given as an example of the Wedge, but when the nail is at rest the resistances on its sides are counterbalanced by friction and not by a Power on the head. The nail is indeed driven into its place by blows on the head; but it does not belong to Statics to investigate the effect of blows in producing motion.

B

W

219. The Screw. The screw consists of a right circular cylinder AB with a uniform projecting thread abcd... traced round its surface, making a constant angle with straight lines parallel to the axis of the cylinder. This cylinder fits into a block C pierced with an equal cylindrical aperture, on the inner surface of which is cut a groove the exact counterpart of the projecting thread abcd...

Thus when the block is fixed and the cylinder is in

troduced into it, the only manner in which the cylinder can move is backwards or forwards by turning round its axis.

220. In practice the forms of the threads of screws may vary, as we see exemplified in the accompanying two figures.

The left-hand figure

most nearly resembles that which is taken for investigation in elementary books; it is usual to disregard the thickness and the breadth of the projecting thread, that is to consider both of these as practically very small. We may form a good idea of the figure of the

thread in the following geometrical manner:

Let ABNM be any rectangle.

Take any point C in BN, and make CD, DE, EF,... all equal to BC.

Join CA and through D, E, F,...draw straight lines parallel to CA, meeting AM at c, d, e,. Then if we conceive

ABNM to be formed into the convex surface of a right cylinder, the straight lines AC, CD, dE, eF,... will form the curve which determines the figure of the

screw.

A

B

Let the angle CAB be denoted by a; then CB= AB tan a; if r be the radius of the right circular cylinder and π express as usual the ratio of the circumference of a circle to its diameter, AB=2πr; thus CB=2πr tan a. CB is the distance between two consecutive threads of the screw measured parallel to the axis. The angle a may be called the angle of the Screw.

221. Suppose the axis of the cylinder to be vertical; and let a Weight W be placed on the Screw. Then the Screw would descend unless prevented by some Power, P. This Power we shall suppose to act at the end of a horizontal arm firmly attached to the cylinder; the length between the axis of the cylinder and the point of application of the Power we shall call the Power-arm.

222. When there is equilibrium on the Screw the Power is to the Weight as the distance between two adjacent threads is to the circumference of a circle having the Power-arm for radius.

Let be the radius of the cylinder, b the length of the Power-arm, a the angle of the screw. The screw is acted on by the Weight W, the Power P, and resistances exerted by the block. These resistances act at various

points of the block, but as the thread is supposed smooth they all act at right angles to the thread; thus their directions all make an angle a with vertical straight lines. Denote these resistances by R, S, T,... Resolve each resistance into two components, one vertical and the other horizontal. Thus the vertical components are R cos a, Scosa, Tcos a,..; and the horizontal components are R sin a, S sin a, Tsin a,.

...

By reasoning as in Arts. 104 and 105 we find that there are two conditions which must hold when the machine is in equilibrium, namely:

The components parallel to the axis must balance each other, thus

W=(R+S+T+...) cos a ;

and the moments of the forces round the axis must balance each other, thus

W b cos a

2πb

distance between two consecutive threads

circumference of circle of radius b

223. The most common use of a Screw is not to support a Weight, but to exert a pressure. Thus suppose a fixed bar above the body denoted by W in the figure of Art. 219; then, by turning the screw, the body will be compressed between the head of the screw and the fixed bar. A bookbinder's press is an example of this mode of using a screw. The theory of the machine will be the same as in Art. 222; W will now denote the pressure exerted parallel to the axis of the screw by the body which is compressed.

EXAMPLES. XVI.

1. A Wedge is right angled and isosceles, and a force of 50 lbs. acts opposite to the right angle: determine the other two forces.

2. A Wedge is in the form of an equilateral triangle, and two of the forces are 40 lbs. each; find the third force.

3. Find the vertical angle of an isosceles Wedge when the pressure on the face opposite this angle is equal to half the sum of the two resistances.

1 4'

4. The tangent of the angle of a Screw is, the radius of the cylinder 4 inches, and the length of the Power-arm 2 feet: find the ratio of W to P.

5. The circumference of the circle corresponding to the point of application of P is 6 feet: find how many turns the Screw must make on a cylinder 2 feet long, in order that W may be equal to 144 P.

Screw is

1

6. The distance between two consecutive threads of a inch, and the length of the Power-arm is 5 feet: find what Weight will be sustained by a Power of 1 lb.

4

7. The angle of a Screw is 30°, and the length of the Power-arm is n times the radius of the cylinder: find the mechanical advantage.

8. The length of the Power-arm is 15 inches: find the distance between two consecutive threads of the Screw, that the mechanical advantage may be 30.

XVII. Compound Machines.

224. We have already spoken of the mechanical advantage of a machine, and have defined it to be the ratio of the Weight to the Power when the machine is in equilibrium; see Art. 166. Now we might theoretically obtain any amount of advantage by the use of any mechanical power. For example, in the Wheel and Axle the advantage is expressed by the ratio of the radius of the Wheel to the radius of the Axle; and this ratio can be made as great as we please: but practically if the radius of the Axle be too small the machine is not strong enough for use, and if the radius of the Wheel be too great the machine becomes of an inconvenient size.

Hence it is found advisable to employ various compound machines, by which great mechanical advantage may be obtained combined with due strength and convenient size. We will now consider a few of these compound machines.

Let AB, BC, CD be three levers, having fulcrums at K, L, M respectively. Suppose all the levers to be horizontal, and let the middle lever have each end in contact with an end of one of the other levers. Suppose the system in equilibrium with a Power, P, acting downwards at A, and a Weight, W, acting downwards at D.

Let Q be the pressure at B between the two levers which are in contact there, and R the pressure at C between the two levers which are in contact there; these pressures may be supposed to act vertically.

T. M.

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