300. Having given the moments of inertia of a body round three rectangular axes, find its moments of inertia round any other axis passing through the origin; and shew that the sum of the moments round any three rectangular axes passing through the same point, is constant. 301. Determine the centre of oscillation of a pendulum consisting of a uniform rod with a weight attached to it, and the effect produced on the time of an oscillation by a small change in the position of the weight. 302. A body descending vertically draws an equal body 25 feet in 2 seconds up a plane inclined at 30° to the horizon, by means of a string passing over a pulley at the top of the plane; determine the force of gravity. 303. State the distinction between accelerating force and 1836 moving force. If a body be projected vertically upwards or downwards, shew that, when it has described a given space, its velocity the velocity of projection that which gravity would generate in the time the body has been in motion. Why may not the last term of this equation be the velocity which gravity would generate in a body falling from rest through the same space? = 304. Two bodies whose common elasticity is e, moving with given velocities, impinge directly on each other; determine their actual and relative velocities after impact. 305. The path of a projectile in vacuo is a parabola with its axis vertical; and the velocity at any point is that acquired in falling from the directrix. 306. If a body oscillate in a circular arc, the accelerating force varies as the sine of its angular distance from the lowest point; prove this, and find the time of a small oscillation. 307. Define the centre of oscillation of a rigid system, and find its position. Two heavy particles connected by a rod without weight are suspended at a given point in the rod; find the time of a small oscillation. 308. A weight P descending vertically draws Q up an inclined plane by means of a string passing over a pulley fixed above the plane; and it is observed, that at the instant when the two parts of the string between the pulley and the weights become parallel, Q rises off the plane. Compare P and Q, having given the positions of the pulley and plane, and the length of the string. 309. A rotatory motion about its axis is communicated to an elliptic cylinder, which is then suddenly laid lengthwise upon a perfectly smooth horizontal plane; determine the subsequent angular motion. Does your solution hold good for all degrees of original velocity? What must be the original angular velocity that the body may assume a position of permanent rest? 310. A uniform chain hangs vertically, its lower end just resting on the earth's surface at the equator; determine its length that it may hang in equilibrium without any fixed support. What is the greatest tension it has to sustain? If the chain were removed so that its lower end should rest on the earth in a given latitude, what would be the form of equilibrium? While hanging in the first position, if an indefinitely small downward motion be given to it, required the velocity with which the last link will strike the ground. (In this and the following question the earth's ellipticity to be considered insensible.) 311. The vertical at any point being defined to coincide with the direction in which a heavy body would begin to descend if let fall; required the nature of the curve which is vertical at all points through which it passes. 312. A body in motion is deflected at equal intervals of time by equal impulses in parallel directions; determine its velocity and direction of motion after any time. Apply the result to find the motion of a body continually deflected by a constant force acting in parallel lines. 313. A uniform heavy rod, moveable freely about a fixed fulcrum, is connected by a hinge at the end of its shorter arm with another, the farther extremity of which slides along a smooth vertical plane; determine the motion, when it takes place in a vertical plane perpendicular to the given one. 314. Shew that the motion of a body, acted on by no forces, about its centre of gravity may be imitated mechanically in the following manner: Describe an ellipsoid, rigidly connected with the body whose equation is Ax2 + By2 + Cz2 = h2, the origin being the centre of gravity, and A, B, C the moments of inertia about the principal axes of the body which are also the coordinate axes, and h a quantity determinable from the initial motion; at the point where this ellipsoid was intersected by the axis of rotation at the commencement of the motion draw a tangent plane, which let remain fixed in space; and upon this plane let the ellipsoid be rolled, so that the angular velocity of the body about a line from the origin perpendicular to the plane may be uniform. 315. Compare the times of titubation of an elliptical cylinder on each of two horizontal planes, of which one is smooth and the other rough. Explain the result when the body degenerates into a circular cylinder. 316. Two bodies are let fall from the same point at an interval of one second, how many feet are they apart at the end of one minute from the fall of the first? 317. When a body moves on a surface of revolution, the force at any point acting in a plane through the axis, its projection on a plane perpendicular to the axis describes areas proportional to the times about the intersection of the axis with the same plane. Shew also how to find the path upon the surface. 318. Enunciate and prove the principle of the conservation of the motions of translation and rotation. 319. Given one principal axis of a body, find the other two. 320. A heavy body is projected vertically upwards in a medium of which the resistance = kv2; determine its velocity when it again arrives at the point of projection. 321. The moment of inertia of a body about an axis through its centre of gravity is less than that about any parallel axis. Find the moment of inertia of a triangle about an axis perpendicular to its plane through its centre of gravity. 322. Find the direction in which an imperfectly elastic ball must be projected from a given point, so that after reflection at a given plane, it may strike another given point. 1821 SECTION XVI. QUESTIONS ON PLANE ASTRONOMY. 1. Why does the apparent distance of two fixed stars increase as they approach the horizon? 2. Shew that the hour of sunrise and sunset together = 12 hours nearly; and find the correction necessary if the sun's declination should have changed by a given quantity. 3. In latitude 45°, find the time of sunrise on the longest day. 4. Explain the method of determining accurately the obliquity of the ecliptic: to what inequalities is it subject? and from what causes do they arise? 5. Given the focal length and aperture of a Herschelian telescope; during what time will the image of a given star be visible in the tube? 6. Explain fully what is meant by the term mean in Astronomy. 7. Explain what is meant by the equation of time, and from what causes it arises: at what times of the year is it nothing, and at what times is its negative and positive value respectively a maximum? 8. Supposing y Draconis to be affected by a sensible annual parallax, in what manner will its apparent place be affected both by aberration and parallax on March 20, June 21, September 23, and December 23, its right ascension being 18 hours nearly? 9. The level of a transit instrument is suspended by hooks from its axis; in what manner are the errors which arise from the axis not being horizontal, or from the level not being parallel to the axis, distinguished from each other, and by what adjustments are they corrected? 10. Explain the method of determining accurately, when the first point of Aries is upon the meridian. 11. Explain the method of determining the sun's meridian altitude by means of a sextant: 1st, on the open sea; 2nd, when the sun's altitude is not less than 60°, but a neighbouring coast is on the same side of the ship with the sun; 3dly, on land. 12. A degree of latitude in latitude 45°, is nearly an arithmetic mean between a degree at the equator and the pole. 13. What are the principal phenomena presented by the sun and earth in the course of a month, to a lunar observer? 14. Explain the mode of constructing a catalogue of the fixed stars. 15. If two straight lines intersect each other in a circle, the sum of the arcs cut off between their extremities is the same as that cut off by any two lines respectively parallel to them, and intersecting each other within the circle. Prove this property, and shew its use in correcting observations made with circular instruments inaccurately centered. 16. When is the planet Venus stationary, and when retrograde? 17. Explain the phases of the earth as they would be seen from the moon. 18. Explain fully the method of finding the longitude at sea by the observed distance between the moon and a star and their altitudes. 19. On board a ship in north latitude, Jupiter is observed on the meridian at 3h. 4m, 56s, and his corrected altitude is 29° 6' 42". One of his satellites is at the same instant eclipsed. His tabulated declination is 5° 4' 35" north, and the tabulated time of the eclipse 7h. 0m. 32s. Required the latitude and longitude of the ship. |