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22. [From the last two Propositions it appears, that if a straight lever be at rest, which is acted on perpendicularly by two weights or other forces P and Q respectively applied at the distances CM and CN from the fulcrum C, then, whether P and Q act on the same or on different sides of the fulcrum, the proportion

is always true.

P: Q CN: CM

When the Lever is used to balance a given weight or force Q by the application of another force P, Q is usually called "the Weight", and P "the Power".

If CM, the perpendicular distance from the fulcrum at which the Power acts, be greater than CN the distance at which the Weight acts, the Power required to balance the Weight is less than it. In this case "force" is said "to be gained" by the application of the lever. But if CM be less than CN, the Power required to balance the Weight is greater than it, and "force" is then said "to be lost". I

23. PROP. IV. To explain the different kinds of levers.

LEVERS are divided into three classes.

(1) Where the Power and the Weight act on opposite sides of the fulcrum,

(2) Where the Power and the Weight act on the same side of the fulcrum, but the perpendicular distance at which the Power acts is greater than that at which the Weight acts.

(3) Where the Power and the Weight act on the same side of the fulcrum, but the distance at which the Power acts is less than that at which the Weight acts.

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Of the FIRST class the poker, when used to raise the coals, is an instance; the bar of the grate on which the poker rests being the Fulcrum, the force exerted by the hand the Power, and the resistance of the coals the Weight. In the common balance, the Power and the Weight are equal forces perpendicularly applied at the ends of equal arms. In the steelyard, the Power, and the Weight are. perpendicularly applied at the ends of unequal arms. Pincers, scissars, and snuffers, are double levers of this kind, the rivet being the fulcrum in each.

Since CM may be either greater or less than CN, the Power in levers of this class may be either less or greater than the Weight, and consequently "Force" may be either "gained" or "lost" by using them.

Of the SECOND class, a cutting blade, such as those used by coopers, moveable round one end by means of a handle fixed at the other, is an example. An oar is also such a lever, the Fulcrum being the extremity of the blade which remains fixed, or nearly so, during the stroke, the muscular strength and weight of the rower being the Power, and the Weight being the resistance of the water to the motion of the boat, which is counteracted and overcome at the rowlocks. A pair of nutcrackers is a double lever of the second class.

Here, since CM is greater than CN, the Power is always less than the Weight, or "Force is gained” by using levers of the second class.

An example of the THIRD class is the board which the turner (or knifegrinder) presses with his foot to put the wheel of his lathe in motion; the Fulcrum being the end of the board which rests on the ground, the Power being the pressure of the foot, and the Weight being the pressure produced at the crank on the axletree of the wheel. Fire tongs, or sugar tongs, are double levers of this kind; the Weight being in either case the resistance of the substance grasped. The limbs of animals are also such levers. Thus, if a weight be held in the hand and the arm be raised round the elbow as a Fulcrum, the weight is supported by muscles fastened at one extremity to the upper arm, and again attached to the forearm, after passing through a kind of loop at the elbow.

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Here, since CM is less than CN, the Power is greater than the Weight, or "Force is lost" by making use of levers of the third class.

24. PROP. V. If two forces acting perpendicularly at the extremities of the (straight) arms of any lever, balance each other, they are inversely as the arms.

Let the forces P and Q, acting perpendicularly at the extremities of the straight arms CM and CN of any lever whose fulcrum

is C, balance each other; p They are inversely as the

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M

N

M'N' through C, and make CM'= CM and CN'=CN. Let a force P', equal to P, act perpendicularly at M', and a force Q', equal to Q, act at N'.

Now since CM' CM and PP, P will produce the same effect as P on the lever; Axiom 11. Art. 18;

But, by supposition, P balances Q,

.. P' would balance Quad Jay

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And since CN' = CN and Q = Q, Q' will produce the same effect on the lever as Q; Ax. III.

But Q would balance P',

.. Q' would balance P';

1

wherefore, by Prop. ir. P': Q: CN': CM', and .. P Q CN CM.

Wherefore, "If two forces, &c." Q. E. D.

25. PROP. VI. If two forces acting at any angles on the arms of any lever, balance each other, they are inversely as the perpendiculars drawn from the fulcrum on the directions, (i. e. the lines), in which the forces act.

Let P and Q be two forces which acting at any angles on the arms CA and CB of any lever ACB, balance each other about the A fulcrum C; They are inversely as

the perpendiculars CM and CN M

drawn from the fulcrum on the lines in which they act.

PY

N

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B

Since a force produces the same effect at whatever point in its line of action it is applied (Art. 15), we may suppose the force P applied at M; and that it may be so applied let a rod (CMA)

without weight be fastened to CA. In like manner Q may be supposed to be applied at N perpendicularly to the part CN of the rod CNB which is added to CB.

And since P acting at M perpendicularly to CM balances Q acting at N perpendicularly to CN, we have by Prop. .,

PQCN : CM;

and therefore when P and Q acting at A and B in the lines AMP, BNQ balance, they are as CN : CM. Wherefore, "If two forces, &c."

t

Q. E. D.

26. PROP. VII. If two weights balance one another on a straight lever when horizontal, they will balance each other in every position of the lever.

Let P and Q be two weights that balance each other round the fulcrum C

on the lever ACB when it

is horizontal; They will ba

lance each other on the lever A M

when it comes into any other

position, as A'CB'.

B

B

Q

Produce QB' to cut AB

in N.

Since weights act perpendicularly to the horizon, A'P and NB'Q are each perpendicular to the horizontal line ACB;

... angle CMA = right angle = angle CNB',

and angle A'CM = opposite angle B'CN; Euc. 1. 15. ... also angle CA'M = angle CB'N, and the triangles are equ angular.

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