Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

66 , M E C H
, Thcory- . point H. 1'hen, lince the forces MD, ND are refolvable
into MH, HD and into NH, HD, and fince MH, HN
deftroy each other, the force upon the back is fuftained
by 2 HD. Confequently, the force upon the back is
to the fum of the refiltances as 2 HD is to 2 MD, or as
HD is to MD. But the angle ADM, which the direc¬
tion of the forces makes with the back of the wedge, is
equal to DMN, and HD is the fine of that angle, MD
being radius, therefore the force upon the back is
to the fum of the refinances as fin. ADM : radius.
Q. E. D.
Corollaries. 128.C0R. I . Since the angle AMD=:MDC-fMCD,
the angle MDC is the difference between MCD the fe-
miangle of the wedge, and AMD the angle which the
direction of the refilling forces makes with the face of the
wedge, and fince HD is the cofine of that angle, MD
being radius, we have the force upon the back to the
fum of the refiftances, as the cofine of the difference be¬
tween the femiangle of the wedge and the angle which
the direction of the refilling forces makes with the face
of the wedge, is to radius.
•
Prop. III.
129. When there is an equilibrium between three
forces a£ting perpendicularly upon the Tides of
a wedge of any form, the forces are to one an¬
other as the lides of the wedge.
This is obvious from Dynamics, $ 144. Cor. 2.
where it is fhewnthat when three forces are in equilibrio,
they are proportional to the fides of a triangle, which
are refpedlively perpendicular to their directions.
Prop. IV.
130. When the power adling upon the back of a
wedge is in equlibrio with the refiftances op-
pofed to it, the velocity of the power is to the
velocity of the refiftance as the refiftance is to
the power.
Tig. 3. Produce DM to K, and draw CK perpendicular to
DK. Then, by Art. 122. the power is to the refin¬
ance as MD : DPI. Let the wedge be moved uni¬
formly from D to C, and DK is the fpace uniformly
defcribed by the refilling force in the direftion in which
it acts j therefore, the velocity of the poAver is to the
velocity of the refiftance as DC : DK ; that is, on ac¬
count of the equiangular triangles DHM, DKC, as
MD : DH 5 that is, as the refiftance is to the power.
Sect. IV. On the Screw.
131. Definition. A fcrerv is a cylinder with an
inclined plane AATapped round it, in fueh a manner, that
the furface of the plane is oblique to the axis of the cy¬
linder, and forms the fame angle with it in every part
of the cylindrical furface. When the inclined plane
winds round the exterior furface of a folid cylinder, it
is called a male fcrew ; but Avhen it is fixed on the in¬
terior circumference of a cylindrical tube, it is called
a female fcrew. In the female lerew, the fpiral grooves
formed by the inclined plane on the furface of the cy¬
lindrical tube, mull be equal in breadth to the inclined
A N I C S.
plane in the male ferew, in order that the one may Theory.
move freely in the other. By attending to the mode ' *
in which the fpiral threads are formed by the circum¬
volution of the inclined plane, it will appear, that if
one complete revolution of the inclined plane is deve¬
loped, its altitude will be to its bafe as the diftance be-
tiveen the threads is to the circumference of the fcreAv.
1 hus, let a b c (fig. 4.) be the inclined plane, whofe Fig. 4.
bafe is a c and altitude b c, and let it be wrapped round
the cylinder MN (fig. 5.) of fuch a fize that the points
a, c may coincide. The furface ab of the plane (fig. 4.)
will evidently form the fpiral thread a d e b (fig. 5.),
and ab the diftance betAA^een the threads Avill be equal to
be (fig. 4.) the altitude of the plane, and the circumfer¬
ence of the fcreAv MN Avill be equal to ac the bafe of the
plane. If any body, therefore, is made to rife along
the plane a deb in fig. 5. or along the fpiral thread of
the fcreAv, by a force acting in a direction parallel to
a deb, there Avill be the fame proportion betAveen the
poAver and the refiftance as if the body afeended the
plane abc (fig. 4.).
132. A male fcrew Avith triangular threads is repre-pj 6 „
fented by AB (fig. 6.), and its correfponding female
fcreAv by AB (fig. 7.). A male fcrew Avith quadrangu¬
lar threads is exhibited in fig. 8. and the female fcreAv Fi g
in which it works in fig. 9. The friftion is confidera- lg’ ’ 9
bly lefs in quadrangular than in triangular threads,
though, when the fcrew is made of Avood, the triangular
threads iliould be preferred. When the ferews are me¬
tallic and large, the threads fliould be quadrangular; but
the triangular form is preferablein fmall fcreAvs. When
the fcreAv is employed in practice, the porver is always
applied to the extremity of a lever fixed in its head.
This is ftiCAvn in fig. 10. where AB is the lever afting
upon the fcrew BC, Avhich works in a female fcrew in Ir
the block F, and exerts its force in bending the fpring
CD ».
Prop. I.
133. If the fcrew is employed to overeome any
refiftance, there will be an equilibrium when
the power is to the refiftance as the diftance be¬
tween two adjacent threads is to the circum¬
ference defcribed by the power.
Let FAKbe a fection of the fcrew reprefented in fig. Fig. 11..
8. perpendicular to its axis 5 CD a portion of the inclined
plane Avhich forms the fpiral thread, and P the porver,
Avhich, when applied at C in the plane ACF, will be
in equilibrium with a weight upon the inclined plane
CD. Then, in the inclined plane, Avhen the direction
of the poAver is parallel to the bafe, avc have (Art. 72.)
P: VV, as the altitude of the plane is to the bafe, or (Art.
131.) as the diftance betAveen two threads is to the Avhole
circumference FKCF. If avcfuppofe another poA\rer PTo
aft at the end of the lever AB, and deferibe the arch
HBG, and that this poAcer produces the fame effeft at
B as the power P did at C, then (Art. 36.), avc lurve
P': P=:CA : BA, that is, as FKCF is to the circumfe¬
rence HBG ; but it Avas fheAvn before, that P : W~ as
the diftance between tA\To contiguous threads is to
FKCF; therefore, by compofition, P: W as the diftance
betAveen two threads is to HBG or the circumference
of a circle whofe radius is AB. f). E. D.
j34. Cor. I. It is evident from the propofition that
the
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