standard masses which must then be placed upon the other pan to bring the pointer again to O. This is the usual method of weighing. It gives correct results, however, only when the knife-edge C is exactly midway between the points of support m and n of the two pans. The method of substitution, on the other hand, is independent of the position of the knife-edge. It is customary to consider that the mass of a body determined as here indicated is a measure of the quantity of matter which it contains.

SUMMARY. The three fundamental measuring instruments are (1) the meter, to measure lengths; (2) the balance, to measure masses; (3) the clock, to measure times.

1 m. = 39.37 in.; 1 in. = 2.54 cm.; 1 km. = .62 mi.; 1 1. 1000 cc. = about 1 qt.; 1 kg. = mass of 1 1. of water at 4° C. = about 2.2 lb.

QUESTIONS AND PROBLEMS

1. Name as many advantages as you can which the metric system has over the English system. Can you think of any disadvantages?

2. Find the number of millimeters in 6 km. Find the number of inches in 4 mi. Which is the easier to do?

3. A new lead pencil is 7 in. long. How many centimeters long is it?

4. The Twentieth Century Limited runs from New York to Chicago (967 mi.) in 20 hr. Find its average speed in kilometers per hour.

5. With a Vickers-Vimy biplane Captain Alcock and Lieutenant Brown completed, on June 15, 1919, the first nonstop transatlantic flight of 1890 miles from Newfoundland to Ireland in 15 hr. 57 min. How many miles did they travel per hour? How many kilometers per hour?

6. From the bed rock upon which the Woolworth Building in New York rests to the top of the tower is 278.3 m. How many feet is it?

7. What must you do to find the capacity in liters of a box when its length, breadth, and depth are given in meters? to find the capacity in quarts when its dimensions are given in feet?

8. Find the capacity in liters of a box .5 m. long, 20 cm. wide, and 100 mm. deep.

9. How many kilograms are there in the 16-pound shot?

10. If you bought a metric ton (= 1000 kg.) of coal, how many more pounds should you have to pay for than if you had bought an English ton?

DENSITY

14. Definition of density. When equal volumes of different substances, such as lead, wood, iron, etc., are weighed in the manner described above, they are found to have widely different masses. The term "density" is used to denote the mass, or quantity of matter, per unit volume.

Thus, for example, in the English system the cubic foot is the unit of volume, and the pound is the unit of mass. Since 1 cubic foot of water is found to weigh 62.4 pounds, we say that in the English system the density of water is 62.4 pounds per cubic foot.

In the centimeter-gram-second (C.G.S.) system the cubic centimeter is taken as the unit of volume, and the gram as the unit of mass. Hence we say that in this system the density of water is 1 gram per cubic centimeter, for it will be remembered that the gram was taken as the mass of 1 cubic centimeter of water. Unless otherwise expressly stated, density is now universally understood to signify density in C.G.S. units; that is, the density of a substance is the mass in grams of 1 cubic centimeter of that substance.

The density of some of the most common substances is given in the following table:

15. Relation between mass, volume, and density. Since the mass of a body is equal to the total number of grams which it contains, and since its volume is the number of cubic centimeters which it occupies, the mass of 1 cubic centimeter is evidently equal to the total mass divided by the volume. Thus, if the mass of 100 cubic centimeters of iron is 740 grams, the density of iron must equal 740 ÷ 100 7.4 grams to the cubic centimeter. To express this relation in the form of an equation, let M represent the mass of a body, that is, its total number of grams; V its volume, that is, its total number of cubic centimeters; and D its density, that is, the number of grams in 1 cubic centimeter; then

=

(1)

This equation merely states the definition of density in algebraic form.

16. Distinction between density and specific gravity. The term "specific gravity" is used to denote the ratio between the weight of a body and the weight of an equal volume of water.*

Thus, if a certain piece of iron weighs 7.4 times as much as an equal volume of water, its specific gravity is 7.4. But since the density of water in C.G.S. units is 1 gram per cubic centimeter, the density of iron in that system is 7.4 grams per cubic centimeter. It is clear, then, that density in C.G.S. units is numerically the same as specific gravity.

Specific gravity is the same in all systems, since it is simply the number obtained by dividing the weight of a body by

*For the present purpose the terms "weight" and "mass may be used interchangeably. They are in general numerically equal, although an important distinction between them will be developed in § 73. Weight is, in reality, a force rather than a quantity of matter.

the weight of an equal volume of water. Density, however, which we have defined as the mass per unit volume, is different in different systems. Thus, in the English system the density of iron (which in the metric system is 7.4 grams per cubic centimeter) is 462 pounds per cubic foot (7.4 x 62.4), since we have found that water weighs 62.4 pounds per cubic foot and that iron weighs 7.4 times as much as an equal volume of water.*

SUMMARY. The density of a substance is its mass per unit volume. Expressed as a formula,

The specific gravity of a substance is the ratio between the weight of any volume of the substance and the weight of an equal volume of water.

The density of a substance in C.G.S. units is numerically equal to its specific gravity.

The density of a substance expressed in pounds and cubic feet is numerically 62.4 times its specific gravity.

QUESTIONS AND PROBLEMS +

1. A ball of yarn is squeezed into one fourth its original bulk. What effect does this produce on its mass? on its volume? on its density?

2. Name three uses made of lead because of its great density, and two uses made of cork because of its small density.

3. A liter of milk weighs 1032 g. What is its density? its specific gravity?

4. How many cubic centimeters are there in a block of brass weighing 34 g.? (See table of densities, p. 8.)

5. What is the mass of a liter of alcohol? (See table, p. 9.)

6. A flask holds 2520 g. of glycerin when filled. What is the capacity of the flask in liters?

* Laboratory exercises on length, mass, and density measurements should accompany or follow this chapter. See, for example, Experiments 1, 2, and 3 of "Exercises in Laboratory Physics" by Millikan, Gale, and Davis.

† Supplementary questions and problems for Chapter I are given in the Appendix.

7. If a wooden beam is 30 × 20 × 500 cm. and has a mass of 150 kg., what is the density of the wood? What is its density in the English system?

8. A contractor has to remove 1000 slabs of marble, each 2 in. thick, 12 in. wide, and 6 ft. long. How many tons must he remove, marble having a specific gravity of 2.7?

9. A piece of gold weighs 772 g. Find the weight of a piece of lead having a volume equal to that of the gold.