1. A gallon of fresh water measures 277 271 cubic inches, and contains 10 lbs. avoirdupois. A ton of sea water measures 35 cubic feet. What is the mass of a gallon of sea water in pounds and decimals?

2. If 100 cubic inches of oxygen, under certain circumstances of pressure and temperature, contains 35 grains, and a cubic inch of mercury contains 0·49 lbs., how many cubic inches of the oxygen would contain the same quantity of matter as a cubic inch of mercury?

3. Of two bodies one has a volume of 5 cubic inches, the other of one-fifth of a cubic foot; the mass of the former is 15 oz., and of the latter 12.8 lb. What is the ratio of the density of the first to that of the second?

4. A flask holds 27 oz. of water. What mass will it hold of an oil whose specific mass is 0.95?

5. A cubic foot of water contains 1,000 ounces. 502 5 ounces of lead of specific gravity 11.5, and 440 ounces of iron of specific gravity 8, are placed in a cistern of the capacity of one cubic foot. Find the quantity of water necessary to fill

the cistern.

6. From the relative densities to water of zinc, iron, tin, copper, lead, find the relative densities to iron of each of the other four metals.

7. The line-density of iron wire of No. 10 Birmingham wire gauge is 4'96 lb. per 100 lineal feet; what is the line-density of copper wire and of brass wire of the same gauge?

8. Sodium has to alcohol the relative density 1.23, and water has to alcohol the relative density 1.26; what is the density of sodium relatively to water?

9. If the specific gravity of a specimen of milk be m, and that of pure milk s; calculate the proportion of water added.

10. What must be the volume of a mass of wood of relative density 0.5, in order that when it is attached to 500 gms. of iron of relative density 7, the mean density of the whole may be equal to that of water?

11. If the price of whisky, the specific gravity of which is 75, be 16s. a gallon, find the price when it is mixed with water so as to have the specific gravity '8.

12. A Prussian dollar, made of an alloy of silver and copper, has the specific gravity 10:05. Determine the relative amount of silver and of copper in it, the specific gravities of these metals being 105 and 87 respectively.

13. A nugget of gold mixed with quartz weighs 10 oz. The specific gravity of gold is 19.35, of the quartz 2·15, and of the nugget 6:45. Find the mass of the

gold and of the quartz contained in the nugget; find also the ratio of their volumes.

14. A mixture is made of 7 cubic centimetres of sulphuric acid (specific gravity, 1.843) and 3 cubic centimetres of distilled water; and its specific gravity when cold is found to be 1'615. Determine the contraction which has taken place.

15. The density of a mixture of two liquids being supposed to be an arithmetical mean between those of the components; determine the ratio of the volumes of the components contained in the mixture.

16. Several liquids which do not alter their volume when mixed are shaken together; determine the specific gravity of the mixture from their specific gravities.

17. Half a pint of a liquid which is half as dense again as water is mixed with a pint of water; what is the density of the mixture?

18. A rod of uniform cross section 18 in. long weighs 3 oz. ; its specific gravity is 88; what fraction of a square inch is the area of its cross section?

19. What is the mass of a cast-iron ball having a diameter of 6 inches; and of a cast-iron cylinder having the same diameter and 4 feet long?

20. A ditch 3 feet deep is dug round a square garden containing one tenth of an acre; find its width in order that the removed earth may raise the garden one foot.

21. Two liquids are mixed first by volume in the proportion of 1 to 4, and second by mass in the proportion of 4 to 1; the resulting specific masses are 2 and 3 respectively. Find the specific masses of the liquids.

SECTION XXV.-MASS-VECTOR.

ART. 135.—Idea of Mass-Vector. The ideas of dynamics differ from those of geometry and kinematics by the introduction of the idea of mass. From the idea of a vector we derive that of a massvector, which is proportional to a vector and to a mass. This term was introduced by Clerk-Maxwell,* and it is expressed in terms of M by L.

A mass-vector can be resolved and compounded in the same manner as a simple vector.

ART. 136.-Centre of Mass. The centre of mass (commonly * Matter and Motion, p. 49.

called centre of gravity) of a number of material particles situated in a straight line is a point such that were the whole mass placed there, the value of M by L would be the same as before. The distance along the straight line from the origin to the centre of mass may be called the equivalent distance. (Compare Art. 31.)

When the particles are situated in one plane, then the centre of mass is a point which satisfies the above condition for two independent axes; and when they are in space, for three independent axes. The vector from the origin to the centre of mass may in a similar manner be called the equivalent-vector. The mass-vector due to the equivalent-vector and the whole mass is the resultant of the several component mass-vectors.

ART. 137. When a body of uniform density is symmetrical with respect to a plane, the centre of mass is somewhere in the plane of symmetry; when it is symmetrical with respect to two planes, the centre of mass lies in the axis of symmetry; and when it is symmetrical with respect to three planes, the centre of mass coincides with the centre of symmetry.

CENTRE OF MASS.

(The body being of uniform density.)

Triangle. From a vertex along two thirds of the line to the middle point of the opposite side.

Semicircle. From the vertex along 5756 of the radius. Pyramid or Cone.-From the apex along three fourths of the axis. Hemisphere. From the vertex along five eighths of the radius.

EXAMPLES

Ex. 1. At the corners of a cube weights are placed of 1, 2, 3, 4, 5, 6, 7, 8 lbs. respectively; determine their centre of mass.

Let the side of the cube be L (Fig. 18.) Then for the direction

The length of the vector from the corner 0 to the centre of mass is

Ex. 2. A uniform rod, 10 feet long, is bent at right angles at a point 4 feet from its end. Find the perpendicular distances of the centre of mass of the rod from the two straight portions of it.

Let the line-density of the rod, which is uniform, be denoted by 1 M per foot; then in the shorter piece there is 4 M, and in the longer piece 6 M. Since either piece is uniform and symmetrical, its centre of mass is at its mid-point.

We have now reduced the mass to two masses of 4 M and 6 M situated at the two mid-points. Their centre of mass is in the joining line, and at a distance of 6/10 of the line from the midpoint of the shorter piece.

The component of the vector from the corner to the centre of mass along the shorter piece is 4/5 feet, and the component along the longer piece is 9/5 fect.

EXERCISE XXV.

1. Find the centre of mass of two spheres of brass, of 1 inch and 2 inches diameter, placed at a distance of 5 inches, the distance being measured from the centres.

2. Where is the centre of mass of a square tin plate? If the plate weighs 5 oz. and a small body weighing 2 oz. is placed at one corner of the plate, where will the centre of mass of the whole be?

3. Find the centre of mass of the figure A when the pieces are of uniform material, and the central piece is half a side piece in length, and is joined at the mid-points of the sides.

4. Find the centre of mass of a T square, the two pieces being of the same material, and equal in length, breadth, and thickness.

5. Find the centre of mass in the case of a wooden F, the principal pieces being of the same length, and the central piece of half that length. Also for an E.

6. Find the centre of mass of the letter Y, the three pieces being uniform, and each one inch in length, and the two upper inclined at an angle of 60°.

7. A wooden vessel, 6 inches square and 6 inches in height, with a neck 2 inches square and 3 inches in height, is full of water. Find the position of the centre of mass of the water.

SECTION XXVI.—MOMENTUM.

ART. 138.-Unit of Momentum. The idea of momentum is derived from the idea of velocity by introducing the idea of mass. The momentum of a body is proportional to its mass and to its velocity; the general unit is M by (L per T). This unit is equivalent to (M by L) per T, when it is understood that the mass remains constant during change of time. Hence the bracket may be dispensed with, and either of these interpretations put upon

M by L per T.

Momentum is a directed quantity, its direction being the same as that of the velocity (or mass-vector) on which it depends. Hence it is resolved and compounded after the manner of directed quantities.

If the speed only of a body is considered, then we consider only its speed-momentum.