Johann Heinrich Jacob Müller.

Principles of physics and meteorology online

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force alone impels the point from a to b in one second ; and if the
action of this force were to cease at the instant the point reaches
by and the point be then solely subjected to the action of the
second force, it would, at the close of another second, reach r.
Hence, if both forces act simultaneously, the point a must, in the
course of a second, reach the same point r.

An illustration will make this more evident. A ship acted

Fig. 5.




upon simultaneously by two forces, the stream and wind, starts
from the point A on the side of a river. Let us assume that the
vessel will be urged obliquely across the river by the action of the
wind alone, in a definite time, say a quarter of an hour, going
from A to B ; and assume it to be borne during the same period
of time by the force of the stream alone, if there were no wind
from A to C ; then it would in the same period of time go from A
to D, if both wind and stream acted simultaneously, that is, it
must reach the point D in a quarter of an hour, when impelled by
the simultaneous action of the two forces, as it would have gone
from A to B in a quarter of an hour, if acted upon solely by the
wind, and from B to D during the next quarter of an hour when
impelled only by the stream.

The line a r (Fig. 6) is the diagonal
of the parallelogram a b r c, which by
means of the law we have mentioned may
be thus expressed.

The resultant of two forces which simultaneously act at any
angle upon a material point is such as to tend to move the point




46 STATICS.

through the diagonal of the parallelogram, which we may con-
struct from the lines corresponding to each of the component or
lateral forces.

As the line which a body passes over in a given time is pro-
portionate to the force which impels it, and as in determining
the resultant we only endeavor to find its direction and relations
of size to both component forces, the law maybe thus expressed:
"If two lines be drawn in the direction of two forces, and
through their point of contact, and their length to be proportionate
to the respective forces, the diagonal of the parallelogram which
is determined by these two lines will represent the resultant both
in magnitude and direction."

As a state of equilibrium must be established between three
forces, if each be equal and opposed to the resultant of the other
two, we may easily, by means of an experiment pertaining to
statics, test the correctness of the law of the parallelogram of
forces.

To the leaf of a table there are attached two vertical rods, each

of which has a movable slide
Flg> 7- bearing a pulley that turns

easily upon its axis in a verti-
cal plane. The rods must be
so screwed on that the vertical
planes of both pulleys coincide.
If now we have a line over the
pulleys, attaching at one end a
weight a, at the other end a
weight c, and lastly, a weight
b between the pulleys, the
whole will be in a state of equilibrium in any definite position of
the threads; we have three forces acting upon the point o in the
directions o p, o q, and o r, and it is easy to ascertain whether
those relations between the amount and direction of the forces
really exist, such as the law of the parallelogram offerees requires.
Supposing by way of illustration, that a = 2 and c = 3 ounces,
how great must be the force at b if the angle p o q be 75 ?
According to the above law the resultant may easily be obtained,
by construction, as in Fig. 8.

If the angle r s t measure 75, and r s = 2 and s t = 3
(some unit being assumed), we shall find that the diagonal




PARALLELOGRAM OF FORCES.



47




s p = 4. Thus, if the angle p o q = 75, the weight 6 must
be equal to 4 ounces ; and if we at-
tach a weight of 4 ounces to the string,
we shall find that the angle p o q will
measure 75; and this we may easily
prove by holding a figure of larger
dimensions behind the thread, r s cor-
responding with o p, and s t with o q.
If b had been made larger than 4, and
all the other parts of the figure were
left unaltered, the angle p o q would
be less than 75 ; and the smaller we
make the weight at 6, the larger will
be the angle p o q.

When both forces are equal, the resultant divides the angle
which they make with each other into two equal parts.

When the two forces are unequal, the resultant divides their
angle into unequal parts, approaching more nearly to the direc-
tion of the larger force.

As we can find the resultant of two forces acting upon a point,
so it is likewise easy to ascertain the resultant of any given
number of forces, nothing more being necessary than to find the
resultant of the two first forces, then their resultant with the third
force, and so on.

As two forces can be replaced by a single force, so conversely,
we may substitute two forces for one ; and we see further, that an
infinite number of different systems of forces may have the same
resultant, and conversely, that one force may be replaced in
innumerably different ways by a sys-
tem of two forces. But if it were
required that the force a r should be
replaced by two other forces, one of
which should have the direction a y,
and the magnitude a c, the problem
is perfectly definite, there being but
one way to complete the parallelogram,
and to find the component or lateral force a b.

From the parallelogram of forces are derived the laws of equi-
librium in all simple machines; and thes we now proceed to
describe.



Fig/9.




48



STATICS.



The Inclined Plane affords a practical illustration of the decom-
position of forces. When a weight rests upon a plane, which
forms an angle x with the horizon, the gravity of the body acting
in the direction a b is no longer at right angles to the plane, and,
consequently, the latter has not to support the full pressure of the

Fig. 10.




weight of the load. In fact, the gravity of the body may be de-
composed into two forces, the one of which acts at right angles
with the plane, causing the pressure, while the other, acting
parallel with the inclined plane, urges the body down it. The
magnitude of these two forces may easily be obtained by con-
struction. If a 6 represent the magnitude and the direction of
gravity, we have only to draw a line at right angles with the in-
clined plane through a, and another parallel with it, then join b
and d, and drop the perpendicular b c. The line a d represents
the amount of pressure which the plane has to support, a c the
amount of force which impels the load down the inclined plane,
or, in other words, the pressure upon the plane, and the force
which tends to move the body parallel to the inclined plane are
to the weight of the body as the lines a d and a c are to a b.

But the triangle a b c is similar to the triangle jR S T and a b:
a c = R S: S T, and, consequently, the force which urges the
body down the inclined plane is to its weight as the height of the
plane is to its length. If we denote by x the angle which the
inclined plane makes with the horizon, then it is evident that
a c = a b sin. x and b c = a b cos. x; and, therefore, if P repre-
sents the weight of the body, the pressure which the plane has to



INCLINED PLANE. 49

support is equal to P cos. x, and the force that urges the body
down the plane is equal to P sin. x.

We will attempt to make this point clearer by the following
illustration. If we lay a load in a little carriage, and place it
upon an inclined plane, it will roll down ; this may, however, be
hindered by attaching to the carriage a line, passing round a
pulley, and having the weight P suspended from its other ex-
tremity. Supposing the little carriage and its load to weigh 100
ounces, and the angle x to be 30, then S T = J R S, and,
consequently, a c = J a b; that is to say, the force which urges
the carriage down the plane is equal to the half of its weight,
and the carriage will, therefore, be prevented from rolling down,
if we make the weight P equal to 50 ounces.

If the angle x were 19 30', then would S T = J R S, and

100

then the weight P need only be = 33 ounces to prevent the

o

carriage from rolling down the plane.

As sin. 14 30' nearly = J, that is to say, when the angle

x = 14 30' S T= i R S, in this case P must = l = 25

4

ounces.

In order to make' experiments with reference to different angles
of inclination, we must use a polished board, which, by means of
a hinge, is so secured to a fixed horizontal board as to admit of
being placed at any angle of inclination that may be required.
The pulley round which the line is passed may be secured to the
board, but we may also easily make use of one of the rods in Fig.
7 for this purpose, as the slide may be pushed up and down to
raise or depress the pulley to the elevation required. Instead of
attaching the weight P directly on the line, we' lay it in a scale
which has been weighed, and, together with its contents, must be
made equal to the computed weight P.

We daily see the practical application of the inclined plane.
Every road leading up an ascent is an inclined plane, on which
weights are lifted from valleys to the summit of hills ; for instance,
in order to draw a loaded wagon up a hilly road, besides the
force necessary to overcome the friction (which is likewise required
upon even ground), we must apply another force to sustain the
equilibrium with that portion of gravity acting parallel with the
inclined plane, and which increases with the steepness of the road.
5



50



STATICS.



For this reason it is preferable to make a road winding circuit-
ously round a hill, instead of carrying it directly upward. It fre-
quently happens in erections of almost every kind that the mate-
rials for building are raised to the required height by means of
inclined planes. This application of the inclined plane was
known to the ancients ; and it is highly probable that the Egypt-
ians availed themselves of it in order to raise the huge blocks of
stone which they employed in constructing their pyramids.



Fig. 11.



Fig. 12.





The Screw is an inclined plane wound round a cylinder. Let a b c,
Fig. 12, be a rectangular piece of paper whose horizontal side, a
5, is equal to the circumference of the cylinder, Fig. 11, the paper
be so wound around the cylinder that a b shall form the periphery
of its base, the hypothenuse a c will wind round the cylinder in
an uniformly ascending curved line, o p q r; if the point a coin-
cide with the point o, b will also coincide with o, and c will be
vertically over o at r. The curved line o p q r, which is repre-
sented in our figure, is termed the thread of the screw ; and its
reverse side has been drawn white in order to show that the entire
curvature of the line from o to r, is the distance of two contigu-
ous threads.

If we imagine a triangle continued along the thread of the screw
round the cylinder, we obtain a screw with a triangular thread, as
shown in Fig. 13; and, if we suppose a
Fig. 13. Fig. 14. parallelogram wound in like manner round
the cylinder, we have a flat-threaded screw,
as represented in Fig. 14. A screw cannot
by itself be applied to remove or lift heavy
weights, or to exercise any strong pressure ;
for to effect these purposes it must be so com-
bined with a screw-box or nut (which is
a concave cylinder, on the interior of which a




THE SCREW. THE WEDGE. 51

corresponding spiral cavity is cut), that the elevations of the one
may accurately fit into the depressions of the other. If we suppose
the screw to be fixed vertically, then every revolution must cause
an elevation or depression of the nut. If a wqjght lying on the nut
should be raised by the turning of the screw, it is evident that the
same principles are at work here, as in an inclined plane of equal
elevation. The steepness of the convolutions of the screw is
inversely proportional to the distance between two contiguous
threads as compared with the circumference of the cylinder.

The screw is used partly to lift heavy weights, and partly to
sustain great pressure, the resistance acting in some cases upon
the screw itself, and in others upon the screw-box. In estimating
the effect of a screw, we must not lose sight of the fact that fric-
tion plays a conspicuous part in its action ; but of this we shall
speak presently. In order to make use of the screw as a power-
ful machine, the turning force is not applied directly to the cir-
cumference, but to a lever, or arm, as we may observe in all
screw-presses.

The Wedge. Another form of applying the inclined plane is
the wedge, which is used to cleave wood and masses of stone. By
thrusting wedges under their keels, ships
are raised for the purpose of being re- Flg> 15 '

paired in the docks. The wedge is the
principal agent in the oil-mill. The seeds
from which the oil is to be extracted are
introduced into hair bags, and placed
between pieces of hard wood. Wedges inserted between the bags
are driven in by allowing heavy beams to fall on them. The
pressure thus excited is so intense that the seeds in the bags are
formed into a mass nearly as solid as wood. All our cutting
implements, as knives, chisels, scissors, are nothing more than
wedges. It must be perfectly clear to every one that the action
of the wedge may be referred to that of the inclined plane.

The Pulley is a round thin disc, hollowed out on its edges, and
turning upon an axis passing through its centre at right angles
with its plane.

We divide pulleys into the fixed and movable. Fixed pulleys
are such as have an immovable axis, and simply allow of things
being turned round them. If a string or line be passed round a
part of the circumference of a fixed pulley, and forces act at either




52



STATICS.



Fig. 16.




extremity, a state of equilibrium will not be brought about unless
the force which stretches the line on the one
side be equal to the force acting on the
> other. Fig. 16 represents a pulley, c, mov-
ing round a fixed axis, and the line stretched
by forces acting in the directions a b and d
e. If we suppose the lines d e and a b pro-
longed to their intersecting point, m, it is
evident that if m were a point connected
with the pulley, we could change the points
of application of the two forces from a and d
to m without altering anything in the action ;
and thus we should have two forces meeting
at m, which could only be in equilibrium if their resultant were
so. If the two forces meeting in m, and acting in the directions
m b and m e, are equal, their resultant will bisect the angle b m e,
and will then pass through the fixed central point c, and we shall
have a condition of equilibrium. If one of the two forces be
greater than the other, the resultant will no longer pass through
the fixed point, and consequently equilibrium will not be main-
tained.

The pressure which
the axis of the pulley has
to sustain must clearly
be equal to the resultant
of the two forces ; and if
the directions of the forces
be parallel, as in Fig. 17,
the pressure upon the
axis is equal to the sum of
the two forces, in which
we might also include the weight of the pulley.*



Fig. 17.



Fig. 18.




* It might be objected that this is arguing in a circle ; for we have already used
the pulley as an experimental illustration of the correctness of the proposition of the
parallelogram of forces, and now we derive the conditions of equilibrium in a pulley
from the parallelogram of forces. This, however, is not so unreasonable as it may
at first sight appear ; for, although the conditions of equilibrium between all the
forces acting on a pulley can only be understood in all their bearings by means of
the theory of the parallelogram of forces, we may easily perceive, even without any
knowledge of these laws, that the powers acting on both ends of a string (the ten-
sion of the string remaining constant) passed round a pulley must be equal if they



THE PULLEY.



53



Fig. 19.




Fig. 20.



A movable pulley cannot be in equilibrium unless the forces
by which the two ends of the string are stretched are equal to one
another, for in this case only does their result-
ant pass through the central point of the disc.
The action of this resultant is not arrested
owing to the fixed condition of the axis, but
owing to there being a third power in the axis
in the direction of the resultant, which is
equal and opposed to it. This third power
is usually applied to a hook fastened on the
block. At Fig. 18 it is represented by a
weight.

When the two ends of the line passing
round the movable pulley are parallel to each
other, as in Fig. 19, it is evident that the force
with which each end is drawn, is half as great
as the weight hanging to the block. When
two groups of pulleys, of which the one is
fixed, and the other movable, are so con-
nected by a line that the latter may pass from
the one to the other, we have a system of pul-
leys.

Fig 20 represents a system consisting of
three fixed and three movable pulleys. The
weight q which is attached to the common
block of the three movable pulleys is sup-
ported by the six lines which connect the
upper and lower pulleys ; and consequently,
as the weight is equally divided between the
lines, each is drawn by one-sixth of the weight
q ; and if sixty pounds weight were suspended
to the bottom, each line would be drawn upon
by a force of ten pounds.

If we observe the external line to the left
side which connects the lowest of the mov-
able pulleys with the highest of those that are




are to be in equilibrium ; for, as each force tends to turn the pulley iu an opposite
direction, a state of equilibrium can only be brought about when these forces are
equal, as must already have been made evident to all in our illustration in Fig. 7.

5*



54



STATICS.



Fig. 21,



fixed, we shall see that this line runs round the top pulley, and
hangs freely down on the right side. Now, in order to establish
a state of equilibrium, it is necessary that the tension of the line
should be equal on the two sides of the upper pulley ; and as we
have seen that the line to the left is drawn with the force of one-
sixth of the weight at q, it is necessary to attach a weight equal
to one-sixth of q to the end of the line, in order to obtain a state
of equilibrium. We may, therefore, again poise a weight of
sixty pounds, by attaching to the line a weight of ten pounds.

As the amount of weight bearing upon the lines depends upon
their number, that is the number of pulleys composing the system,
it follows that another relation will be established between the
forces and weights, but this can readily be obtained by a similar
mode of deduction.

The Lever. Suppose a line passed round a pulley, to the end
of which the weighty (Fig. 21) is attached ; whilst, on the other
side, the line is drawn in the direction a 6, with a force equal to

the weight p. Here, how-
ever according to the theory
of the parallelogram offorces,
we may decompose the forces
meeting at a, and acting in
the direction a 6, into lateral
forces, one of which acts in
the direction of d from a,
being a prolongation of the
direction of the radius m a,
while the direction of the
other force a f is parallel
with g p.

If the pulley be fixed, the
action of the force a d will be
counteracted by the resist-
ance of the fixed central
point m ; we may, therefore, entirely remove the component force
acting in the direction a d, without disturbing the equilibrium,
and we may replace the active force a b by its component force
acting in the direction of a f.

If the line a c represent the force p acting in the direction a b,
then the line a /will give the amount of the component force P,




THE LEVER. 55

and, without further working out the relations of size between a c
and a f or p and P, we see at once that P must be larger than
p; we might, therefore, without disturbing the equilibrium, re-
place the force p, acting in the direction a b by another force P,
likewise acting at a, but in a vertical direction.

Instead of letting the force P act directly at , we may, without
disturbing the equilibrium, choose any part of the line a f as the
point of application ; we may, for instance, let the force P act at
the point A, where the lines af and g m intersect each other, and
thus we have two rectangular forces p and P in a state of equili-
brium, at the ends of a straight line h g revolving round m.

The two forces are unequal, as their Fi 22

respective points of application at h and g
are at unequal distances from the fulcrum
m. We have now to ascertain the relation
which exists between the magnitude of the
forces p and P, and the lengths h m and
g m. The triangles c af, Fig. 21, and a A m are similar to each
other, and hence a c : af = hm : a m. But the lengths a c and
a f are to each other as the forces p and P; thus we have

p : P = h m : a m,
and since a m = g m,

p: P = h m : gm,
or

p:P=L:l . . . . (1),

if we make the length h m = L and g m = I. Or, to express
the same fact inwards, we may say that the forces P and p bear
an inverse ratio to the distances of their points of application from
the fulcrum m.

A straight, inflexible rod turning on a fixed point is called a
lever. If two opposite forces at right angles to its direction be
applied at two different points of a lever, a state of equilibrium
will be established when the above condition has been fulfilled.
The distance of the point of application of a force from the fulcrum
is called the arm of the lever ; and we may, therefore, thus ex-
press the condition of equilibrium in the lever : Two forces tend-
ing to draw the lever in opposite directions are in equilibrium
when they bear an inverse proportion to the corresponding arms
of the lever.

If, for instance, the arm h m (Fig. 22) was half the length of



56 STATICS.

g m, then P must be twice as large as p. A force p may be in
equilibrium with a hundredfold larger force P if the arm m g be
100 times as long as the arm h m.

From the proportion (1), it follows that P L = p I, that is
to say, in order that two forces in a lever shall be in equilibrium,
it is necessary that the products of the force and the distance
at which it acts from the fulcrum be equal for both forces. If,
for instance, the force p = 6 ounces, and the arm be 12 inches,
it would be necessary, in order to bring them to a state of
equilibrium, to have on the other side an arm three times shorter,
that is, 4 inches, acted on by a force three times greater, that is,
3 X 6 = 18 ; it is evident that the product 6 12 is equal to the
product 4 x 18.

The product obtained by multiplying the force by the arm of
the lever is called the static moment of the force. We may also
define the static moment of a force as that force which, acting at
an arm of one unit on the opposite side of the fulcrum, shall pre-
serve the state of equilibrium.

In Fig. 23, if we assume that the force to the right = 6, and

the arm of the lever = 5, the
23 - static moment of the force will be

-i \ H^ 5 x 6 = 30 ; then, if the force on
the left hand is to be in a state of
equilibrium with the former, the
static moment of the two must be
equal, and the force acting on the left
side on an arm equal to 3 must have a value of 10. But, instead
of letting the force 6 act on the arm of length 5, we might, with-
out disturbing the equilibrium, apply a force of 30 on the arm of
length 1 ; and, in like manner, the force 10 acting on the other
side of the lever, which equals 3, may be replaced by a force ot
30 acting at an arm equal to 1.

When several forces act on each side of the fulcrum, a state of
equilibrium will be established, if the sums of the static moments
on each side be equal. For example, in Fig. 24 m is the fulcrum,

and on one side the force 5

Fig. 24.

acts on the arm 2, the force 2

I I I J^ ' I ' I i I on the arm 4, and the force

IB ran pj| Hi ill 4 on the arm 6, while on the

s ^ other side the forces 10 and



l



STATICS. 57

3 act on the arm 3 and 4. Now, all these forces will be in a
state of equilibrium, for the sums of the static moments of both
sides are equal. The sum of the static moments on the one side is
5x2 + 2x4 + 4x6 = 42, and the sum of the same forces
on the other side is 10 x 3 + 3 x 4 = 42. Instead of the force
5, which acts at the distance 2, we might have the force 10 at
the distance 1 ; thus also the forces 2 and 4, acting at the dis-
tances 4 and 6, may be replaced by two other forces, 8 and 24,
acting at right angles to arm 1. We may likewise substitute the
forces 10, 8, and 24, acting at the distance 1, for the forces 5, 2,



Online LibraryJohann Heinrich Jacob MüllerPrinciples of physics and meteorology → online text (page 3 of 55)