20. A conical cup revolves with an angular velocity w about its axis, which is vertical; supposing its depth to be d, (greater than 2 w2 tan2 a' a being the semivertical angle of the cone); find the greatest amount of water which the cup will hold. 21. Two hollow cones filled with water, are connected together by a string attached to their vertices which passes over a fixed pulley; prove that, during the motion, if the weight of the cones be neglected, the total pressures on their bases will always be equal, whatever be the forms and dimensions of the cones. If the heights of the cones be h, h', and heights mh, nh, be unoccupied by water, the total normal pressures on the faces during the motion will be in the ratio n2 + n + 1 : m2 + m + 1. 22. A solid triangular prism, the faces of which include angles a, ß, y, is placed in any position entirely within an inelastic gravitating fluid; if P, Q, R, be the pressures on the three faces, which are respectively opposite to the angles a, ß, y, prove that Pcosec a +Qcosec B+ R cosec y is invariable so long as the depth of the centre of gravity of the prism is unchanged. 23. A heavy sphere is placed in a vertical cylinder, filled with atmospheric air which it exactly fits. Find the density of the air in the cylinder when the sphere is in a position of per manent rest. 24. Water is gently poured into a vessel of any form: prove that, when so much water has been poured in that the centre of gravity of the vessel and the water is in the lowest possible position, it will be in the surface of the water. OPTICS. REFLECTION AT A PLANE SURFACE. 1. SHEW that the path of a ray of light incident from one point and reflected by a plane surface to another, is shorter than it would have been according to any other law of reflection. 2. A ray of light is incident from a point A, and reflected at a given plane surface to another point B; supposing B fixed, find the locus of A when the whole length of the incident and reflected ray is constant. 3. A man 6 feet high stands before a vertical mirror of the same height, at a distance of 4 feet from the mirror; a lamp is placed behind him a height of 11 feet from the ground, and at a distance of 12 feet from the mirror; find what portion of his person is illuminated, and determine his distance from the mirror when his feet are just illuminated. 4. A ray is incident on a plane reflecting surface, shew that it is equally inclined to any straight line in the surface before and after reflection; and hence that a ray reflected by two intersecting mirrors is equally inclined to the line of their intersection before the first and after the second reflection. REFLECTION AT A SPHERICAL SURFACE. 1. A mirror collects solar rays to a point at a distance of 6 inches from it; where will be the image of an object placed in front of it at a distance of 12 feet? 2. In reflection at a spherical surface the conjugate foci lie on the same side of the principal focus. 3. The foci of incidence and reflection are on opposite sides of a concave mirror of given radius, and the focus of reflection twice as far from it as the focus of incidence. Determine their actual distances. 4. A luminous point is placed in the axis of a concave mirror of one foot radius, at the distance of 3 feet from it; find the focus of reflected rays. 5. Two parallel rays are incident on a spherical reflector at the same side of the axis; shew that the angle between the reflected rays is equal to twice the difference between the angles of incidence. 6. Find the distance of the point, to which rays diverging from a distance of 20 feet are made to converge by a concave mirror of two feet radius, from the principal focus of the mirror. 7. Given that the distance between the conjugate foci of a concave mirror is equal to the radius, find the focus of incidence. 8. Rays are incident upon a convex mirror of 3 feet radius from a distance of 16 feet, required the nature of the reflected pencil. 9. If any circle be drawn through any two conjugate foci, prove that in general two other conjugate foci will lie on the same circle. 10. If a small convergent pencil of rays be incident directly on a concave spherical mirror, and the convergence be estimated by the angle of the cone, prove that the convergence of the reflected is greater than that of the incident pencil by a constant quantity. COMBINED REFLECTIONS. 1. Find the total number of images formed, when a luminous point is situated symmetrically with respect to two plane mirrors inclined at an angle of 11° 15'. 2. Determine the arrangement of a luminous point and two mirrors inclined at an angle to each other, when the images are situated in the corners of a regular hexagon. 3. At what angle must two mirrors be inclined, so that a ray incident parallel to one of them may after reflection at each be parallel to the other? 4. If an object be placed between two parallel plane reflectors, which are moved parallel to themselves, their distance. remaining constant, shew that the images formed by an even number of reflections will remain stationary, and the other images will move in the same direction as the reflectors with twice their velocity. 5. A small pencil of rays diverges from a given point within a polished sphere, the axis of the pencil coinciding with a diameter; find the geometrical focus after two reflections. 6. A luminous point is equidistant from two plane parallel mirrors; find the path of the axis of the small pencil of rays, by which an eye placed in a given position between the mirrors, can see the third image proceeding from either side, and shew that its length is equal to the distance of the image from the eye. 7. BAD, BCE are two plane reflectors inclined at an angle of 15o. A is a given luminous point in one of them. Find at what angle a ray from A must be incident on the other reflector, in order that after three reflections it may be parallel to BA. 8. There are three plane reflectors, two of which are at right angles to each other, and a ray of light is incident upon the third, and reflected successively by each of them; it is required to shew that the angle between the first incident and last reflected rays is equal to twice the angle of incidence upon the first surface. 9. A luminous point is in the centre of an equilateral triangle; shew by considering the course of a ray parallel to one side, that the distance of the image from the luminous point for 2n reflections is na, and for 2n +1 reflections a (n2+n+1)*, a being a side of the triangle. 10. Four plane mirrors are all perpendicular to one plane; determine the position of a luminous point in that plane in order that its four images formed by one reflection at each mirror respectively may lie in a straight line. 11. A luminous point is situated at the middle point of the base of a hollow perfectly-reflecting vertical cylinder of very small radius, and a horizontal screen is held over it at a height above its upper end which is half as great again as the height of the cylinder. Prove that a series of alternately bright and dark rings is formed on the screen, the breadths of which are equal to the radius and diameter of the cylinder respectively. REFRACTION AT A PLANE SURFACE. 1. Find the thickness of a plane glass mirror, silvered at the back, that the distance of the image from the first surface may be twice as great as in a mirror of inconsiderable thickness. 2. At the bottom of an empty hemispherical basin a crown piece is placed, and an eye is so situated as just to see the edge of the crown piece over the rim of the basin. When the basin is filled with water the whole crown piece becomes visible. Find the radius of the basin. 3. Is it necessary to aim above or below in order to strike with a bullet a fish swimming in the water? 4. A glass plate of sensible thickness is in contact with the surface of still water; find how much the image of a point in the water will be elevated. Find also the area of the upper surface of the glass which is effective in transmitting rays. 5. A ray, passing through a point Q, is incident on a refracting plate, q is the intersection of the emergent ray produced backwards with the normal to the plate through Q: if G. P. 9 |