THE PULLEY.

If

A pulley is a small wheel, which turns on an axis passing through its centre and fixed in a block, receiving its motion from a cord, that passes round its circumference. The pulley is either single or combined: it is also, either fixed, or movable. a power sustain a weight by means of a single, fixed pulley-a cord passing over it, with the weight attached to one end and the power to the other-the power and weight are equal: and if the pulley be put in motion, the velocity of the power, and the veloci

ty of the weight will also be equal. But, if the fixed pulley be combined with one movable pulley-as represented in the first set of pulleys above-the weight is equal to twice the power which sustains it; and if the pulley be put in motion, the velocity of the power will be equal to twice the velocity of the weight. Thus, every cord going over a movable pulley, adds 2 to the powers, and hence, in a system of pulleys, we have the following proportion.

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

14. In the second set of pulleys represented above, three of the pulleys are fixed, and three are movable. If a power of 45 pounds were applied to the cord, what weight would it balance?

15. What power must be applied to a cord that runs over 2 movable pulleys, in order to balance a weight of 800 pounds?

16. What power must be applied to a cord that runs over 6 movable pulleys, to balance a weight of 2000 pounds? 23*

17. If a cord, which runs over 3 movable pulleys, be attached to an axle 4 inches in diameter, the wheel of the axle being 38 inches in diameter, and a power of 20 pounds be exerted at the circumference of the wheel, what weight would be raised under the pulleys?

THE INCLINED PLANE.

An inclined plane is a plane making an angle with the horizon. For instance, a plank presents an inclined plane, when one end is resting upon the level ground, and the other end is raised to any height less

than that which would render it vertical. A convenient use of the inclined plane is exemplified in rolling casks from a cellar, upon sloping pieces of timber, or planks.

On an inclined plane, as the perpendicular height of the plane is to the length of the plane, so is the power to the weight.

18. A certain inclined plane is 16 feet in length, and 7 feet in perpendicular height. What weight might be drawn up this plane, by a power, which, if exerted on a cord over a single, fixed pulley, would raise 25 pounds?

19. What power would be necessary to sustain a rolling weight of 1000 pounds, upon an inclined plane of 75 feet length, and 38 feet perpendicular height?

20. What must be the length of an inclied plane, whose perpendicular height is 15 feet, that the exertion of the power of 42 pounds shall draw up 200 pounds?

21. On a rail-road, there is an inclined plane of 80 rods in length, rising to a perpendicular height of 50 feet. What power must be exerted on the summit, to draw up a train of cars weighing 62000 pounds?

22. Suppose a set of pulleys, 3 of which are movable, to be applied to a weight upon an inclined plane of 50 feet length, and 14 feet perpendicular height; what weight upon the plane, would be sustained by 40 pounds at the power cord of the pulleys?

THE WEDGE.

The wedge may be viewed as a moving inclined plane; the head of the wedge, where the power is applied, answering to the perpendicular height of the plane. In the wedge, however, the inclined plane is double, and the force produced by its advance is divided into

two equal parts, acting at right angles with each side. As the breadth of the head of a wedge is to the length of its side, so is the power acting against the head, to the force produced at the side.

Observe, that the force mentioned in the above proportion, respects one side of the wedge, only. If the forces against both sides be required, then, only half the breadth of the head must be taken into the proportion.

In the common mode of applying the wedge, the friction against the sides is very great- at least equal to the force to be overcome. Therefore, not less than one-half of the power is lost; and for this loss there is no allowance made in the above proportion. The wedge, however, has a great advantage over all the other mechanical powers, arising from the force of percussion or blow with which the head is struck, by a mallet. The power thus obtained is incomparably greater than that of any dead weight or pressure, such as is commonly employed on other instruments.

23. Suppose a power of 50 pounds to be applied to a wedge, the head of which is 2 inches broad, and the side 12 inches long, what weight of force would be effected on either side; if there were no friction to resist?

24. If a force of 1000 pounds is to be effected on the side of a wedge, that is 14 inches long, and 3 inches broad at the head, what power must be applied to the head; allowing nothing for friction? Again, allowing the friction, which is to be overcome, to be equal to the force effected, what power will be necessary?

The screw is a spiral thread or groove, cut round a cylinder, and every where making the same angle with the length of the cylinder. In one round of the spiral, it rises along the cylinder, the distance between two threads. Therefore, if the surface of the cylinder, with the spiral thread on it, were unfolded and stretched into a plane, the spiral would form a

straight inclined plane, whose length would be to its height, as the circumference of the cylinder is to the distance between two threads of the screw. The inclined plane being thus recognised in the screw, the following proportion is obvious.

As the distance between two threads of a screw is to the circumference of the circle described by one revolution of the power, so is the power to the weight.

The length of the lever to which the power is applied, being one-half of the diameter of the circle round which the power revolves, the circumference may be found from the lever, as taught in page 173.

In the common use of the screw, about one-third of the power is expended in overcoming friction; and for this loss, no allowance is made in the above stated proportion.

25. If the threads of a screw be 1 inch apart, and a power of 50 pounds be exerted at the end of a lever 70 inches long, what weight of force will be produced at the end of the screw; allowing nothing for friction.

26. If the threads of a screw be .2 of an inch apart, and a power of 40 pounds be exerted at the end of a lever 30 inches long, what will be the force at the end of the screw; allowing of the power to be lost in overcoming friction?

27. Suppose a power of 48 pounds is to be employed to effect the weight of 5000 pounds, by means of a screw, whose threads are 1.3 inches apart; what must be the length of the lever; allowing of the power to be lost in overcoming friction?

28. Suppose the end of a screw, whose threads are .8 of an inch apart, and whose lever is 7ft. long, to be set upon a wedge, that is 15in. long at the side, and 2 inches broad at the head; what weight of force would be effected on either side of the wedge, by applying 100 pounds' power to the lever; allowing of the force on the screw, and of that on the wedge to be lost in friction?

XLI.

MISCELLANEOUS QUESTIONS.

1. What vulgar fraction is that, which being multiplied by 15, will produce 2?

2. What decimal fraction is that, which being multiplied by 15, will produce .75?

3. What quantity is that, which being divided by gives the quotient 21 ?

3

42

4. What vulgar fraction is that, from which if you take 3, the remainder will be?

59

5. What vulgar fraction is that, to which if you add 2, the sum will be ?

6. What quantity is that, which being multiplied by 3, produces the fraction ?

7. What quantity is that, from which if you take of itself, the remainder will be 12?

3

8. What quantity is that, to which if you add 2 of 11 of itself, the sum will be 61?

9. A farmer carried to market a load of produce, consisting of 780lb. of pork, 250lb. of cheese, and 154lb. of butter; he sold the pork at 6 cents, the cheese at 8 cents, and the butter at 15 cents per lb.; and agreed to take in pay, 60lb. of sugar at 10 cents per lb., 15 gallons