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2. In the first and second kinds there is an advantage of power, but a proportionate loss of velocity; and in the third kind there is an advantage in velocity, but a loss of power.

3. When the weight x its distance from the fulcrum = the power × its distance from the fulcrum, then the lever will be at rest, or in equilibrio; but if one of these products be greater than the other, the lever will turn round the fulcrum in the direction of that side whose product is the greater.

4. In all the three kinds of levers, any of these quantities, the weight, or its distance from the fulcrum, or the power or its distance from the fulcrum, may be found from the rest, such, that when applied to the lever, it will remain at rest, or the weight and power will balance each other.

Weight its distance from fulc.

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9. In the first kind of lever, the pressure upon the fulcrum = sum of weight and power; in the second and third = their difference.

10. If there be several weights on both sides of the fulcrum, they may be reckoned powers on the one side of the fulcrum, and weights on the other.

Then, if the sum of the product of all the weights their distances from the fulcrum be to the sum of the products of all the powers their distances from the fulcrum, the lever will be at rest; if not, it will turn round the fulcrum in the direction of that side whose products are greatest.

11. In these calculations the weight of the lever is not taken into account; but if it is, it is just reckoned like any other weight or power acting at the centre of gravity.

12. When two, three, or more levers act upon each other in succession, then the entire mechanical advantage which they give, is found by taking the product of their separate advantages.

13. It is to be observed in general, before applying these observations to practice, that if a lever be bent, the distances from the fulcrum must be taken, as perpendiculars drawn from the lines of direction of the weight and power of the fulcrum.

Example. In a lever of the first kind, the weight is 16, its distance from the fulcrum 12, and the power is 8; therefore by No. 7 of this chapter, 16 × 12

& fulcrum.

24 the distance of power from the

In a lever of the second kind, a power of 3 acts at a distance of 12; what weight can be balanced at a distance of 4 from the fulcrum? Here, by No. 6, 3 x 12 = 9 weight.

4

In a lever of the third kind the weight is 60, and its distance 12, and the power acts at a distance of 60 × 12

9 from the fulcrum; therefore, by No. 5,

80 the power required.

9

If there be a lever of the first kind, having three weights, 7, 8, and 9, at the respective distances of 6, 15, and 29, from the fulcrum on one side, and a power of 17 at the distance of 9 on the other side of the fulcrum, then a power is to be applied at the distance of 12 from the fulcrum, in the last-mentioned side; what must that power be to keep the lever in balance?

Now, it is clear that the

Here (6 × 7) + (15 x 8) + (29 × 9) 423 the effect of the three weights on the one side of the fulcrum, and 17 × 9 = 153 = the effect of the power on the other side. effect of the weight is far greater than the effect of the power; and the difference, 423-153 273, requires to be balanced by a power applied at the distance of 12, which will evidently be found by dividing 270 by 12, which gives 22.5, the weight required.

==

14. The Roman steel-yard is a lever of the first kind, so contrived that only one movable weight is employed.

TABLE showing the Effects of a Force of Traction of 100 pounds, at different Velocities, on Canals, Railroads, and Turnpike Roads.*

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1,800

1;350

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10,800 1,800

1,350

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10,800 1,800

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1,350

3.66 55,500 39,400 14,400 10,800 4.40 38,542 27,361 14,400 5-13 28,316 20,100 14,400 | | 5.86 21,680 15,390 14,400 10,800 1,800 1,350 7.33 13,875 9,850 14,400 | 10,800 1,800 | 1,350 9,635 6,810 14,400 10,800 1,800 1,350 7,080 | 5,026 14,400 10,800 | 1,800 | 1,350 5,420 3,848 14,400 10,800 1,800 1,350 4,282 3,040 14,400 | 10,800 14.66 3,468 2,462 14,400 10,800 13.5 19.9 1,900 1,350 14,400 10,800

4

5

6

8.80

10.26

8

9

11.73
13.20

10

1,800 | 1,350

1,800 1,350

1,800 | 1,350

*The force of traction on a canal varies as the square of the velocity; but the mechanical power necessary to move the boat, is usually reckoned to increase as the cube of the velocity. On a railroad or turnpike, the force of traction is constant, but the mechanical power necessary to move the carriage increases as the velocity.

TABLE of the Tractive Power of the Locomotive Engine, when the adhesion is from one-fifth to one-fifteenth that of the insistent weight of the Driving Wheels.

Insistent weight on driv.

wheels, in tons.

TRACTION IN LBS. WHEN THE ADHESION IS IN THE FOLLOWING RATIOS.

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5

2240

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6

2688

1600 1400 2440 1920

1680

1244.4 1493.3 1344

1120

1018.1 1221.8

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933.3 1120 1033.8 960 1306.6 1206.1 1120 1493.4 1378.4 1280 1680 1550-7 1440

861.5

800

746.6

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10

4480

3733.3 3200 2800

2489

2240

2036.3

1866.6 1723

1600

1493.3

11

4928

4106.6

3520 3080

2737.7

2464

2240

2053.3

1895.4 1760

1642.6

12

5376

4480

3840 3360 2986-6

2688

2443.6

2240

2067.7 1920

1792

13

5824

4853.3 4160 3640 3235.5 2912

2647.2

2426.6

2240

2080

1941-3

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