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(A CB Co)

A B, that is (100 — 36) — 8

=

A B, the base of the plane; then 10: 8: 600 lbs.: 480 lbs, the pressure against the plane.

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67. If the power (Fig. 33), instead of acting in a direc tion A C, parallel to the plane, should act in a direction D E, making any angle EDC with it, then the power, weight, and pressure against the plane, are respectively as DE, E B, and D B ;* for the weight W may be considered as kept in equilibrio by three forces acting in these directions.

Therefore the power P: W :: DE: EB:: sin D BE or sine C A B+: sine E D B.

P: pressure against the plane P: DE: DB:: sine CAB: sine D E B.

Also, the weight W: pressure against the plane P';: EB: DB:: sine E D B: sine D E B.

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Cor. 1.

When the power acts in the direction D e, parallel to the base of the plane, the three above proportions become

P: W:: Dee B:: BC: A B :: sine CA B: cos. CA B. P: P: De: DB:: BC: A C:: sine C A B : radius. W: Pe B; DB::AB: AC: cos. CAB; radius.

*DB is perpendicular to A C, the same as in last article. The triangles A C B and D C B are right-angled triangles, and are evidently similar.

The triangles A C B and D e B are also similar,

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Cor. 2. The least power will be necessary to sustain a given weight when it acts in a direction parallel to the

plane; for, by form. 1, P =

W sine C A B
sine ED B

and since

W and sine C A B are both given, therefore P is propor

tional to

1
sine EDB'

and will evidently be the least pos

sible when sine E D B is the greatest; that is, when ED B is a right angle, or E D coincides with C D.

Cor. 3. The pressure against the plane will be greatest when the power acts in a direction parallel to the base of P sine DEB the plane; for, by form. 2, P/ sine CA B

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and sup

posing P and the sine C A B given, then P' is proportional to sine D E B, and is therefore greatest when sine DE B is the greatest; that is, when D E B is a right angle, or when D E coincides with D e.

Example 1.

If a weight of 150 lbs. be sustained on an inclined plane by a power acting in a direction parallel to the base of the plane, the length of the plane being 10 feet, and the base 8 feet, required the pressure against the plane.

By Art. 67, Cor. 1, form. 6, we have P' =

WXAC

AB

Here W == 150, A C 10 feet, A B = 8 feet; hence

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Example 2.

Compare the pressures against an inclined plane in the two following cases:

1st, When a body is sustained on an inclined plane by a power acting parallel to the plane.

2d, When a body is sustained on an inclined plane by a power acting parallel to the base of the plane.

Let p represent the pressure against the plane in Case 1st, and P' that in Case 2d. Then, by Art. 66,

Wp: AC: A B.

And by Art. 67, Cor. 1, we have

P: W:: AC: A B.

P/

ex æquo P': p :: A C2: A B2.

That is, the pressure in the latter case : the pressure in the former :: the square of the length of the plane : the square of the base of the plane.

Let the inclination of the plane be 30°, then the length is to the base as 1:3; hence the above pressures are in proportion to each other as 1:, or as 4:3; and if the inclination of the plane is 60°, the length of the plane is double the base; therefore the above pressures are, in this case, as 2o: 1o, or in proportion to each other as 4 : 1.

Example 3.

Two inclined planes A B and B C have the same height BD, and upon these planes two weights P and W keep each other in equilibrio by means of a string going over a pulley fixed at B, the parts of the string B W and B P being parallel to the planes. Prove that P: W:: BC: A B. (Fig. 34.)

By form. 1, the power necessary to sustain W on the WX BD plane A B = and in the same manner the " A B

power necessary to sustain P on the plane BC=PxBD

BC

Now whatever power is exerted at B to sustain W on the

plane A B, the very same power must be exerted at that point to sustain P upon the plane B C in the case of an WX BD PX BD equilibrium; therefore

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A B

=

BC

.. W x B D x B C = P x B D x A B, or W x B C PX A B. Hence, P: W:: BC: A B.

THE WEDGE.

68. The Wedge is an instrument made of iron or some hard substance. Its form, in the most useful cases, is that of a prism contained between two isosceles triangles, as AC B. (Fig. 35.)

69. In the wedge A C B, if the power acting perpendicularly to the back A B is to the force acting perpendicularly against either side A C or B C, as the breadth of the back A B is to the length of the side A C or B C, the wedge will be in equilibrio.

For, by Art. 29, Cor. 2, when three forces are in equilibrium, they are as the corresponding sides of a triangle drawn perpendicular to the directions in which these forces act. But A B is perpendicular to the direction of the force. against the back, and A C, B C are perpendicular to the forces acting against them; therefore the three forces are as A B, A C, B C.

Cor. If we take into the account the resistance at both sides of the wedge, then, if there is an equilibrium, the power at D is to the whole resistance as the back A B is to the sum of the sides A C, B C, or as A B to 2 AC or 2 B C.

In general, the wedge is used for splitting or cleaving wood, and separating the parts of hard bodies, by a blow from a hammer or mallet. The force impressed by percussion, or a blow on the back of the wedge, produces an effect incomparably greater than any dead weight or pressure, such as is employed in the other mechanical powers. And it may also be observed, that the wedge is seldom

L.

urged otherwise than by percussion; and very little can be gathered from the theory, but that the thinner the wedge is, the greater is its power.

THE SCREW.

70. The Screw (Fig. 36) is a spiral groove or thread, winding round a cylinder, so as to cut all the lines drawn on its surface parallel to its axis at the same angle. The spiral may be either on the convex or concave surface of the cylinder, and it is called accordingly either the screw or the nut.*

71. If the power be applied parallel to the base of the screw, and perpendicular to the radius of the cylinder, and if the weight press perpendicularly on the axis, an equilibrium is produced when the power is to the weight or resistance, as the distance between two threads of the screw to the circumference described by the point to which the power is applied,

For the screw is nothing more than the inclined plane A B C wrapped round the cylinder, the base A B of the plane being equal to the circumference of the cylinder's base, and coinciding with it, and the height B C of the plane equal to the distance between two of the threads; and since the power in this case acts parallel to the base, we have, by Art. 67, Cor. 1, P: W:: BC: AB: the distance between two of the threads: the circumference described by the power, which, in this case, is the circumference of the cylinder.

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But when the power is applied at any other point, as P, the effect of that power will be increased in the proportion of O P to O A, or as the circumference described by the point P to the circumference described by the point A ; therefore P: W:: the distance between two of the threads

*Or they are sometimes called the exterior and interior

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