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(5) A CYLINDER is a solid described by a rectangle (ABCD) revolving round D

one of its sides (AB) which remains fixed.

A

C

B

[The side (AB) that remains fixed is the length of the cylinder, and is called its axis.

The surfaces described by the two sides (AD and BC) of the rectangle which are adjacent to the fixed side, are circles in form, and are parallel to each other; they are called the ends, or bases, of the cylinder.]

17. DEF.

A LEVER is a rigid and inflexible rod, moveable in one plane round a point in the rod called the FULCRUM.

In the following propositions the thickness of the rod is neglected, and the lever is considered to be a geometrical line; it is also supposed to be without weight.

18. The investigation of the properties of the Lever will be made to depend on the following Axioms, which the mind readily admits as true.

AXIOM I. If two equal weights, or forces, act perpendicularly upon a horizontal straight lever, they will balance round a fulcrum placed at the middle point between the points at which they are applied; and the pressure on this fulcrum will be the sum of the weights.

AXIOM II. If any two weights, or forces, acting perpendicularly on a horizontal straight lever, balance each other, the pressure on the fulcrum is equal to the sum of the weights.

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AXIOM III. The efforts made by two equal forces, acting perpendicularly at the extremities of equal arms of a lever, to turn the lever round, are equal.

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19. PROP. A horizontal prism, or cylinlinder, of uniform density, will produce the same effect by its weight as if it were collected at its middle point.

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Let P and P be two equal weights acting perpendicularly at M and N on the straight horizontal lever AB which is move

2PA

M
A

N

B

able round any fulcrum

P

2P

in it; They produce the

same effect as if they were applied at C the middle point between M and N.

At C let a weight 2 P act perpendicularly downwards, and also let a pressure equal to 2 P act upwards at C. Since these two forces are equal and opposite they will counteract each other, and the tendency of the lever to turn round the fulcrum will remain unaltered.

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Now since P at M and P at N would produce a pressure downwards of 2P on a fulcrum at C, (Axiom I.), the pressure upwards exerted by such a fulcrum would be 2 P. The pressure therefore of 2P, which we have supposed to act upwards at C, will serve instead of such a fulcrum, and will therefore balance P at M and P at N. Wherefore these three forces, (namely, P at M, P at N, and 2P acting upwards at C), will have no tendency whatever to turn the lever about, but its tendency to turn about, (which has been shewn to be the same as when the

lever was acted on by no other forces than P at M and Q at N), arises entirely from a force 2P applied at C, and acting downwards.

Now supposing that a horizontal prism (or cylinder) of uniform density, would produce the same effect by its weight, as if the whole matter composing it were collected into a number of small equal weights placed at equal distances along a horizontal straight line of the same length as the prism (or cylinder), then, since it appears from what has been proved above that every pair of weights equidistant from the center of the line would produce the same effect as if it were collected at the center, the whole prism therefore, or cylinder, (being made up of such pairs of equal weights), will produce the same effect by its weight as if it were collected at its center. Q. E. D.

COR. Hence it follows that a horizontal prism, or cylinder, of uniform density will balance on its middle point, and the pressure on a fulcrum placed there will be the weight of the prism, or cylinder.

20. PROP. II. If two weights, acting perpendicularly on a straight lever, on opposite sides of the fulcrum, balance each other, they are inversely as their distances from the fulcrum ; and the pressure on the fulcrum is equal to their

sum.

* A uniform prism, or cylinder, may be divided, by means of equidistant planes parallel to its ends, into equal small prisms or cylinders lying on each side of the plane passing through its center. Now if these be supposed to be condensed and placed at equal distances along a straight line equal to the length of the prism, or cylinder, the supposition made in the text will not appear unreasonable.

Let there be an uniform cylinder whose length is AB, and whose weight A M E C N

is P+Q. If it be placed

horizontally it will balance

P

A

-B

on a fulcrum C placed at its middle point, (Prop. 1. Cor.), and the pressure on C will be the weight of the cylinder, namely P+Q.

Let AB be divided in E so that d

*

AE : AB :: P: P+Q;\ bax N

then, since the cylinder is supposed to be of uniform density,

weight of the part AE of the cylinder: weight of the whole

:: AE: AB

:: P: P+Q, by construction;

but weight of the whole P+ Q,

=

.. weight of part AE = P ;

and weight of EB = Q.

But if the parts AE, EB be collected at their middle points M and N, and act perpendicularly on the lever, they will produce the same effects by their weights as before; Prop. 1. Therefore the weights P and Q, acting perpendicularly at M and N, will balance about C, and the pressure on C will bẹ P + Q.

Now

P AE AB-BE 2BC-2 BN BC-BN CN QEB AB-AE 2AC-2 AM AC-AM CM3

=

or PQ CN : CM.

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

21. PROP. III. If two forces, acting perpendicularly on a straight lever in opposite directions and on the same side of the fulcrum, balance each other, (1) they are inversely as their distances from the fulcrum, and (2) the pressure on the fulcrum is equal to this difference.

AR
AC

Let the forces P and Q, acting perpendicularly at M and N on the straight PA lever MC in opposite directions and on the same side of M the fulcrum C, balance each M other; Then P: Q:: CN: CM, PV and, (Q being the force which

VR

is the nearer to the fulcrum), the pressure on the fulcrum Q-P.

=

Let the fulcrum at C be removed, and the place of its resistance (R) supplied by a force acting perpendicularly to the lever in the direction of P's action.

Since P and R are counterbalanced by Q, they must produce a pressure at N equal and opposite to Q. Let Q be removed, and its place supplied by a fulcrum on the contrary side of the lever to that on which Q acted, sustaining the pressure (namely Q) which P and R produce. The equilibrium therefore still existing round N, we have by Prop. II.,

PR: CN: NM

:

.. P: P+R CN CN + NM :: CN: CM. But Q, the pressure on the fulcrum at N, is (by Prop. 11.) P+ R.

=

PQ CN : CM.

Also, RQ- P.

Wherefore," If two forces, &c."

Q. E.-D.

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