(-4) 9. Ex. 692. If a body slides down a gentle incline of 1 foot vertical to m horizontal, show that the acceleration very nearly equals And if m=100 show that the error equals about 이이이이 part of the whole. 1 20000 m μ Ex. 693. A train moving at the rate of 24 miles an hour comes to the top of an incline of 1 foot in 350; the resistances are 8 lbs. per ton; the steam is cut off at the top of the incline, and the train comes to rest at its foot; determine-(1) the retardation of the train's velocity; (2) the length of the incline; (3) the time of motion. Ans. (1) 175. (2) 27,104 ft. (3) 1540 sec. Ex. 694. At the slide at Alpnach the first declivity has an inclination of 22° 30′ and is 500 feet long; being kept continually wet the limiting angle of resistance is 14°; in how many seconds would a tree descend this first declivity were it not for the resistance of the air ? Ans. 14.3 sec. Ex. 695. A body slides from rest down a plane whose inclination is 1 and length L; it passes with the velocity acquired during the descent of the first plane to a second whose inclination i is less than the limiting angle of resistance ; if l is the space through which it slides before coming to rest show that Ex. 696. A body weighs w lbs.; it is pulled up an inclined plane by a force P that acts parallel to the plane, show that the accelerative effect g, where a is the angle of inclination and the equals( sin (a + 4) cos limiting angle of resistance. Ex. 697. Let AC, CB be two planes sloping downward in contrary directions from the point c, and inclined to the horizon at angles A and B respectively; a weight P slides down ca and draws a weight Q ир св by means of a fine cord which passses over c and is tied to each weight; if the limiting angle of resistance between the weights and the planes is 4, show that Ex. 698. In the last case if the inclines are equal and small, being 1 in m, show that Ex. 699. If the resistances are 8 lbs. per ton and the incline 1 in 140, and a set of full trucks is required in their descent to pull up the incline an equal number of similar empty trucks, show that the contents of each truck should on the average be more than double the weight of the truck. Ex. 700. If a circle be placed with its plane vertical, and through its highest point any chord be drawn, a body will descend along that chord (supposed to be smooth) in the same time as down the vertical diameter. Ex. 701. If through any point there is drawn a vertical line and any number of inclined planes on the same side of the line, and having a common limiting angle of resistance ; then if bodies begin to slide from the point down these planes at the same instant, show that after any interval they will be found in the arc of the segment of a vertical circle cut off by the vertical line which subtends at the centre an angle equal to - 2 ф. 130. The Work accumulated in a Moving Body. The following Examples depend on the principle proved in Art. 119. Ex. 702. A train moving at the rate of 15 miles an hour comes to the foot of an incline of 1 in 300, resistances 8 lbs. per ton; if the steam is cut off, how far will it go before stopping? Ans. 1095 ft. [If w is the weight of the train in lbs. the number of units of work accumulated in it is w (22)2 -; now if l is the horizontal length of the plane 2g the units of work required to draw w over this length is by Example 522 ...1= the number of feet required. wl 1 1) 280 300 + 222 x 300 x 280 580 x 2g The same answer can be obtained by the principle exemplified in the last Article.] Ex. 703. A body slides down an inclined plane the height of which is 12 feet and length of base 20 feet; find how far it will slide along a horizontal plane at the bottom, supposing the coefficient of friction on both planes to be, and that it passes from one plane to the other without loss of velocity. Ans. 52 ft. [From Ex. 522 it appears that the body arrives at the bottom of the plane with a number of units of work accumulated in it equal to w {12-201 Ex. 704. If the velocity of a moving body (whose weight is w lbs.) changes from v tov, show that the number of units of work accumulated Ex. 705. A train weighing 90 tons comes to the foot of an incline of 1 in 160 with a velocity of 30 miles an hour; the resistances are 7 lbs. per ton, the length of the incline 2 miles; the train has at the top of the incline a velocity of 20 miles an hour; how many units of work have been done by the steam in getting the train up the incline? and through how great a S distance would an expenditure of the same number of units have taken the train with a uniform velocity along a horizontal line? Ans. (1) 16,570,400 units. (2) 26,302 ft. Ex. 706. If a train begins to descend the incline in the last Example with a velocity of 20 miles an hour, how far will it descend by its own weight before acquiring a velocity of 30 miles an hour? Ans. 5378 ft. Ex. 707. There are two points A and B on a railroad 4 miles apart on the same horizontal line; the railroad is in two equal inclines, one up and the other down, of 1 in 160; the train, which weighs 50 tons and experiences resistances equal to 7 lbs. per ton, has a velocity of 30 miles an hour at A and B, and a velocity of 20 miles an hour at the top of the incline; the velocity being supposed to change uniformly from 30 to 20 and again from 20 to 30, and when the latter velocity is attained further acceleration is checked by putting on the break; determine-(1) the loss in units of work in consequence of the inclines; (2) the loss of time in consequence of the inclines. Ans. (1) 1,810,000 units. (2) 72 sec. Éx. 708. A chest 6 feet long and 2 feet square stands on its end on the deck of a ship, one face being perpendicular to the direction of the motion; the ship is suddenly brought to rest; what must have been its velocity if the chest is just overthrown, it being supposed that all sliding is prevented ? Ans. 2.2 miles per hour. W [If w is the weight and v the required velocity, the number of units of work accumulated in it must be v2; and to overthrow the chest re2g quires W (103) units of work.] Ex. 709. Show from the principles of the present Article that the velocity acquired by the bodies in Ex. 697 while moving from rest over a length I of the planes is given by the formula Ex. 710. There is an inclined plane of 1 in 90 along which a train weighing 80 tons is made to descend for a distance of 300 feet; to the train is attached a rope which, after passing round a pulley at the top of the incline, is fastened by the other end to a lighter train weighing 16 tons; the rope is so long that the light train is at the foot of the incline when the heavy one is at the top; find (1) the velocity with which the heavy train reaches the foot of the incline; (2) if the heavy train is disconnected from the light one at the foot of the incline, find the distance to which it will run before stopping on the horizontal plane, resistances on the incline being 7 lbs. per ton, on the level 8 lbs. per ton. Ans. (1) 9.07 ft. per sec. (2) 359.7 ft. 131. Mass, density, momentum. - If a body when placed in one pan of a perfectly just balance exactly counterpoises a second body when placed in the opposite pan, the bodies are said to contain equal quantities of matter. If the bodies were both lead they would contain equal quantities of lead, if both platinum they would contain equal quantities of platinum; if one were lead and the other platinum, we can say that they contain equal quantities of matter the word matter being a general term, denoting lead, platinum, wood, stone, water, air, &c. When any two bodies contain equal quantities of matter they possess certain qualities in common, thus : -if they are successively placed in the same relative position to a third body, the mutual attractions between them severally and the third body will be equal; e.g. if the third body is the earth, this is the fact indicated by their counterpoising each other in a perfectly just balance. Again, if they both move with equal uniform velocities of translation, they will have accumulated in them the same number of units of work. If they move with equal uniform velocities in equal circles they must be acted on by equal forces tending to the centres of those circles. Other physical properties might be named which they have in common. It is on account of all bodies possessing these and other properties in common that there is need for the use of the term matter in the science of mechanics. The quantity of matter in a body is called its mass. Now, any substance which is an exact counterpoise to the standard pound contains as much matter as the standard pound. We may therefore take the standard pound as the unit of mass, and may speak of a body whose mass is three, four, or any number of pounds. If this is done, it must be borne in mind that we are using the word pound in a different sense from that in which it denotes a unit of force the sense in which we have hitherto used it. That the senses are really different is evident on a little consideration, thus: - Suppose a perfectly just balance to be in equilibrium with a standard pound in one pan and an equal mass in the other, the equilibrium would be maintained if the force of gravity were altered by any amount, provided it continued to act equally on both bodies and along the same vertical lines. But if the force of gravity were altered, the pressure exerted by the standard pound on a horizontal plane supporting it would no longer be the same, and for this reason a particular place is mentioned in defining the unit of force, viz., the mutual attraction in London between the earth and a pound of matter is a pound of force.* (Art. 23.) The definition of mass leads to the following definitions : Def. A body is said to be of uniform density when any equal volumes taken from any parts of the body contain equal quantities of matter. Def. When a body has a uniform density, the quantity of matter in the unit of volume is called its density. Def. The momentum of a body is the product of its mass and its velocity. It will be remarked that the momentum of the body is referred to its mass, and not to its weight. The momentum of a moving body does not depend on the force of gravity, e.g. a cubic inch of lead moving with a given velocity would strike the same blow whether the accelerative effect of gravity were 32-1912 or had any other value; that is to say, its momentum must not be made to depend on the weight, which varies with the force of gravity (g), but on the mass which is irrespective of g. 132. The Third Law of Motion, and the Unit of Force. -The third law of motion is this :- When a force acts on * The variations in the weight of a given body-the variations in the mutual sensible attraction between the earth and that body-at different points on the earth's surface, could be observed directly by means of a delicate spring. Herschel's Outlines of Astronomy, Art. 234. (See Ex. 715.) |