8. A string can just support the weight of 100 lbs. without breaking. A body of 1 lb. is attached to one end of the string a yard long, and made to revolve uniformly about the other end as centre; find the maximum velocity which can be given to the body. Find also the kinetic energy in foot-pounds of the body when revolving at its maximum speed. 9. Detine simple harmonic motion and prove that the motion of a spiral spring stretched by the weight of a body attached to it and made to oscillate is S. H. M., if the mass of the spring may be neglected. Find also the time of oscillation. 10. Define the moment of inertia of any body about an axis. Find that of a rectangle about an edge and thence deduce that of a triangle about an axis coinciding with one of the sides. 11. The right ascension and north polar distance of a star being given, find its latitude and longitude. 12. The mass of the earth being taken as unity, find that of the sun. SECOND PAPER. 1. Define the centres of oscillation and percussion of a rigid body, and shew that they coincide. 2. To find the law of thickness of a vessel of cylindrical bore, in order that when filled with fluid of uniform density, with its axis vertical, the tendency to burst, as measured by the tension at every point, may be the same throughout. 3. A cone with its axis vertical and base downwards is filled with fluid; find the normal pressure on the curved surface, and compare it with the weight of the fluid. 4. Define the limiting ratio of variable quantities, and the prime ratio of nascent quantities. Find the limit of +3÷ 1 + 3x, (1) when a becomes gradually smaller and ultimately vanishes, (2) when becomes indefinitely large. 5. Find, according to Newtonian methods, the area enclosed by the arc of a parabola, a diameter, and an ordinate. 6. Find also the volume of a paraboloid of revolution. 7. Find also the centre of mass of a parabolid of revolution. 8. Define curvature and find the radius and chord of curvature through the pole at any point of an equianangular spiral. 9. Every body which moves in any curve line described in a plane, and describes areas proportional to the times of describing them about a point either fixed or moving uniformly in a straight line, by radii drawn to that point, is acted on by a centripetal force tending to the same point. 10. If a body is moving in a parabola under the action of a force towards the focus; find the law of force. 11. At a point in a horizontal dial a style OA is fixed so that its angle of inclination AOB is equal to the latitude 44° 13′ 26′′, and the direction OB is due north; what angle will OB make with the shadow of OA in the horizontal plane a 1 p.m. 12. Find the geocentric latitude y', and the radius of the earth, at the geographical latitude 4. |