Encyclopaedia Britannica > Volume 8, DIA-England

376
DYNAMICS.
Of Accelf1- Fig* 20.
rating and
{j\
B\ I
2J
101. (2.) Let us suppose that the body is impelled from
A (hg. 20), towards the point C, by a force proportional
to its distance from that point. This force may be repre¬
sented by the ordinates DA, EB, eb, &c. to the straight
line DC. We may take any magnitude of these ordinates ;
that is, the line DC may make any angle with AC. It
will simplify the investigation if we make the first force
AD AC. About C describe the circle AHa, cutting
the ordinate EB in F; let eb be another ordinate, cutting
the circle in/very near to F; draw CH perpendicular to
AC, and make the arch H/t — /F, and draw be parallel to
HC ; join FC and DH, and draw F^ perpendicular to/6.
Let IML be another ordinate.
The area DABE is to the area DAKL as the square
of the velocity at B to the square of the velocity at K.
But DABE is the excess of the triangle ADC above the
triangle EBC, or it is half of the excess of the square of
CA or CF above the square of CB, that is, half the square
of BF. In like^ manner, the area DAKL is equal to half
the square of KM; but halves have the same ratio as the
integers; therefore the square of BF is to the square of
KM as the square of the velocity at B to the square of the
velocity at K; therefore the velocity at B is to the velo¬
city at K as Bh is to KM. The velocities are proportional
to the sines of the arches of the quadrant AFH described
on AC.
Cor. 1. The final velocity with which the body arrives
at C is to the velocity in any other point B as radius to
the sine of the arch AF.
Cor. 2. The final velocity is to the velocity which the
body would acquire by the uniform action of the initial
force at A as 1 to ^2; for the rectangle DACH ex¬
presses the square of the velocity acquired by the uniform
action of the force Dx\ ; and this is double of the triangle
DAC; therefore the squares of these velocities are as I
and 2, and the velocities are as ^/l, and ^2, or as 1
to \/2.
102. Cor. 3. The time of describing AB is to the time
of describing AC as the arch AF to the quadrant AFH.
For when the arch 1/is diminished continually, it is plain
that the triangle/zF is ultimately similar to CFB, by rea¬
son of the equal angles Ctb (or CFB) and fiF, and the
right angles CBF and/Fi; therefore the triangles ^F
and CBF are also similar. Moreover, Bb is equal to Fy,
F/is equal to AH, which is ultimately equal to cC; there¬
fore, since the triangles/^F and CFB are similar, we have
F<7 : 1/~ bB : Cb — FB : HC ; therefore B6 is to cC
as FB to HC, that is, as the velocity at B to the velocity
at C ; therefore B6 and cC are described in equal mo¬
ments when indefinitely small; therefore equal portions
Ff, AH, of the quadrant correspond to equal moments of Of 4
the accelerated motion along the radius AC ; and the rating
arches AF, FM, MH, &c. are proportional to the times Retar
of describing AB, BK, KH, &c. Fore
Cor. 4. The time of describing AC with the unequally ''"’’V
accelerated motion, is to the time of describing it uni¬
formly with the final velocity as the quadrantal arch is to
the radius of a circle; for if a point move in the quad¬
rantal arch so as to be in F, / M, H, &c. when the body
is in B, A, K, C, it will be moving uniformly, because the
arches are proportional to the times of describing those
portions of AC ; and it will be moving with the velocity
with which the body arrives at C, because the arch AH is
ultimately — Cc. Now if two bodies move uniformly with
this velocity, one in the arch AFH, and the other in the
radius AC, the times will be proportional to the spaces
uniformly described; but the time of describing AFH is
equal to the time of the accelerated motion along AC,
therefore the proposition is manifest.
103. Cor. 5. If the body proceed in the line Ca, and
be retarded in the same manner that it was accelerated
along AC, the time of describing AC uniformly with the
velocity which it acquires in C is to the time of describ¬
ing ACa with the varied motion, as the diameter of a
circle to the circumference ; for because the momentary
retardations at K', B', &c. are equal to the accelerations at
K and B, &c. the time of describing AC« is the same with
that of describing AHa uniformly with the greatest velo¬
city ; that is, to the time of describing AC uniformly as
AHa to AC, or as the circumference of a circle to the
diameter; therefore, &c. N. B. In this case of retard¬
ing forces it is convenient to represent them by ordinates
K'L, B'E, aD', lying on the other side of the axis ACa;
and to consider the areas bounded by these ordinates as
subtractive from the others. Thus the square of the ve¬
locity at K' is expressed by the whole area DACK'L'C,
the part C'K'L' being negative in respect of the point
DAC. This observation is general.
Cor. 6. The time of moving along KC, the half of AC,
by the uniform action of the force at A, is to that of de¬
scribing ACa by the varied action of the force directed to
C, and proportional to the distance from it, as the diame¬
ter of a circle to the circumference ; for when the body
is uniformly impelled along KC by the constant force IK,
the square of the velocity acquired at C is represented by
half the rectangle IKCH, and therefore it is equal to the
velocity which the variable force generates by impelling
it along AC (by the way, an important observation). The
body will describe AC uniformly with this velocity in the
same time that it is uniformly accelerated along KC.
Therefore by Cor. 5 the proposition is manifest.
Cor. 7. If two bodies describe AC and KC by the ac¬
tion of forces which are everywhere proportional to the
distances from C, their final velocities will be proportional
to the distances run over, and the times will be equal.
For the squares of the final velocities are proportiona
to the triangle ADC, LKC, that is, to AC2, KC2, and
therefore the velocities are as AC, KC. The times of
describing AC and KC uniformly, with velocities propor¬
tional to AC and KC, must be equal; and these times
are in the same ratio (viz. that of radius to j of the cir¬
cumference) to the times of describing AC and KC with
the accelerated motion. Therefore, &c.
Thus, by availing ourselves of the properties of the
circle, we have discovered all the properties or characters
of a motion produced by a force always directed to a fixed
point, and proportional to the distance from it. Some of
these are remarkable, such as the last corollary; and they
are all important, for there are innumerable cases where
this law of action obtains in nature. It is nearly the law
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