Art. 297.—To find what weight may be balanced by a given power.

RULE

As the distance between the body to be raised, or balanced, and the fulcrum, or prop, is to the distance between the prop and the point where the power is applied, so is the power to the weight which it will balance.

1. If a man, weighing 160 lbs., rest on a lever 10 feet long, what weight will he balance on the other end, supposing the prop to be 1 foot from the weight?

1:9:160: 1440 lbs. Ans. 2. If a weight of 1440 lbs. were to be raised by a lever 10 feet long, the prop being 1 foot from the weight, what power must be applied to the other end, to balance the weight?

Ans. 160 lbs. 3. At what distance from the prop must a power of 160 lbs. be applied, to balance 1440 lbs., 1 foot from the prop ?

Ans. 9 feet. 4. At what distance from a weight of 1440 lbs. must a prop be placed, so that a power of 160 lbs., applied 9 feet from the prop, may balance it?

Ans. 1 foot.

OF THE WHEEL AND AXLE.

Art. 298.-The wheel and axle are here represented with the weight attached to the circumference of the axle, and the power applied to the circumference of the wheel. The principle of the lever is employed in the wheel and axle.

RULE.

As the diameter of the axle is to the diameter of the wheel, so is the power applied to the wheel, to the weight suspended on the axle.

1. If the diameter of the axle be 6 inches, and that of the

wheel be 60 incnes, what weight applied to the wheel will balance 10 lbs. on the axle ? 60:6:10: 1 lb. Ans.

2. If the diameter of the wheel be 60 inches, what must be the diameter of the axle, so that 1 lb. on the wheel may balance 10 lbs, on the axle ? Ans. 6 inches.

3. If the diameter of the axle be 6 inches, what must be the diameter of the wheel, so that 10 lbs. on the axle may balance 1 lb. on the wheel? Ans. 60 inches.

THE PULLEY.

Art. 299:-The pulley is a small wheel, moveable about its axis by means of a cord, which passes over it.

When the axis of a pulley is fixed, the pulley only changes the direction of the power; if moveable pulleys are used, an equilibrium is produced, when the power is to the weight as one to the number of ropes applied to them. If each moveable pulley has its own rope, each pulley will be double the power.

Art. 300.-The number of moveable pulleys and the power given, to find what weight may

be raised.

RULE.

As 1 is to twice the number of moveable pulleys, so is the power to the weight.

OBS.-Reverse the rule, to find the power.

1. What weight would balance a power of 45 lbs., applied to a cord that runs over 3 moveable pulleys?

2. If a cord, which runs over 2 moveable pulleys, be attached to an axle 3 inches in diameter, the wheel of the axle being 28 inches in diameter, and a power of 10 lbs. be exerted

at the circumference of the wheel, what weight would be raised under the pulleys?

Thus, 3:28::10×3×2: 560 lbs. Ans.

OF THE SCREW.

Art. 301.-The screw is a spiral thread, or groove, cut round a cylinder, and everywhere making the same angle with the length of the cylinder.

The power is to the weight which is to be raised, as the distance between two contiguous threads of the screw is to the circumference of a circle, described by the power applied at the end of the lever.

RULE.

Multiply twice the length of the lever by 3.1416, which will give the circumference of the circle; then say, as the circumference is to the distance between the threads of the screw, so is the weight to be raised to the power which will raise it.

1. The threads of a screw are 1 inch asunder; the lever, by which it is turned, is 30 inches long, and the weight to be raised is 1 ton=2240 lbs. What power must be applied to turn the screw ?

30×2=60, and 60 × 3.1416-188.496 inches, the circumference. Then 188.496 : 1 Then 188.496:1:: 2240: 11.88 lbs. Ans.

2. If the lever be 30 inches, the circumference of the circle described by the power 188.496, the threads of the screw 1 inch asunder, and the power 11.88 lbs., what weight will be raised? Ans. 2240 lbs.

3. If the weight be 2240 lbs., the power 11.88 lbs., and the lever 30 inches in length, what is the distance between the threads of the screw? Ans. 1 inch. 4. If the power be 11.88 lbs., the weight 2240 lbs., and the threads 1 inch asunder, what is the length of the lever? Ans. 30 inches, nearly.

INCLINED PLANE.

Art. 302.-An inclined plane is a plane which makes an acute angle with the horizon.

To find the power that will draw a weight up an inclined

plane.

RULE.

As the length of the plane is to the perpendicular height of the plane, so is the weight to the power.

1. An inclined plane is 40 feet in length, and 8 feet in perpendicular height. What power is sufficient to balance a weight of 2000 pounds? Ans. 400 lbs. 2. A certain railroad, 200 rods in length, has a perpendicular elevation of 20 feet. What power is sufficient to sustain a train of cars weighing 100,000 pounds 2 Ans. 606323.

THE WEDGE

Art. 303.-The wedge is composed of two inclined planes, whose bases are joined.

When the resisting forces, and the power which acts on the wedge, are in equilibrium, the weight will be to the power as the height of the wedge to a line drawn from the middle of the base to one side, and parallel to the direction in which the resisting force acts on that side.

Art. 304.-To find the force of the wedge

RULE.

As the breadth, or thickness, of the head of the wedge, is to one of its slanting sides, so is the power whick acts against its head, to the force produced at its side.

Suppose 100 lbs. to be applied to the head of a wedge, 2 inches broad, and 20 inches long, what force would be effected on each side? Ans. 1000 lbs.

MATHEMATICAL PROBLEMS.

Art. 305.-PROB. I. The sum and difference of two numhers given, to find those numbers.

RULE.

Subtract the difference from the sum, and divide the remainder by 2. The quotient will be the smaller number. Then add the given difference to the smaller number, and this sum will be the larger number.

EXAMPLE.

An assembly of 344 persons is convened in two rooms, one of which has 142 persons more in it than the other. How many are in each ?

Operation.

344-142=202; then 202-2=101 persons, in one room; then 101+142=243, in the other.

Art. 306.-PROB. II. The sum of two numbers, and the difference of their squares given, to find those numbers.

RULE.

Divide the difference of their squares by the sum of the numbers, and the quotient will be their difference. We then have their sum and difference, to find each number, by Prob. I.

EXAMPLE.

A. and B. played at marbles, having at first 14 each; but after playing several games, B. having lost some of his, would not play any longer, and it was found that the difference of the squares of what each then had, was 336. How many did B. lose?

Thus 336-14+14=12 difference; 14 half sum, and 12÷2=6—half difference. Then +6=20, A. retired with ; and 14—6—8, B. retired with: then 14-8=6, B. lost.

Art. 307.-PROB. III. The difference of two numbers, and the difference of their squares given, to find those numbers.

RULE.

Divide the difference of their squares by the difference of their numbers, and the quotient will be their sum; then proceed by Prob. I.