To balance the weight,

The power must be to the weight as the distance between the tops of the coils is to the circumference described by the lever.

1. A certain screw has a triangular thread, the coils of which are 1 inch apart; what weight will be balanced on it by a power equal to 40 lbs., applied at the end of a lever 4 ft. from the centre of the screw?

Ans. 12,063.7 lbs., about. Note. The circumference described by the lever, in example 1, is a mere trifle more than the circumference of a circle of which the lever is the radius, because the lever rises 1 inch, in turning round.

2. If a screw has a square thread with the tops of the coils of an inch apart, and a lever 36 in. long, what power applied to the lever will balance a weight of 1,500 lbs. ? Ans. 5 lbs. 4 oz., about. 3. There is a weight of 7,540 lbs. to be sustained by a screw, to which a power equal to 25 lbs. can be applied by means of a lever 6 ft. long; the thread being square, what distance must the tops of the coils be apart?

Ans. 1 inch, nearly.

4. There is a downward pressure of 5,000 lbs. to be exerted by a power equal to 40 lbs., through the medium of a screw having a triangular thread the coils of which are of a ft. apart; how long must the lever be?

Ans. 3.18 ft., about.

Observations on the Screw. When the power balances the weight, the power must be increased in order to raise the weight; of the power being usually lost in overcoming friction. The power may also be diminished about before the weight will overcome the friction and descend.

5. There is a weight of 1,508 lbs. to be raised 5 ft. high by means of a screw which has a lever 5 ft. long, and a square thread with the tops of the coils of an inch apart; what power will be necessary ? Ans. 4 lbs. 6. What weight will a power equal to 35 lbs. raise by means of a screw with a lever 4 ft. long, the thread being triangular, and the coils 14 inch apart?

Ans. 5,277.9 lbs., nearly.

If the power is applied at the end of a lever, what is necessary to bal

ance the weight?

What is said of the friction of the screw ?

Figure 32.

LESSON 201.

THE WEDGE.

If you hold the weight in figure 30 firmly, and raise it by pushing along the inclined plane, the plane becomes a wedge; now to balance the weight, the power must evidently be to the weight as before, that is, as the height of the plane is to its length, or as the thickness of the head of the wedge is to the length of the slanting side. The only difference will be in the greater friction, and difficulty in moving the plane or wedge instead of the weight. A common wedge is two inclined planes placed together, as the dotted line in figure 32 represents.

Therefore, to balance the resistance,

The power must be to the resistance, as the thickness of the head is to the length of both the slanting sides.

1. There is a wedge with a head 2 in. thick, the length of each side being 10 in.; what amount of resistance will a power equal to 50 lbs. applied to the head balance ?

Ans. 500 lbs. 2. What power applied to a wedge 3 in. thick at the head, and 12 in. long on one side, will balance a resistance equal to 1,800 lbs. ? Ans. 225 lbs.

3. A power equal to 80 lbs. is to be exerted on a wedge to balance a resistance amounting to 1,200 lbs. ; what must the thickness of the head be, the length of a side measuring 15 in. ? Ans. 2 in.

Observations on the Wedge. Friction greatly modifies the operation of the wedge; thus, if the power be a weight

Explain how the inclined plane in figure 30 becomes a wedge. Now to balance the weight, how must the power evidently be to the weight? What will the only difference be? What is a common wedge?

To balance the resistance, how must the power be to the resistance?

placed on the head of a common wedge, it often will not overcome part as much resistance as we should expect from the preceding proportion, whereas if the wedge be struck, it many times will overcome 10 times this resist

ance.

The operation of an ordinary wedge, therefore, can only be estimated or guessed at by those accustomed to its use. The principle of the wedge, however, is employed in many machines, like presses used in printing and manufacturing, in which friction operates regularly, and diminishes the effect of the power about .

By different combinations of these simple machines, all others, however complicated, are formed; and the preceding rules enable us to calculate the effect of any machine whatever.

There is no power gained by using these simple machines, or any machine; on the contrary, some of the power is always lost; still, they enable us to perform what would be difficult or impossible from a direct application of the power. For instance, a man cannot lift a millstone, but by working 20 minutes with a screw and lever, he can raise it several feet; the sum of his exertions, however, is about greater than the power employed by 20 men who lift it the same height in 1 minute, because of his labor is expended in overcoming friction.

What thing greatly modifies the operation of the wedge? Explain this modification.

How can the operation of an ordinary wedge, therefore, be obtained? In what is the principle of the wedge employed? How does friction operate in them?

What are formed by different combinations of these simple machines? What do the preceding rules enable us to calculate?

Is there any power gained by the employment of these simple machines, or of any machine? What on the contrary is true? What do they enable us to perforin? Give an instance. What is said of the sum of his exertions?

PROMISCUOUS QUESTIONS

IN

MENSURATION, SQUARE ROOT, CUBE ROOT, &c.

LESSON 202.

To be performed in the mind.

1. 729 cubic ft. of clay were dug out of a cubic pit ; what was the depth of the pit ?

2 How many square yards of carpeting will cover a floor 20 ft. long and 18 ft. wide ?

3. I bought 8,100 sq. ft. of land, to be laid out in an exact square; how long will one side be?

4. A lever of the first kind is 12 ft. long, and the fulcrum is 2 ft. from the weight; what power applied to the lever, will balance a weight of 60 lbs. ?

5. A weight of 100 lbs. is to be raised by a wheel and axle, the axle being 6 in. in diameter; what must the diameter of the wheel be so that a power equal to 10 lbs. shall balance the weight?

6. What weight can I raise by means of pulleys with a power of 40 lbs., the weight being supported by 6 ropes ?

7. What power will balance an iron cylinder weighing 70 lbs. on an inclined plane, the length of which is 12 ft., and the height 3 ft. ?

8. There is a circle 12 ft. in diameter ; if you reckon the circumference as 3 times the diameter, how many square feet will it contain?

9. A marble monument in the shape of a pyramid is 2 ft. square at the base, and 6 ft. high; how many cubic feet does it contain ?

10. How many square rods are there in a triangular piece of land, the base of which is 10 rods, and the altitude 8 rods ?

11. What is the square root of 400? 12. What is the cube root of 125,000?

LESSON 203.

1. There is a block of granite in the shape of a frustum of a square pyramid, the top is 2 ft. square, the base 5 ft. square, and the height is 4 ft.; what is its weight? Ans. 8,775 lbs.

2. What is the cube root of 74,088 ?

Ans. 42. 3. There is a house 33 ft. long, 24 ft. wide, 19 ft. from the sill to the eaves, and 30 ft. from the sill to the top of the roof; how much will it cost to paint the outside at 25 cts. a sq. yd., no allowance being made for windows and doors? Ans. $671⁄2.

4. I own a piece of ground 250 ft. long, the widths of which, measured every 50 ft., are as follows; 35 ft., 18 ft., 16 ft., 24 ft., 10 ft., and 20 ft.; how many square feet of land are there in the piece? Ans. 4,775 sq. ft.

5. The quantity of water flowing over a mill dam, July 1st, was found to be 28 cubic ft. a second at 6 o'clock in the morning, 23 cubic ft. at 9 o'clock, 19 cubic ft. at 12 o'clock, 18 cubic ft. at 3 o'clock, and 22 cubic ft. at 6 o'clock in the evening; what quantity, on an average, passed over the dam in a second? Ans. 21 cubic ft. 6. There is a square garden containing 2,500 sq. ft. ; how far is it between the opposite corners ?

Ans. 70.71 ft., or 70 ft. 81⁄2 in., about. 7. How many feet, board measure, are there in a stick of timber 2 ft. square, and 22 ft. 9 in. long?

Ans. 1,092.

8. What is the surface of a circle 15 ft. in diameter ?

Ans. 176.714 sq. ft., about. 9. There is a pine log 4 ft. in diameter, and 27 ft. long, the ends of which are cut off obliquely, but parallel to each other; how many cubic ft. does it contain ?

Ans. 339.29, about. 10. How many square inches of leather will cover a foot ball 6 in. in diameter ? Ans. 113 sq. in., nearly.

LESSON 204.

1. How large a cube will 90,800 ounces of lead make? Ans. A cube 2 ft. square. 2. How many gallons of milk are there in a cask of which the length is 26 in., the head diameter 17 in., and the bung diameter 22 in., the staves curving but little ? Ans. 29 gals., nearly.

3. What is the carpenters' tonnage of a single decked vessel, the keel measuring 60 ft., the breadth of the beam 20 ft., and the depth of the hold 9 ft. ? Ans. 120 tons.

4. There is a weight of 730 lbs. to be raised by a lever of the first kind 10 ft. long, to which a power can be applied equal to 146 lbs. ; how far must the fulcrum be