go. Find the time of a complete oscillation, assuming that when the length of the string is not greater than l it exerts no force on the particle. 4. Find an expression for the velocity in a planetary orbit, in terms of the distance from the centre of force, the major axis, and the strength of the centre. 5. AB, BC, CD are three equal heavy stiff rods, smoothly jointed at B and c. A, C and B, D are joined by light inelastic strings of length ✓ 3 times that of a rod, and the frame rests in a vertical plane with A, D on a smooth horizontal floor. Find the reactions at the joints and the tension of the strings. 6. Find necessary and sufficient conditions for the equilibrium of a given system of forces on a rigid body, 7. Summarize the chief results (i) in the theory of couples, (ii) in the reduction of a system of forces to simple equivalents. 8. A sphere of radius r whose centre of mass is at a distance a from its centre, rests on two rough planes, each making an angle 45° with the horizontal. Shew that when slipping is about to take place the line a makes an angle 0 with the vertical, where ✓2a sin 0=r sin 2, and a is the angle of friction. MIXED MATHEMATICS.-PART III. The Board of Examiners. 1. A particle moves in a plane under a central force varying as the distance from the centre, and a force in a fixed direction, varying as a simple harmonic function of the time. Find the complete integrals of the motion. a 2. In a planetary orbit find expressions for the radius vector and true anomaly in terms of the time correct to the second power of the eccentricity. |