(243.) We have hitherto considered the power and weight as acting on
the lever, in directions perpendicular to its length and parallel to
each other. This does not always happen. Let A B, _fig. 83._,
be a lever whose fulcrum is F, and let A R be the direction of the
power, and B S the direction of the weight. If the lines R A
and S B be continued, and perpendiculars F C and F D
drawn from the fulcrum to those lines, the moment of the power will be
found by multiplying the power by the line F C, and the moment of
the weight by multiplying the weight by F D. If these moments be
equal, the power will sustain the weight in equilibrium. (185).
It is evident, that the same reasoning will be applicable when the
arms of the lever are not in the same direction. These arms may be of
any figure or shape, and may be placed relatively to each other in any
position.
(244.) In the rectangular lever the arms are perpendicular to each
other, and the fulcrum F, _fig. 84._, is at the right angle. The
moment of the power, in this case, is P multiplied by A F, and
that of the weight W multiplied by B F. When the instrument is in
equilibrium these moments must be equal.
When the hammer is used for drawing a nail, it is a lever of this kind:
the claw of the hammer is the shorter arm; the resistance of the nail
is the weight; and the hand applied to the handle the power.
(245.) When a beam rests on two props A B, _fig. 85._, and
supports, at some intermediate place C, a weight W, this weight is
distributed between the props in a manner which may be determined by
the principles already explained. If the pressure on the prop B be
considered as a power sustaining the weight W, by means of the lever of
the second kind B A, then this power multiplied by B A must
be equal to the weight multiplied by C A. Hence the pressure on
B will be the same fraction of the weight as the part A C is of
A B. In the same manner it may be proved, that the pressure on A
is the same fraction of the weight as B C is of B A. Thus, if
A C be one third, and therefore B C two thirds of B A,
the pressure on B will be one third of the weight, and the pressure on
A two thirds of the weight.
It follows from this reasoning, that if the weight be in the middle,
equally distant from B and A, each prop will sustain half the weight.
The effect of the weight of the beam itself may be determined by
considering it to be collected at its centre of gravity. If this point,
therefore, be equally distant from the props, the weight of the beam
will be equally distributed between them.
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