9. Which is the more elastic, copper or brass? Express the ratio.

10. What force is needed to double the length of a steel rod whose diameter is 2 mm., assuming perfect elasticity?

11. If a force of 2000 kgm. produces an elongation of 2 mm. in a specimen of brass, what force will produce a similar change of length in a cast-iron specimen of the same dimensions? Assume perfect elasticity in both.

VIII. ELASTICITY OF BENDING: FLEXURE.

1. If a rod whose width is 2 cm. bends 1 cm. under a force of 1 kgm., how much will a rod twice as wide bend under a force of 7 kgm.?

2. If a rod 100 cm. long, 2 cm. broad, and 3 cm. thick is deflected 0.5 cm. when placed horizontal and loaded at the middle with a certain weight, what would be the deflection, under the same load, of a rod 50 cm. long, 2 cm. broad, and 1 cm. thick? (Harvard Entrance Examination.)

3. A certain beam 4 ft. long, placed horizontal and supported at the ends, is bent downward 0.5 inch by a load placed at the middle. How far would it be bent by the same load if it were 8 ft. long? (Harvard Entrance Examination.)

4. If a certain beam 16 ft. long, 4 in. wide, and 6 in. thick is bent 1 in. by a load of 500 lbs. placed at its middle, how much would a beam 10 ft. long, 8 in. wide, and 12 in. thick be bent by the same load? (Harvard Exam.) 5. A certain bar is 100 cm. long, 2 cm. wide, and 2 cm. thick.

(a) Find width of a bar of the same material and the same length which is 3 cm. thick, but will bend exactly as much under a given force as the first bar.

(b) If another bar is made as long, find how thick it must be to bend like the first.

(c) If the first bar is made twice as long and its width 3 times as great, find the new thickness to retain the original flexure.

(d) Find the new force required to produce the same flexure when the width is 4, the length 200, and the thickness 1.

6. Compare the elasticity of the first of two materials with the second from the following data: First, a force of 2 kgm. bends a rod 70 cm. long, 2 cm. wide, and .5 cm. thick, 1.5 cm. Second, a force of 15 kgm. bends a rod 80 cm. long, 1 cm. wide, and 3 cm. thick, 2 cm.

7. Which will increase the stiffness of a rod most, to multiply the width by 6 and the length by 2, or to multiply the thickness by 8 and divide the width by 12?

8. A man wishes to make a uniform rectangular beam 30 ft. long out of 1800 lbs. of iron and have it as stiff as possible. Which is the better arrangement, to make the width 10 and the thickness 2, or to make the width 1 and the thickness 4?

IX. BREAKING STRENGTH.
Transverse Forces.

1. A rod has its thickness changed from 3 to 5. Express the ratio of the increase of stiffness to the increase of strength resulting.

2. If a beam 10 ft. long and 2 in. square breaks under a weight of 2000 lbs. at the middle, find the force required to break a similar beam 15 ft. long, 2.5 in. wide, and 1 in. thick.

3. How much thicker must a beam be made to retain its strength when its width is divided by 3?

4. Which produces the greater change in strength of a beam: first, doubling its width, halving its thickness, and dividing its length by 3; or, second, halving its width, multiplying its thickness by 1.5, and dividing its length by 4?

X. ELASTICITY OF TORSION.

1. A wire of diameter .2 mm. is twisted through an angle of 10° by a force of value 3. Through what angle will a wire of the same material be twisted by a force 5 if its diameter is .3 mm. ?

2. A piece of shafting 6 ft. long and 2 in. diameter is twisted through an angle of 50' by a certain force. How much larger must the shaft be made that the torsion may be reduced to 30'?

3. It is desired to make the shaft of Prob. 2 three times as long and of twice as large diameter as in the first instance. Find the angle of torsion resulting.

4. If a wooden rod of diameter .3 in. and length 30 in. is twisted 20° by a force of 8 lbs., find a force needed to deflect a rod of similar material through an angle of 15° if it is 40 in. long and .4 in. in diameter.

5. Which is the more elastic, a wire of diameter .4 in. and length 30 cm. which is twisted 7° by a certain force, or one which has a diameter of .3 and length 20 cm. and is bent 8° by the same force?

NOTE. -Observe that elasticity varies inversely as the amount of strain produced by a given force.

6. If a wire whose length is 7 and diameter d is bent through an angle a by a force f, express in terms of these letters the value of the force which will give three times the angle of torsion with a wire whose length and diameter are twice those of the first.

CHAPTER V.

MECHANICS OF FLUIDS.

Definitions.-Fluid; liquid; gas; pressure at a point.
NOTE.-1 cu. cm. of water weighs 1 gram.

1 cu. ft.

66

66

66

62.5 lbs. (nearly).

In the following problems the pressure due to the atmosphere is not to be considered.

XI. PRESSURE IN LIQUIDS.

1. What is the total pressure on the bottom of a vessel one meter square when it is filled with water to the depth of one centimeter?

2. What is the total pressure on the bottom of a vessel 10 cm. square and 10 cm. deep when filled with water?

3. In each of the two preceding questions what is the pressure per square centimeter ?

4. A vertical tube has an area in cross-section of 1 sq. cm. It is filled with water to a depth of two meters. What is the total pressure of the water at the bottom?

5. What is the total pressure at a point 1 m. from the bottom? at a point 150 cm. from the bottom?

6. What is the average pressure throughout the length of the tube?

cm.

7. Suppose the tube to have a cross-section area of 4 sq. How would this affect the total pressure at the bottom, and would it cause the pressure per sq. cm. to be greater or less than before?

8. State a general rule for calculating the total pressure

24

on a horizontal surface covered with water when its dimensions and the depth of the water are given in the metric system.

9. How must this rule be given if the English units of length and weight are used?

10. How must the rule be given if the liquid used is heavier or lighter than water?

11. A rectangular tank filled with water and having a closed top is 80 cm. by 40 cm. on the bottom, and 30 cm. high; what is the total pressure on the lowest surface when it stands on the bottom?

12. What is the pressure per sq. cm. over this surface? 13. What is total pressure on the lowest surface and pressure per sq. cm. when it stands on a side ?

14. What when it stands on an end?

15. State a rule for finding pressure per unit area at any depth and in any liquid. Does this rule depend on the direction in which the pressure is considered as acting?

16. What is the pressure on a sq. cm. of area at a depth of one kilometer of water? Of alcohol which is as heavy as water?

17. A cubical box 2 ft. on an edge is sunk in water till its top is 10 ft. from the surface. If this top is horizontal, find total pressure on the top, on the bottom, and on each side.

18. To what depth must a horizontal surface be immersed in water in order that the pressure may be 500 lbs. per square foot?

19. The reservoir of a city water-supply is 28.8 feet higher than the outlet in a certain house. What will be the pressure at the mouth of a faucet of one square inch cross-section ?

20. A dam is 80 ft. long and holds back a pond 15 ft. deep and 800 ft. long. Find total pressure in pounds on the face of the dam.

21. How would the pressure on the dam be affected if