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Now area ABCD=AB(AD+BC), but AB=T, AD=V, BC=V +ƒT,

.. area ABCD={T(V+V+ƒT)=VT+}ƒT.

Also area ABFD=AB (AD+BF), where AB and AD have the same values as before, and BF=V—ƒT,

.. area ABFD={T(V+V−ƒT)=VT—{ƒT. Therefore generally, if S represent the space described after T seconds,

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in which formula the sign is to be taken according as the motion is accelerated or retarded.

47. TO DETERMINE A RELATION BETWEEN THE SPACE DESCRIBED AND THE VELOCITY ACQUIRED BY A BODY WHICH IS PROJECTED WITH A GIVEN VELOCITY, AND WHOSE MOTION IS UNIFORMLY ACCELERATED OR RETARDED.

Letv be the velocity acquired after T seconds, then by (v-V). equation (34), v = V ±ƒT, .. T: 土 f

Now area ABCD=AB(AD+ BC), where

C

P

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(v-V)

P

B

, AD=V, BC=v,

.. area ABCD=1

·V)

f

· 1

(V + v) = ¿ (v2 — V3),

f

area ABFD=‡AB(AD+BF), where AB=T=_ (~—V),

AD=V, BF=v.

.. area ABFD=-1

(v_V) (v+V)
f

=

f

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Therefore generally, if S represent the space through which

the velocity v is acquired, then S

:.v2—V2=±2ƒS

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in which formula the sign is to be taken according as the motion is accelerated or retarded.

If the body's motion be retarded, its velocity v will eventu ally be destroyed. Let S, be the space which will have been

described when v thus vanishes, then by the last equation 0—V2= — 2ƒ'S..

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(37),

where V is the velocity with which the body is projected in a direction opposite to the force, and S, the whole space which by this velocity of projection it can be made to describe.

If the body's motion be accelerated, and it fall from rest, or have no velocity of projection, then v2-0=+2ƒS,

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Let S, be the space through which it must in this case move to acquire a velocity V equal to that with which it was projected in the last case, therefore V' 2fS,. Whence it follows that S, S,, or that the whole space S, through which a body will move when projected with a given velocity V, and uniformly retarded by any force, is equal to the space S, through which it must move to acquire that velocity when uniformly accelerated by the same force.

In the case of bodies moving freely, and acted upon by gravity, f equals 32 feet and is represented by g; and the space S, through which any given velocity V is acquired, is then said to be that due to that velocity.

WORK.

48. WORK is the union of a continued pressure with a continued motion. And a mechanical agent is thus said to WORK when a pressure is continually overcome, and a point (to which that pressure is applied) continually moved by it. Neither pressure nor motion alone is sufficient to constitute work; so that a man who merely supports a load upon his shoulders, without moving it, no more works, in the sense in which that term is here used, than does a column which sustains a heavy weight upon its summit; and a stone, as it falls freely in vacuo, no more works than do the planets as they wheel unresisted through space.*

49. THE UNIT OF WORK.-The unit of work used in this country, in terms of which to estimate every other amount

* Note (j) Ed. App.

of work, is the work necessary to overcome a pressure of one pound through a distance of one foot, in a direction opposite to that in which a pressure acts. Thus, for instance, if a pound weight be raised through a vertical height of one foot, one unit of work is done; for a pressure of one pound is overcome through a distance of one foot, in a direction opposite to that in which the pressure acts.

50. The number of units of work necessary to overcome a pressure of M pounds through a distance of N feet, is equal to the product MN.

For since, to overcome a pressure of one pound through one foot requires one unit of work, it is evident that to overcome a pressure of M pounds through the same distance of one foot, will require M units. Since, then, M units of work are required to overcome this pressure through one foot, it it evident that N times as many units (i. e. NM) are required to overcome it through N feet. Thus, if we take U to represent the number of units of work done in overcoming a constant pressure of M pounds through N feet, we have

U=MN..

(39).*

51. TO ESTIMATE THE WORK DONE UNDER A VARIABLE

PRESSURE.

Let PC be a curved line and AB its axis, such that any one of its abscissæ AM,, containing as many

C

equal parts as there are units in the space through which any portion of the work has been done, the corresponding ordinate M,P, may contain as many of those equal parts, as there are in the pressure under which it is then being done. Divide AB into exceedingly small equal parts, AM,, MM, &c., and draw the ordinates M,P,, M,P,, &c.; then if we conceive the work done through the space AM, (which is in reality done under pressures varying from AP to M,P,), to be done uniformly under a pressure, which is the arithmetic mean between AP and M,P,, it is evident that the number of units in the work done through that small space will equal the number of square units in the trapezoid APP,M, (see Art. 45.), and similarly with the other trape

*Note (k) Ed. App.

zoids; so that the number of units in the whole work done through the space AB will equal the number of square units in the whole polygonal area APP ̧Ð ̧Ð ̧, &c., CB.

2

But since AM, M,M,, &c., are exceedingly small, this polygonal area passes into the curvilinear area APCB; the whole work done is therefore represented by the number of square equal parts in this area.

Now, generally, the area of any curve is represented by the integral fyda, where y represents the ordinate, and æ

the corresponding abscissa. But in this case the variable pressure P is represented by the ordinate, and the space S described under this variable pressure by the abscissa. If therefore U represent the work done between the values S, and S, of S, we have.

S2

2

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Mean pressure is that under which the same work would be done over the same space, provided that pressure, instead

E

B

D

of varying throughout that space, remained

the same: thus, the mean pressure in respect to an amount of work represented by the curvilinear area AEFC, is that under which an amount of work would be done represented by the rectilineal area ABDC, the area ABDC being equal to the curvilinear area AEFC; the mean pressure in this case is represented by AB. Thus, to determine the mean pressure in any case of variable pressure, we have only to find a curvilinear area representing the work done under that variable pressure, and then to describe a rectangular parallelogram on the same base AC, which shall have an area equal to the curvilinear area.

If S represent the space described under a variable pressure, U the work done, and p the mean pressure, then U

PSU, herefore p=.*

P

52. To estimate the work of a pressure, whose direction is not that in which its point of application is made to move. Hitherto the work of a force has been estimated only on

*Note (1) Ed. App.

the supposition that the point of application of that force is moved in the direction in which the force operates, or in the opposite direction. Let PQ be the direction of a pressure, whose point of application Q is made to move in the direction of the straight line AB. Suppose the pressure P to remain constan, and its direction to continue parallel to itself. It is required to estimate the work done, whilst the point of application has been moved from A to Q.

Resolve P into R and S, of which R is parallel and S perpendicular to AB. Then since no motion takes place in the direction of SQ, the pressure S does no work, and the whole work is done by R; therefore the work R. AQ.

Now R=P. cos. PQR, therefore the work P. AQ cos. PQR. From the point A draw AM perpendicular to PQ, then AQ cos. PQR=QM; therefore work=P. QM. Therefore the work of any pressure as above, not acting in the direction of the motion of the point of application of that pressure, is the same as it would have been if the point of application had been made to move in the direction of the pressure, provided that the space through which it was so moved had been the projection of the space through which it actually moves. The product P. QM may be called the work of P resolved in the direction of P.

The above proposition which has been proved, whatever may be the distance through which the point of application is moved, in that particular case only in which the pressure remains the same in amount and always parallel to itself, is evidently true for exceedingly small spaces of motion, even if the pressure be variable both in amount and direction; since for such exceedingly small variations in the positions of the points of application, the variations of the pressures themselves, both in amount and direction, arising from these variations of position, must be exceedingly small, and therefore the resulting variations in the work exceedingly small as compared with the whole work.*

*Note (m) Ed. App.

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