20. In the last Problem, supposing the pressure of the air at first to be 14lbs. on the square inch, what is the pressure on a square inch of the inner surface of the Receiver after 4 strokes of the piston?

By Art. 81, the elastic force of air varies as its Density; therefore, in this case, since the density is about one-half its original density, the pressure will be about one-half the original pressure, that is, about 7 lbs. on the square inch.

21. Find the pressure of the air in the Receiver of a Condenser after 10 descents of the piston, the content of the Receiver being 10 times that of the Barrel.

It is plain, that by 10 descents of the piston, the quantity of air in the receiver is exactly doubled; therefore its density is doubled, and the elastic force, or pressure, will likewise be doubled. So that, if the instrument was filled with common air at first, of which the pressure is about 14 lbs. on the square inch, the pressure of the condensed air will be about 28 lbs. on the square inch.

22. If the atmospheric pressure be 14 lbs. on the square inch, when the Barometer stands at 28 inches, what will it be, when the Barometer stands at 30 inches?

By Art. 87, "the pressure of the atmosphere is accurately measured by the weight of the column of mercury in the Barometer”. And in the same Barometer the weight of the column of Mercury will be proportional to its height. Therefore here,

atmospheric pressure in 1st case: atmospheric pressure in 2nd, :: 28: 30,

:: 14 : 15,

or 14 lbs. required pressure :: 14 : 15,

.. required pressure = 15 lbs.

23. The height of the mercurial Barometer being 30 inches, required the height of a Barometer, of which the column above the cistern contains equal weights of mercury (s. G. 136), and of proof spirit (s. G. 0.93).

In any Barometer the weight of the column in the vertical tube is equal to the atmospheric pressure, which in this case is measured by 30 inches of mercury. In the compound Barometer in question,

half the column, in weight, is mercury; and therefore its height is 15 in.

The length of the column of proof spirit is thus found:The weight of the column being equal to that of 15 inches of mercury, and the thickness of the column the same, since W=MS

1

always, and here M the height of the column

height of proof spirit: 15 in. :: 13'6 0'93,

.. height of Barometer=15+21911=23411 inches.

24. Two bodies of different Specific Gravities balance in a common pair of scales, when the Barometer stands at 28 inches. Will they balance also, when the Barometer stands at 30 inches? If not, which will preponderate?

The two bodies, being of different s. G., but equal in weight by the first trial, must be unequal in bulk; and the greater will be that which has the lesser s. G.

Let M be the number of cubic inches in larger body,

m

.... smaller ...

S the s. G. of air, when barometer stands at 30 in.,

In either case the weight lost by the body is the weight of an equal bulk of air. The larger body loses MS in one case, and Ms in the other, the difference being therefore M(S-s). The smaller body loses ms in one case, and ms in the other, the difference being m(S-s). And since M is greater than m, the greater body suffers the greater loss of weight by being transferred to air of greater density. Therefore the lesser body will preponderate and the bodies will not balance.

25. A Condenser at first is full of air, the same as that of the surrounding atmosphere; and the content of the Barrel is one-tenth that of the Receiver. From a flaw in its construction, the Receiver will only sustain a pressure equal to three-fourths that of the atmosphere. During what descent of the piston, and at what part of the descent, will the Receiver burst?

Let B, and 10B, be the contents of Barrel, and Receiver, respectively;

P the pressure, and D the density, of the atmosphere,

P. the pressure, and D, the density, of the air in the Receiver after a descents of the piston. Then,

:. 10B×D+x×B×D=mass of air in receiver after a descents,

3P, or

7

as soon as

74

P>

.P, or 70>40+4x, or x>7, that is, in the

the Condenser will burst, when P, exceeds P+P, or

10+x
10

middle of the eighth descent.

P, that is,

26. Shew that 95o of Fahrenheit's Thermometer denotes the same temperature as 35° of the Centigrade.

where C and F are the numbers of degrees on the Centigrade, and Fahrenheit, Thermometer, respectively, for the same temperature.

27. The sum of the numbers of degrees indicating the same temperature on the Centigrade, and Fahrenheit, Thermometer being equal to 0, find the number of degrees on each.

Here C+F=0, by the question, .. C= - F.

EXAMPLES AND PROBLEMS,

WITH ANSWERS*.

i. DENSITY, AND FORCE. (Arts. 8...16.)

(1) Two bodies, each of uniform density, are of the same size. Shew that by weighing them it may be determined whether they are of the same density.

(2) Eight cubic inches of a substance weigh 5 lbs., and one cubic inch of another substance weighs half a pound. Compare the densities of the substances.

(3) A cubic foot of a substance weighs 5 cwt., and a cubic foot of another substance weighs 14 qrs. Compare the density of the latter substance with that of the former.

(4) Half a cubic foot of water cubic foot of zinc weighs 7000 oz. and zinc.

weighs 500 oz. avoirdupois, and a Compare the densities of water

(5) A quart of water weighs 2lbs., and fifty gallons of proof spirit weigh 372 lbs. Compare the density of proof spirit with that of water.

(6) A rectangular log of wood is 16 feet long, 2 feet wide, 14 feet deep, and weighs 3000 lbs. Compare the density of the wood with that of a metal, a cubic foot of which weighs 480 lbs.

(7) Equal bulks of two substances weigh 4 lbs. and 9 lbs. Find the ratio between two bulks of those substances, which contain the same quantity of matter.

(8) of alcohol 10 pints weigh 8lbs. ; and of oil 10 quarts weigh 18 lbs. Compare the densities of alcohol and oil.

The Answers are placed together at the end of the Section,

(9) A body weighs four times as much as another body which is thrice its bulk. Compare their densities.

(10) Half a cubic foot of a substance weighs 2 cwt.; what bulk of another substance that is six times as dense will weigh 3 cwt.?

(11) A mass of a substance weighs 8 lbs.; what will be the weight of a mass, twice its size, of a substance whose density is thrice as great?

(12) A block of wood weighs 60 lbs.; and a piece of iron, onethird its size, weighs 10 st. 10 lbs. Compare the quantities of matter in the two substances, and their densities.

(13) If a force of 5 lbs. be represented by a line 1 ft. 3 in. in length, what force will a line 2 ft. in length represent ?

(14) The measure of a force, when the unit of weight is 1 lb., is 3; what is the measure of the same force, when the unit of weight is 5 lbs. ?

ii. THE LEVER. (Arts. 17...29.)

[NOTE.-In the following Examples, unless the contrary be expressly stated, the LEVER is supposed to be straight, and to rest horizontally in equilibrium; and the weights, or other forces, to be applied to it perpendicularly.]

(1) Half a cwt. acts vertically at the extremity of one of the arms, 10 inches long, of a horizontal straight Lever. What number of pounds attached to the extremity of the other arm, 14 inches long, will balance it?

(2) Two weights, of 5 lbs. and 7 lbs. respectively, keep a horizontal straight Lever at rest. The length of the arm at which the larger weight acts is 2 feet 1 inch. Find the length of the other

arm.

(3) A weight of 14 lbs., suspended 2 inches from the fulcrum of a horizontal straight lever, is balanced by a weight of 8 oz. Find the length of the arm at which the latter weight acts.

(4) A straight Lever is kept at rest by two forces acting perpendicularly on it on opposite sides of the fulcrum, and at distances from the fulcrum of 9 inches and 15 inches; the greater force being 6 st. 6 lbs., find the other.