THE VOLUME OF A SPHERE equals is the diameter.

πD3
6

where = 3.1416 and D

23. The diameter of a steel bicycle bearing-ball is 3.1 mm.; find its volume.

24. How many cu. cm. in a sphere whose radius is 3 cm. 25. Find the diameter of a ball whose volume is 1000 cu. cm. ?

26. How large a sphere can be made from 1728 cu. inches of putty?

27. How much greater volume has a sphere whose diameter is 6 than one whose diameter is 4?

28. A rectangular mass of clay is 4 cm. by 8 cm. by 10 cm. Find the diameter of a sphere that can be formed with it.

29. Find the height of a cylinder of the same diameter as the sphere of question 28 which could be made of the same amount of material.

NOTE. The student will find that a review of areas and volumes, in some arithmetic or geometry, will be useful at this point.

CHAPTER IV.

SOME PROPERTIES OF SOLIDS.

Definitions. Breaking strength; tenacity; elasticity; strain ; stress; deformation; limit of elasticity.

VI. TENACITY.

1. An iron wire whose area of cross-section is .251 sq. mm. breaks under a force of 10 kgm. Find the force necessary to break a wire of same material and 1 sq. mm. crosssection area.

2. How large must a wire of the same material be to sustain 1000 kgm.?

3. It is found by experiment that a rod of oak one inch square will withstand a tension of 15000 lbs. Find the weight which a round rod of oak whose diameter is inch will sustain.

4. Which has the greater tenacity, a material which will sustain a stretching force of 2000 lbs. to the square inch or one which will sustain a force of 50 lbs. applied to a round rod whose diameter is of an inch?

5. A hemp rope whose cross-section area is one square inch will sustain 12000 lbs. How many pounds will a rope whose diameter is inch sustain ?

6. Find the relative strength of steel and iron from the following data:

An iron wire whose diameter is .72 mm. breaks under a stress of 31.6 kgm.

A steel wire whose diameter is .4 mm. breaks under a stress of 12.8 kgm.

7. A certain wire is known to have a tenacity of 80 kgm. per sq. mm. cross-section area. If it weighs 7 kgm. per 1000 meters, how long must a piece be to break under its own weight?

8. Solve the preceding problem for a wire whose diameter is .1 mm.

9. Which material has the higher tenacity, one which breaks under a force of 200 lbs. to the square inch, or one which breaks under a force of 60 kgm. to the square centimeter ?

10. If the breaking strength of a rod 2 cm. square is 25 kgm., what will it be if the rod is made 3 cm. square?

11. If d is the diameter of a round rod and B its breaking strength, find an expression for the breaking strength B' when the diameter is d'.

12. The tenacity of lead is 2.07 kgm. per sq. mm. crosssection area; that of copper is 40 kgm. How large must a lead wire be to hold as much as a copper wire whose diameter is .81 mm. ?

9. Elasticity of a solid is defined as that property by virtue of which a solid tends to recover its shape and size after a deformation:

Four principal cases of deformation of a solid may be mentioned:

(a) Elongation.

(b) Bending.

(c) Breaking strength under transverse forces.

(d) Torsion.

The following formulæ express the dependence of each upon the various conditions of force and size. If F force applied, L length, W width, T thickness, D = diameter, then for a given material—

=

=

=

(This is for the case of a rod with free ends supported on knife-edges.)

When a rod is submitted to the action of a transverse force (i.e. force acting in a plane at right angles to the longest dimension of rod) sufficient to break it, the dependence of the breaking strength of the rod upon its dimensions is expressed by

Formula (c)......Breaking Strength α

Formula (d)...... Angle of Torsion o

WT2
L

FL

D'

NOTE. The student should understand the exact significance of these formulæ and be able to read them as Physical Laws. They are of course obtained as the results of direct experiment.

VII. ELASTICITY OF STRETCHING: ELONGATION.

1. If a certain wire whose length is 4 m. stretches 3 mm. under a certain force, how much will it stretch under a force 15 times as great?

2. A force of 2 kgm. stretches a certain steel wire 2 mm. It is found that a wire 1 m. long stretches .51 mm. under a pull of 1 kgm. Find length of first wire.

3. A force of 1 lb. is found to produce a deformation of 2 per cent of its original length in a certain wire whose length is 15 cm. and cross-section area .6 sq. mm. Find the per

cent of deformation produced by the same force on a wire of the same material 75 cm. long and of cross-section area 1 sq. mm.

4. If a wire 12 ft. long stretches inch under a force of 4 lbs., how long is the wire of 3 times the cross-section area which stretches inch under a stress of 9 lbs.?

5. The results of an experiment on the stretching of a No. 28 spring brass wire were as follows:

7. A solid is found to be elongated 14 mm. by a force of 4 kgm. By making a certain change in the length of the specimen treated it was found that the elongation for the same force rose to 21 mm. Find the ratio of the length in the first case to that in the second, and find also what change in the diameter would have produced the same effect on the elongation.

8. An iron and a brass wire have each the length 15 cm. when each is stretched by a force of 1 kgm. The length of the iron wire becomes 15.4 cm. under a stress of 3 kgm., and that of the brass wire becomes 15.6 cm. under a stress of 7 kgm. Compare the elasticity of iron with that of brass.

NOTE. -Young's Modulus of Elasticity is the weight in kilograms which applied to a bar 1 mm. square would double its length if the material were perfectly elastic.

The moduli (in round numbers) of several materials are here given in kgm. per sq. mm.;