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217. A second's pendulum is carried to the top of a mountain, and there loses 48" 6 in a day; determine the height of the mountain, supposing the earth's radius to be 4000 miles.
218. What is the numerical value of the measure of the 1832 force of gravity ? and by what experiments is it determined ?
219. When a body is accelerated in a straight line by a uniform force, and sets off with a given velocity, find the velocity acquired and the space described in a given time.
220. Two weights are connected by a string passing over a fixed pulley, and slide along two inclined planes ; shew that the velocities of the weights at any time are inversely as the cosines of the angles which their strings make with the planes.
221. A body acted on by gravity moves on the convex surface of a cycloid, the velocity at the highest point being V2gh; determine the point where it will leave the curve, and the latus rectum of the parabola afterwards described.
222. A and B are two points on opposite sides of a plane curve, and a body goes from A to B in the least time possible, its velocity on the two sides of the curve being in the ratio of
Shew that the path of the body cuts the curve in angles whose cosines are in the same ratio.
223. A uniform rectangular parallelopiped is supported on two linges placed on one of its edges which is vertical ; determine the magnitude and direction of the pressure on the hinges. Also if the body be struck by a given horizontal force perpendicularly to one of its faces at a given point, determine the angular velocity communicated to it, and the consequent pressure on the hinges.
224. State the second law of motion, and the experiments by which it is established.
225. What is the meaning of the terms velocity and accelerating force in variable motion ? Investigate the equation
dt 226. In a system consisting of any number of points in the same plane moveable about an axis perpendicular to that plane, a force P acts to turn the system ; apply D'Alembert's principle to find the effective accelerating force on any point.
227. If a pendulum oscillating in a small circular arc be acted upon, in addition to the force of gravity, by a small horizontal force (as the attraction of a mountain) in the plane in which it oscillates, having given the number of seconds gained in a day, find the horizontal force.
228. A cylinder descends vertically by unrolling itself from a string the end of which is fixed; and at any point of its descent a weight suspended by a string is attached to the cylinder at the point where the former string is a tangent; shew that this weight will not alter the motion of the cylinder.
229. If a uniform straight rod oscillate about one extremity through a small angle, in a medium of very small density, of which the resistance varies as the square of the velocity, find the difference between the angles described on opposite sides of the vertical; and shew that it varies as the length of the rod when its thickness and density are given.
230. A door oper.ed through a given angle is to be shut by means of a weight attached to a string passing over a pulley and acting horizontally on the door, the pulley being in the post against which the door shuts; to determine the motion.
231. Of all right cones having the same volume, determine that which will oscillate about an axis through its vertex and perpendicular to the axis of the figure in the least time possible.
232. Find the time of an oscillation in the cycloid, when the resistance varies as the velocity.
233. Define the moment of inertia, and that of a globe revolving round its axis.
234. State the properties of the principal axes of rotation, and find the moment of inertia about any axis in terms of the moments about the principal axes.
235. A small weight is attached to an uniform rod which oscillates about a given point in it; determine the simple pendulum, and shew that there are two points, one above, another below the point of suspension, where a small change in the position of the weight does not affect the length of the simple pendulum.
236. Find the equation to the curve described by a projectile acted on by gravity; and determine its greatest distance from a plane passing through the point of projection.
What laws of motion are employed in this investigation, and in what manner?
237. Find the centre of oscillation of any body.
238. Two bodies whose common elasticity is e, moving with 1833 given velocities, impinge directly on each other; determine their velocities after impact.
239. Compare the space described by a projectile in the direction of projection, with its vertical fall in the same time. If the velocity be given, determine the angle of projection, that the focus may lie in the horizontal line through the point of projection.
240. Define accelerating force, and state how it is measured. Describe Atwood's machine, with the nature and objects of the experiments made by it.
241. Find the time of oscillation in a small circular arc.
242. Having given the moment of inertia of any body about an axis through its centre of gravity, find the moment of inertia about an axis parallel to the former, at a known distance from it. Also having given the moment of inertia of a plane figure about each of two axes in it, at right angles to one another, find its moment about an axis through their intersection perpendicular to them.
243. A given plane area revolves with an uniform angular velocity about a horizontal axis fixed in its own plane and at a given altitude above a horizontal plane. Determine the point at which a sphere of given mass should be opposed to its impact that it may be projected to the greatest possible distance on the horizontal plane.
244. A pendulum is observed to make n vibrations in a certain time at a place of known latitude, and, by calculation from an assumed approximate value of the earth's ellipticity, the number of vibrations in the same time performed by the same pendulum, at the place of observation and the equator,
ought to be n' and N respectively. Shew that a nearer value of the ellipticity is obtained by multiplying the assumed one by
n' - N 245. An inflexible straight rod is set in motion round a vertical axis passing through one extremity, about which it is capable of revolving freely in a horizontal plane. Determine the motion of a ring sliding freely along it; prove that the whole vis viva of the system is constant.
246. A groove in form of a cycloid, with its vertex downwards and base horizontal, is cut in a solid vertical plane ; determine the time of oscillation of a heavy body moving along it, while the vertical plane itself is capable of moving freely along a smooth horizontal plane, and the curve which the body describes in space.
247. From two points in the same vertical line, and at given distances from a fixed horizontal plane, two equal elastic balls are dropped ; determine the successive points of meeting, and the times in which each will return to its original position.
248. If on a rough horizontal plane revolving uniformly about a vertical axis a rough sphere be placed; determine its initial motions, and shew that its path in space will be a circle.
249. A particle attracted to two centres of force varying inversely as the square of the distance, will oscillate in the arc of an hyperbola of which they are the foci, supposing it to have been originally at rest in such a position as to be attracted equally by each.
250. A semi-cycloid is placed with its axis vertical and vertex downwards, and from different points in it a number of heavy bodies are let fall at the same instant, each moving down the tangent at the point from which it sets out; prove that they will reach the involute all at the same instant.
251. A rocket ascends vertically in a medium of which the resistance varies as the velocity; the inflammable composition contained in it being supposed to produce a constant moving force, and to be exhausted in n at an uniform rate of consumption; determine the height to which the rocket will ascend.
252. Determine the motion of two heavy particles connected by an inflexible rod without weight, one of which moves on a surface of revolution, and the other is constrained to move in the axis of the surface which is vertical. Find the velocity of the particle on the surface, when the other continues stationary.
253. A body acted on by gravity descends from rest down a given circular arc, the tangent to which at the lowest point is horizontal; compare the initial accelerating force with that down the chord.
254. Find the centre of spontaneous rotation when a body acted on by no forces is struck by another, and determine its path, neglecting the inertia of the striking body.
255. State the principle of least action, and apply it to find the path described by a projectile when acted on by gravity.
256. Having given the centres of gravity and oscillation of any number of bodies revolving round a common axis, determine the centre of oscillation of the system.
257. When a body is moveable about a fixed axis, define the centre of percussion, and investigate the coordinates which determine its position. Shew that it coincides with the centre of oscillation, when the axis of rotation is parallel to one of the principal axes through the centre of gravity.
258. When a vibration is propagated along a cord fixed at one end, prove that the reflected wave returns with a velocity equal to that of the incident, but at the opposite side of the axis.
259. Determine the time in which a cylindrical body rolls down a given inclined plane.
260. A body moves on any curved surface acted on by any forces ; to find the pressure.
261. Explain how velocity and accelerating force are mea- 1834 sured. If a heavy body, in vacuo, fall from rest through 1670 feet in 1", determine the force of gravity.
262. Determine the path of a heavy body projected obliquely, and shew distinctly at what points of the investigation the 1st and 2nd laws of motion are applied.