32 PHYSICS. 
more and more from that of the circle, in proportion to ihe increase of the 
velocity. . 
e. Of the Pendulum. 
A body which is capable of oscillation about an axis, neither vertical 
nor passing through the centre of gravity, is called a pendulum. Suppose 
(pl. 16, fig. 20) a material point, B, to be attached in such a manner to the 
extremity of a weightless line, AB, that the line can swing freely about the 
other extremity, A, we shall have a simple or mathematical pendulum; and 
the combination of a small heavy sphere with a thin thread, to which it is 
suspended, may be regarded without serious error as a simple pendulum. 
If such a simple pendulum be brought from its vertical position, AB, which, 
from the laws of statics, it must assume, into the position, AB’, and left to 
itself, it will, by reason of the attraction of gravitation, be brought back 
towards B, and describe the arc, BB’, lying in the same plane with AB. It 
will arrive at B with a velocity corresponding to the depth of fall, that is, 
to the segment of the radius, AB, obtained by letting fall a perpendicular 
from B’ upon this radius. With the velocity thus attained. the resistance 
of the atmosphere and of friction being now left out of the question, the 
material point will endeavor to continue its path in the arc, BB”, on the 
other side of B, until this*velocity previously attained has become zero. 
This point is evidently at B’ when BB’’=B’B. At B” the same state of 
things occurs as at B’, and the pendulum must incessantly perform equal 
oscillations in the arc, B’BB’. In descending the velocity must constantly 
increase, and in ascending decrease, being greatest at the point of equili- 
briurm or the lowest point of the are. The motion of the pendulum from 
B’ to B”, is called the oscillation; that part of it from B to B’ or B”, is the 
ascending semi-oscillation; and from B’ or B” to B, the descending semi- 
oscillation. The amplitude is the arc corresponding to the oscillation ex- 
pressed in degrees, minutes, and seconds: the time necessary to describe 
this are is the duration of the oscillation. The fact that in the material 
pendulum, the duration and amplitude of the oscillation continually decrease, 
results from the friction at the point of suspension and the resistance of the 
atmosphere. The pendulum being thus retarded, cannot reach the height, 
B”, and the altitude attained becomes less and less at each successive 
oscillation. 
The laws of oscillation for the pendulum are as follows :—1. The duration 
of minute oscillations is independent of their amplitudes; they are 
isochronous ; and a pendulum swings through an arc of 5° in neither greater 
nor less time than through an arc of 1°. 2. The duration of an oscillation 
is independent of the material and the weight of the ball, one of lead moving 
no faster than one of cork. 3. The oscillations of two unequal pendulums 
are to each other as the square roots of their lengths. 
When it is said as above that the weight of the pendulum has no influence 
upon the duration of oscillations, it is to be understood as applying only to 
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