1
T
WO
H
UNDRED
Y
EARS
OF
A
NALY
ZING
R
OBERVAL
’
S
B
ALANCE
John F. Hall
Professor of Civil Engineering, Emeritus
Caltech
Caltech Civil Engineering Report No. 2024
-
01
June 2024
r,
·
e
'
'.J·J·
2
T
able
of
contents
Page
Section 1. Introduction
3
Section 2. Roberval’s balance
4
2.1 Description
4
2.2 Conditions for the weights to balance
5
Section 3.
The analyses
6
3.1
First account: p
rinciple of virtual
velocities
6
3.2 Philippe de
L
a Hire
’s flawed analysis
8
3.3
Antoinne Parent’s correct analysis
10
3.4
Lowest elevation for center of gravity
1
3
3.5
Decomposition of forces
14
3.6 Couples
1
5
3.7
Free
-
body method
18
3.8
More efficient free
-
body method
21
Section 4. Considerations for a practical Roberval balance
23
4.1 Design aspects
23
4.2 Analyses needed for a practical
device
25
Section
5
. Conclusions
29
Section
6
. References
3
1
Section
7
. Figures
34
The figure on the front page is taken from a 1713 paper
(6
)
by Antoine Parent
whose analysis of Roberval’s balance may be the
second
ever
structural analysis of
a complex structure
by the free
-
body method.
3
Section
1
.
Introduction
Gilles Personne de Roberval
(1602 to 1675)
was
a French mathematician and
scientist
who
in 163
3
became the chair of mathematics
at
Collège de France.
He
worked on problems solvable by using limits or infinitesimals before the invention
of calculus (
1
).
In 1669, Roberval made a presentation to the
Académie Royale des Sciences
on a
new type of weighing scale.
Th
is
scale had the peculiar property that if two
weights, each
attached to
one of the scale
’
s two platforms, were in balance, either
one could be shifted right or left on its platform and th
e state of
balance would be
maintained. Roberval explained this behavior by eluding to a principle that was
later to be
come
know
n
as
vitesses virtuelle
, or the principle of virtual velocities
;
see the 1725 edition of Pierre Verignon book
Nouvelle mécanique ou statique
(2)
where a 1717 letter
from Johann Bernoulli to Verignon is discussed.
For solving problems of equilibrium, the principle of virtual velocities was, at the
time, a
competing
tool to the
rule
s of static
equilibrium
that was more convenient
for some situations because
it
main
ly involved geometric considerations
.
Additionally
, b
ecause Roberval’s
invention
resembled a lever in appearance
,
the
property whereby the weights could be
moved
without
creat
ing
an imbalance
seemed to violate the
rules of statics, specifically: the sum of the moments about
the fulcrum of a lever equals zero
.
S
o
,
some doubt was created about the use of
such
rules
to solve problems of equilibrium.
Within a few decades of Roberval’s presentation to the
Académie
,
two
individuals
attempted to analyze the device using the rules of statics, both realizing that it
could not be
viewed
as a simple lever. The first
attempt
was unsuccessful
,
but
the
second
succe
eded
, becoming
one of
the first
analys
e
s of
a complex structure
using
the rules of static
equilibrium
.
Since then, many others published analyses
using
a
variety of
approaches
.
4
The purpose of this report is to
survey
200 years of analyses of Roberval’s balance,
and
representative
samples are
presented in Section 3. The following
S
ection
2
describes
the device and gives the conditions for two weights to balance.
In order to avoid confusion, the term
“balance” will be used in this report when
referring to the state of the two weights being in equilibrium (balancing). The term
“scale” will refer to Roberval’s balance or its variations
, except that the phrase
“Roberval’s balance” will be retained.
Section
2
. R
oberval’s balance
2.1
Description
The
scale
as originally
described
by Roberval is shown in Figure
1
, which is taken
from
reference 3
.
There are two main components: the frame
퐴퐵퐸퐷
and the
vertical post
퐶퐹
along with its pedestal. The frame is divided into six pieces:
T
op
bar
퐴퐵
is continuous through
퐶
where it is attached to the post with a pin
ned
connection that allows the top bar to rotate. The extensions shown in the figure
beyond
퐴
and
퐵
are superfluous and are usually omitted by most authors.
B
ottom
bar
퐷퐸
is continuous through
퐹
where it is attached to the post with a pin
ned
connection similar to that at
퐶
. Left vertical bar
퐴퐷
is pinned to the top bar at
퐴
and pinned to the bottom ba
r at
퐷
. Right vertical bar
퐵퐸
is pinned to the top and
bottom bars at
퐵
and
퐸
, respectively.
The figure omit
s
the pins at
퐴
and
퐵
, but
these should be present like they are at
퐷
and
퐸
.
Left
horizontal
platform
퐺퐻
is
rigidly connected to the left bar at
퐼
, and the right
horizontal
platform
퐿푀
is rigidly
connected to the right bar at
푁
.
A rigid connection is one that does not allow any
relative rotation to occur between the connected components.
The left and right halves of the top and bottom bars are referred to as arms.
Roberval’s balance ha
s
equal arms; that is,
퐴퐶
̅
̅
̅
̅
=
퐶퐵
̅
̅
̅
̅
=
퐷퐹
̅
̅
̅
̅
=
퐹퐸
̅
̅
̅
̅
, where the
overbar denotes length.
Lengths of
the
vertical bars are also equal:
퐴퐷
̅
̅
̅
̅
=
퐵퐸
̅
̅
̅
̅
.
So,
5
t
he frame
퐴퐵퐸퐷
is a mechanism that can change its shape from a rectangle
to a
parallelogram in which the top and bottom bars
ar
e inclined. Because of the way
the frame is supported
on the post
, the left and right bars remain vertical
, and the
platforms remain horizontal.
See Figure 1b.
Weights
푃
and
푄
are attached to the left and right platforms, respectively, either
sitting directly on them or suspended from them. Of interest are the conditions
under which
푃
and
푄
balance, i.e., when the s
tructure
is in a state of equilibrium.
One characteristic of the structure as described is that it is statically indeterminate.
This means that in order to determine the internal forces in all the components,
knowledge of the elasticities of the components
along with their cross
-
sectional
dimensions
is
required
, and this concept was not understood in Parent’s time.
However, it turns out that this indeterminacy does not prevent a
determin
ation of
the conditions under which
푃
and
푄
balance.
Analyses of pinned structures such as Roberval’s balance
typically assume that the
pins fit tightly. However, as a practical matter
for construction
,
tight
-
fitting pins
will cause binding as the
arms of the
rectangle
, with
the
six pinned connections,
incline,
unless the geometry is perfect; this is
related to
Roberval’s balance
being
statically indeterminate. To avoid this problem in construction, a vertical slot
can
be
used for the pin hole at
퐹
, allowing some vertical movement but restraining
horizontal movement. This also makes the
structure
statically determinate. So,
an
analysis of Roberval’s balance can either use perfect geometry with all pins tight
fitting (indeterminate) or with a vertical slot at
퐹
(determinate).
Both have the
same balance condition.
Several
of the
analys
e
s
that are
discussed in
Section
3
consider
more general
versions of Roberval’s balance. Most common is a
scale
with
unequal arm
s
in
which
퐴퐶
̅
̅
̅
̅
=
퐷퐹
̅
̅
̅
̅
but different from
퐶퐵
̅
̅
̅
̅
=
퐹퐸
̅
̅
̅
̅
.
2.2
Conditions for the weights to balance
6
The pertinent variables
include whether the
scal
e has unequal or equal arms, the
position of the
arms
(
horizontal or inclined
), whether the
pin
hole at point
퐹
is
slotted
vertically
(determinate structure) or
un
slotted (
in
determinate),
the locations
where the platforms attach to the vertical bars,
and
where
the two weights are
position
ed
on their respective platforms
.
With proper analysis
and neglecting
friction in the
connections
,
it can be shown that if two weights satisfy
푃
푄
=
퐶퐵
̅
̅
̅
̅
퐴퐶
̅
̅
̅
̅
,
(
1
)
then t
hey will balance
in all of the listed cases. Note that Equation 1
reduces to
푃
=
푄
if the arms are equal
length
.
It should be noted that for the scale with the unslotted hole
at
퐹
, bars
퐴퐵
and
퐷퐸
share the load from the weights and each transfers a portion to the post
퐶퐹
. The
actual proportions are
statically
indeterminate
.
The
analysis leading to Equation 1
omit
s
the
self
-
weight
of the scale.
Of course,
a
ny scale should
be
balance
d
when the two weights are not present. This is not a
problem when the scale has equal arms
, but f
or a scale with unequal arms, a small
additional weight permanently attached to one of the platforms
would
be
necessary.
This additional weight is not included in
푃
or
푄
.
Section
3
.
The
analys
e
s
The papers discussed below
have been
selected because of
their
significance
;
they
do not represent a complete set of analyses in the 200
-
year time period. The
presentation is in chronological order.
Joint friction is neglected, and all consider
the case without a vertical slot where the lower horizontal bar connects to the post
,
i.e., the
statically
indeterminant case
with perfect geometry
.
3.
1
First account
: principle of virtual
velocities
7
In 1670, after Roberval’s presentation
to the
Académie Royale des Sciences
, an
article appeared in the
Journal des
savant
s
(
3
)
entitled
“
Nouvelle
manière
de
balance inventée
par M. de Roberval,
Profess
eu
r Royal des Mathématique
s
dans
l
’Université de Paris
”.
The un
identified
author says the invention is different from
other
s
cales
and seems, at first, to overturn the
principles of statics.
The article describes Roberval’s balance with reference to
the
drawings copied into
Figure
1
,
and
the arms are not specified to be equal
. The main statement is that if
two weights, equal or unequal, balance each other
in a given position
,
then
the
weights can be
relocat
ed arbitrarily on their respective platforms and balance
is
maintained.
This difference in behavior compared to a
simple lever
is
attributed
to
the
difference in how a weight moves when the arms are
rotat
ed
.
In Roberval’s
balance, this motion is the same at any point of attachment on a platform.
The
language used is that b
alance is maintained because a weight makes no more effort
being suspended at one
location
compared to another because the platform moves
parallel to itself
, and this is demonstrated in the figure.
A special point is made
that two weights can balance each other even if they are on the same side of the
post, as suggested
by
Figure
1
b
where the possible suspension point
퐿
for weight
푄
on the right platform is to the left of the post
퐶퐹
.
As mentioned, t
he basis for the statement in the article about balance not
depending on the
location
s of the weights on their respective platforms is that a
weight always makes the same effort
as the platform moves regardless
of where it
is attached.
This
reasoning
is rather vague
,
but
it
is
in line with
the
then current
thinking of what was to be
later
called
the principle of virtual velocities.
One
omission in the article
is that
no condition is given for
when
two weights
푃
and
푄
are
in balance, such as Equation 1.
However,
a direct application of the
principle of virtual velocities
,
푃
·
푣
푃
=
푄
·
푣
푄
,
(
2
)
8
where
푣
푃
and
푣
푄
are the
vertical components of the
velocities of the attachment
points of
푃
and
푄
when the mechanism is active
, directly results in Equation 1
since
the ratio of
푣
푃
to
푣
푄
equals the ratio of the arm lengths
퐴퐶
̅
̅
̅
̅
to
퐶퐵
̅
̅
̅
̅
. I
t is
evident that the formulation applies
for unequal arm lengths, horizontal or inclined
arms, arbitrary locations of the weights on the platforms
,
and arbitrary attachment
locations of the platforms to the vertical bars.
(The formulation also applies for
unslotted or slotted pin hole at
퐹
.)
3.
2
Philippe
d
e La Hire’s flawed analysis
D
e La Hire (1640
-
1718) was a French mathematician
, astronomer
and architect
who published an early book on mechanics (
4
) in 1695
:
Traité de
mécanique
, ou
l'on explique tout ce qui est nécessaire dans la pratique des arts, & les propriétés
des corps pesants lesquelles ont un plus grand usage dans la physique
.
In the section
of
his
book
on the lever, de La Hire devotes his
40
th
proposition to a
discussion of Roberval’s balance, which he refers to as
a
paradox of mechanics.
De La Hire’s drawing
s are
copied into Figure 2, where
퐸퐴퐵퐹
is the frame.
He
takes the arms to be equal and seeks the conditions under which equal weights
balance. His weights are denoted by
푅
, attached to point
퐾
of the left platform
퐾퐼푀
, and
푃
, attached to point
퐷
of the right platform
퐷퐶퐿
, and where
퐷
happens
to be to the left of the post
푆푇
.
퐾
and
퐷
are meant to be arbitrary locations on their
respective platforms
, and the platforms can be mounted at any positions on the
vertical bars.
De La Hire presents two arguments to show that equal weights balance at arbitrary
locations
on their respective platforms
.
The second argument is the briefer of the
two. He says that since any point on the right platform
퐷퐶퐿
moves the same a
s
point
퐴
of the top bar
퐸퐴
as the arms rotate
, the effect of weight
푃
is the same as if
it were a
ttach
ed directly to point
퐴
. Similarly,
he says
weight
푅
can be considered
to be a
ttach
ed
directly to point
퐸
of the top bar.
Th
ese statements are
consistent
9
with
the principle of virtual velocit
ies, but de La Hire continues is a different
direction
by saying that the top
bar acting as a lever carries all of the two weights
.
Although incorrect
in general because of the indeterminacy
,
t
his
allows him
to
conclude
using moment equilibrium
that equal weights
푅
and
푃
balance when the
arms are of equal length, i.e.,
퐸퐺
̅
̅
̅
̅
=
퐺퐴
̅
̅
̅
̅
, which is the correct result.
Then, in a
generalization to the case of unequal arms,
de La Hire
s
a
ys that unequal
푅
and
푃
balance when they are
in
reciprocal ratio to the
arm lengths
, which is also correct,
being equivalent to Equation 1.
The first argument is more complex and involves
many
missteps even though it
also
ends up with the correct answer. The diagram shown as Figure
2
b
is
employed.
Point
푂
is at the intersection of the top bar and the line of action of
weight
푃
. De
L
a Hire begins
by identifying
퐷퐵퐴
as an angular lever
, although
퐷퐵
is not a
physical
component. Apparently, he is considering the right platform and
the right vertical bar as a solid body (
퐷퐶
plus
퐴퐵
) in equilibrium under th
r
e
e
force
s:
푃
applied at
퐷
and force
s
applied at
퐴
and
퐵
.
Lines of action of the latter
two are assumed to be along
푂퐴
and
푂퐵
, respectively,
without justification. With
this assumption, magnitudes of t
hese three forces
acting on the body
퐷퐶
/
퐴퐵
are
proportional to the lengths of the sides of the triangle
푄푉푅
,
whose sides are
perpendicular to the
three
lines of action. [It appears in the figure that
푄푉
has been
drawn perpendicular to
퐷퐵
; it should be perpendicular to
푂퐴
.
And
푄푅
is
perpendicular to
푂퐵
.
]
So,
weight
푃
is to the force applied at
퐵
along
푂퐵
as
푅푉
̅
̅
̅
̅
is
to
푄푅
̅
̅
̅
̅
.
Next, de
L
a Hire
identifie
s
another
angular lever
퐴푂퐻
,
but
it is unclear what solid
body this lever is supposed to represent.
푂퐻
is not a physical component, and
푂퐴
is part of the top bar
퐸퐴
, which is fixed
against translation
at
퐺
and continues past
푂
w
h
ere additional forces are applied.
De
L
a Hire considers the lever
퐴푂퐻
in
equilibrium under th
r
e
e
force
s: force
푋
acting
on
푂퐴
at
퐴
,
force
푍
acting
on
푂퐻
at
퐻
, and a force at
푂
. Lines of action are assumed to be downward, along
퐻퐵
, and
along
푂퐵
, respectively, again without justification. With this assumption, t
he three
10
forces are again proportional to the lengths of the sides of the triangle
푄푉푅
, so
푋
is
to the force applied at
푂
along
푂퐵
as
푅푉
̅
̅
̅
̅
is to
푄푅
̅
̅
̅
̅
.
De La Hire equates the
force applied at
퐵
along
푂퐵
to the
force applied at
푂
along
푂퐵
, and then concludes from the preceding that
푃
=
푋
. Repeating the same
process for the other side of the scale
gives
푅
=
푌
, where
푌
is the force acting
downward on point
퐸
.
It follows that
푋
=
푌
since
푅
=
푃
, and
so
the top bar must
be in equilibrium since its two arms
퐸퐺
and
퐺퐴
have equal length. Thus,
de La
Hire finds that equal forces
푅
and
푃
balance when
arbitrarily located on their
respective platforms
, and his argument applies to the arms being in horizontal or
inclin
ed positions.
This conclusion is correct, but the derivation is
illogical
. It shows that, although de
La Hire was well acquainted with some basic tools of statics, he
had no clear
concept of how to apply them to a structure with some complexity. The reason
that de La Hire’s argument leads to the correct conclusion is that he knew the result
ahead of time.
3.3
Antoine Parent
’s correct analysis
Parent (1666
-
1716) was a French mathematician who wrote extensively on
mechanics
as well as many other subjects
. His textbook
Eléments
de
m
écanique
et
de
p
hysique
(
5
) appeared in 1700
,
and it was followed by several volumes of
collected works.
Parent presents his analysis in
“
XV Mémoire
:
D
e
l’
a
dmirable
b
alance
Roberval
line
”
contained in his 3
rd
volume of
Essais et
r
echerches de
mathématique
et de
p
hysique
(
6
)
published in
1713.
The
purpose of this work is to demonstrate
that Roberval’s balance is no different from other machines in that it can be
analyzed using established mechanical principles, namely,
rules of static
equilibrium
based on
the lever. In the opening paragraph of the memoire, Parent
expresses opposition to the use of
what he calls
hypothetical recipro
cal
11
movements
, which
is
a
reference to the principle of virtual velocities. Parent
labels
th
e
method, or at least how it was being portrayed at the time, as nonsensical
.
Although he admits this method works for Roberval’s balance, he says it leaves the
mechanics unknown.
In
Mémoire
XV
, Parent
goes so far as to say
that
Roberval’s
balance was invented to
destroy
levers
, which refers to
the
mechanics based on
levers
rather than levers themselves
.
[Parent’s dis
trust
of the principle of virtual velocities is understandable since
, at the
time, it was primarily an observation that seemed to give correct results and not
something that had been rigorously shown to be true. The principle is, after all,
based on work concepts, which
had not
then
been
developed. In “
Nouvelle
démonstration physique de l'équilibre dans la pesanteur”
in the 2
nd
volume of
Essais
(
7
), Parent talks about challenging Joseph Saurin on the subject as well as
trying to come up with a physical explanation of the
virtual velocities
principle
.]
The diagram that Parent uses is shown in Figure
3, which is redrawn for clarity in
Figure 4
. The
arm
s of the
scal
e are
showed in
an
inclin
ed position and are
unequal
in
length;
the horizontal distance between the left
vertical
bar
퐸퐹
and the
post
퐺퐻퐾
is
푎
, and that between the
post
and right
vertical
bar
퐶퐷
is
푏
.
These
distances depend on the
inclination
of the
arms
.
The platform
푄푉
is
connected
t
o
left vertical
bar
퐸퐹
at
arbitrary point
푀
, and platform
푇푍
is
connected to
right
vertical
bar
퐶퐷
at
arbitrary point
퐿
.
Weights are denoted by
퐴
on the right
platform
and
퐵
on the left platform
.
Parent focuses on the top and bottom bars of the parallelogram,
퐸퐺퐶
and
퐹퐻퐷
,
which he views as levers with fulcrums at
퐺
and
퐻
, respectively. He
understands
that weight
퐴
,
which he locates at point
퐿
along the line of
the right vertical bar
퐶퐷
, is carried partly by the top bar at
퐶
and partly by the bottom bar at
퐷
, although
the proportions are unknown.
Weight
퐵
is
suspended from
a
n
arbitrary
point
푄
to
the left of the
left
vertical bar
퐸퐹
.
12
The effect of weight
퐵
is decomposed into a force acting on
퐸
and another force
acting on
퐹
, represented by the lines
푃퐸
and
푃퐹
, respectively, and where the
location of point
푃
is unknown
except that it must lie along the line of action of
weight
퐵
.
Essentially, Parent is considering the vertical bar
퐸퐹
and the platform
푄푉
as a free body
subjected to
weight
퐵
and forces at
퐸
and
퐹
. Thus, their lines of
action
intersec
t
(
at
point
푃
),
and their magnitudes can be represented by
퐸퐹
̅
̅
̅
̅
,
푃퐸
̅
̅
̅
̅
and
푃퐹
̅
̅
̅
̅
, respectively.
[Parent omits unit conversion factors for force and length
since such factors cancel out at the end.]
Since the top bar
퐸퐺퐶
is in equilibrium, the first part of weight
퐴
times its lever
arm
푏
is equal to the force represented by
푃퐸
times its lever arm
푁퐺
̅
̅
̅
̅
. Similarly for
the bottom bar
퐹퐻퐷
, the second part of weight
퐴
times its lever arm
푏
is equal to
the force represented by
푃퐹
times its lever arm
푂퐻
̅
̅
̅
̅
.
Parent sums these two
equations to get
퐴
·
푏
=
푃퐸
̅
̅
̅
̅
·
푁퐺
̅
̅
̅
̅
+
푃퐹
̅
̅
̅
̅
·
푂퐻
̅
̅
̅
̅
.
(
3
)
He
then uses geometry to show that
the right side of Equation
3
is
equal to twice
the area of triangles
푃퐸퐺푃
and
푃퐻퐹푃
, which is equal to twice the area of triangle
퐸퐺퐹퐸
, which is equal to
퐸퐹
̅
̅
̅
̅
·
푎
. Thus, combining this result with
Equation
3
and
noting that
퐸퐹
̅
̅
̅
̅
=
퐵
gives
퐴
·
푏
=
퐵
·
푎
, or
퐴
퐵
=
퐸퐺
̅
̅
̅
̅
퐺퐶
̅
̅
̅
̅
,
(
4
)
since the ratio of
푎
to
푏
equals the ratio of
퐸퐺
̅
̅
̅
̅
to
퐺퐶
̅
̅
̅
̅
. This is the balance
condition, and it is equivalent to Equation 1. It
applies for arbitrary location
of
weight
퐵
on the left platform
, and
it can be
easily
shown to apply
for arbitrary
location
of
weight
퐴
on the right platform.
Parent’s
solution
,
which
employs free bodies and the rules of static
equilibrium
,
is
correct
and adequate to explain the
peculiarity
of Roberval’s balance.
A similar
solution that focuses on the equilibrium of the top and bottom bars of the
parallelogram, considered as levers, appears in a book by
Issac
Todhunter in 18
67
;
see
subs
ection
3.
8
. While Parent appears to be the first to
analyze
Roberval’s
13
balance successfully
, at least in terms of what he consider
s
real mechanics,
his
analysis
was
largely forgotten. Only one of the studies included in this report
mentions it
,
that by
Samuel
Clark in his book of 1
764
; see
the following
subs
ection.
3.
4
Lowest elevation for center of gravity
An English mathematician, Samuel Clark
sometimes
identifie
s
himself in his
published works as a teacher of mathematic
s.
Nothing
else was found about him.
In 1764, Clark published
An Easy Introduction to the Theory and Practice of
Mechanics;
Containing a Variety of Curious and Important Problems Investigated
with the Greatest Facility by the Application of One General Property of the
Center of Gravity without Having Recourse to the Composition and Resolution of
Forces
(
8
).
The pr
operty
to which Clark
refers was formulated by
Evangelista
Torricelli during the preceding century (
9
)
. It
states
that
the equilibrium position
for a
group
of
bodies whose movement
s
are
subject to certain constraints
is
characterized by the combined center of gravity being at the lowest
elevation
possible.
T
h
is
principle
later became
recognized as potential energy being a
minimum in the equilibrium state
.
Clark applies the principle to Roberval’s balance as one of many problems he
solves in
his
book. His diagram, shown as Figure
5
, is similar to that of Parent
(Figure
3
)
except that
Clark add
s
the horizontal line
푀푅푁
as a reference for
expressing the
location of the
center of gravity
,
and he
hang
s the weight
퐴
from
point
푍
.
Arms of the scale are
of
unequal
length
.
The following quantities are
defined:
푎
=
퐸퐺
̅
̅
̅
̅
=
퐹퐻
̅
̅
̅
̅
;
푏
=
퐺퐶
̅
̅
̅
̅
=
퐻퐷
̅
̅
̅
̅
;
푐
=
퐾퐹
̅
̅
̅
̅
;
푑
=
퐿퐷
̅
̅
̅
̅
;
푝
=
푅퐻
̅
̅
̅
̅
;
푥
is the sine
of
the inclination
angle
푆퐹퐻
; and the lengths of the cords supporting weights
퐴
and
퐵
are
푠
and
푟
, respectively. The distance from line
푀푅푁
of the combined center
of gravity of weights
퐴
and
퐵
is given by
퐴
(
푝
+
푏푥
−
푑
+
푠
)
+
퐵
(
푝
−
푎푥
−
푐
+
푟
)
퐴
+
퐵
,
(
5
)
14
where
푥
is the position variable. Using Newton’s calculus, Clark takes the fluxion
of this expression and sets it to zero:
퐴푏
푥
̇
−
퐵푎
푥
̇
=
0
,
(
6
)
which gives
Parent’s Equation
4
.
Clark correctly concludes that if this equation is
satisfied, the
scal
e is in equilibrium, independent of where the weights are attached
to the platforms and the value of the position variable
푥
.
Clark includes a translation of Parent’s solution in order to make a comparison. He
remarks that while Parent clearly deduced the property of Roberval’s balance,
Parent’s solution appears much inferior, presumably because of the greater amount
of
effort required
.
3.
5
Decomposition of forces
Jean le Rond d’Alembert (1717
-
1783)
was a French mathematician, physicist and
philosopher who
was a
co
-
collaborator on
the
Encyclopédie, ou dictionnaire
raisonné des sciences, des arts et des métiers
, par une
société de gens de lettres
.
Volume 9, published in 1765, contains an entry on the lever written by d’Alembert
that discusses Roberval’s balance (
10
). He describes the
scal
e as a paradox of
mechanics that is often used to embarrass beginners. He adds that equilibrium of
the
scal
e ha
d
been poorly demonstrated in most previous works that have dealt
with it, and that he is unaware of any treatment as rigorous as the one he gives.
The only previous work that he mentions directly is one by
John
Desaguliers
(
11
),
who use
s
the principle of virtual velocities.
D’Alembert begins with the special case where the
scal
e has equal
arms,
and the
platforms are attached to the vertical bars at mid
-
height. His diagrams are
reproduced in Figure
6
. He seeks to show that balance is maintained when the
equal weights are not equidistant from the post.
The method employed by d’Alembert is decomposition of forces, and he
use
s the
left part of Figure
6
(clarified in the redrawn insert)
, which shows the
scal
e in an
15
inclined position. The weight
퐼
on the right side is represented by the line
푄퐼푂
,
and this weight
is replaced by t
wo
equivalent forces represented by lines
푐퐼
(down
and to the right) and
퐼푑
(down and to the left). Then, force
푐퐼
is replaced by the
downward component
푐푛
and the right directed component
푐푄
, and force
퐼푑
is
replaced by the downward component
푛푑
and the left directed component
푂푑
.
Finally, components
푐푄
and
푂푑
are each replaced by components directed along
bars
푓푐
and
푒푑
, respectively, which go straight into the supports
퐵
and
퐴
, and
vertical components, which a
re equal and opposite with the same line of action.
Thus, regarding a state of balance, the effect of weight
퐼
in its original position is
equivalent to the same weight acting along the vertical bar
푐푑
, i.e., the sum of the
forces
푐푛
and
푛푑
.
In identical manner, weight
퐻
on the left side is seen to be
equivalent to the same weight acting along the vertical bar
푓푒
. D’Alembert
concludes that since weights
퐼
and
퐻
are equal and since their effect is the same as
being placed equidistant from the post, which is a position of equilibrium, the
weights in their arbitrary position
s
must also be in equilibrium.
D’Alembert’s argument a
ssumes
that equal weights
applied along the vertical bars
푓푒
and
푐푑
equidistant from the post are in balance
. While this is certainly true
when the top and bottom bars are horizontal, because of symmetry,
and it is also
true when these two bars are inclined
,
balance
does not seem obvious in the
latter
case
. Other than this point
, h
is logic is sound. He also notes
that the same type of
analysis w
ould
hold when the platforms are connected to the verticals at points
other than mid
-
height. Toward the end of the
entry,
he seems to consider the
conditions under which unequal weights balance, i.e., a
scal
e with arms
of unequal
length
. But his
logic
here is
unclear.
3.
6
C
ouple
s
Louis Poinsot (1777
-
1859) was a French mathematician and physicist who
introduced the concept of a couple in his 1803 book
Eléments de statique
(12)
.
The book went through
many
editions, and in the third (1821), Poinsot added a
discussion of Roberval’s balance
, which he
notes is
an ancient paradox because it
16
was thought to
violate the principle of the lever
, i.e., moment equilibrium
.
However, Poinsot
remarks
th
at Roberval’s
balance is not really a lever at all, but a
compound machine, and so
it
should not be viewed as a paradox. He adds that by
his theory of couples, an explanation of Roberval’s balance does not offer any
difficulty. He also states that prior attempts by using decomposition of forces
did
not give a true explanation, and he specifically mentions d’Alembert’s
encyclopedia article (see
subs
ecti
on
3.5
).
The diagram used by Poinsot
is shown in Figure
7
. Points
퐹
and
푓
are
the
fixed
points around which the top and bottom bars
퐴퐵
and
푎푏
rotate freely. Platforms
퐺퐻
and
퐾퐼
are connected at mid
-
height to the verticals
퐴푎
and
퐵푏
.
Poinsot
first
considers
Roberval’s balance to have
equal arms.
He aims to show that two equal
weights
푃
and
푄
balance
no matter where they are
at
tached to their
respective
platforms.
Poinsot
attache
s
the
equal weights
푃
and
푄
at
arbitrary
points
퐼
and
퐺
,
respectively, as shown in the figure.
Weight
푃
is then moved in to point
퐾
at the
connection to the right vertical bar
, and to compensate for this movement, a
clockwise couple is added whose moment is the product of
푃
and the distance
퐾퐼
̅
̅
̅
.
According to couple theory, this couple can be expressed
in various ways, such
as
two equal and opposite forces of magnitude
푃
′
:
one applied outward at point
퐵
parallel to
the top
bar
퐴퐵
and the other
applied
inward at point
푏
parallel to
the
bottom
bar
푎푏
.
푃
′
is
found as the product of
푃
and
퐾퐼
̅
̅
̅
divided by the
perpendicular distance
푚푛
̅
̅
̅
̅
between the top and bottom bars. Since the two forces
푃
′
and
⎼
푃
′
are directed
either
toward or
away from the fixed points
퐹
and
푓
, they
have no
effect on
the state of
balance
. The same can be done with weight
푄
on the
left
side
.
T
hus,
regarding
a
possible
state of balance
, weights
푃
and
푄
attached at
points
퐼
and
퐺
are equivalent to being attached
a
t
points
퐾
and
퐻
along
the vertical
bars
.
It remains to show that equal weights
푃
and
푄
supported at
퐾
and
퐻
balance, i.e.,
are in equilibrium; the issue is when the
scal
e is in an inclined position. Poinsot
17
says that the top bar
퐴퐵
alone
carrie
s
these two weights, acting as a simple lever
.
Since the lengths
퐴퐹
̅
̅
̅
̅
and
퐹퐵
̅
̅
̅
̅
of the arms
are equal,
balance occurs when
푃
=
푄
,
and this statement
must then
hold
for any locations of these two weights on their
respective platforms.
However, Poinsot neglects to consider that
the top and
bottom
bars share in carrying the weights in an indeterminate way, a
n aspect dealt
with
by Parent many years earlier (
subs
ection 3
.3
); see also
subs
ections
3.7
and
3.8
following
.
Poinsot continues
on
to mention more general cases:
arms of
unequal
lengths
,
forces with arbitrary directions replacing the weights, and platforms attached to the
vertical bars at points other than mid
-
height and at other than right angles. He says
that given two forces in equilibrium
on Roberval’s balance
, they can be moved
parallel to themselves as one wants and equilibrium will be maintained.
This
statement follows from Poinsot’s
initial
analysis.
Furthermore, Poinsot
state
s
in
words a variation of Equation
1
that applies for a given general case
:
values of
푃
and
푄
that balance would be in a ratio that depends on the distances from point
퐹
to the lines of action
of
푃
and
푄
after the
ses lines of action
have been moved to
intersect points
퐵
and
퐴
at the ends of the top bar.
Only one of Poinsot’s generalizations changes the balance condition given
in
Equation 1, the replacement
of
the weights with non
-
vertical forces.
For this case,
which is not of practical interest,
Poinsot does not provide a derivation, but
the
following account is consistent with
the
approach
that he use
s
for the weights.
In Figure 8, the
force
s
푃
and
푄
make angles
훽
and
훾
with respect to vertical.
[
푃
and
푄
have been interchanged
to make Figure 8 consistent with Figure 1.]
These
two forces are moved parallel to themselves to the points
퐴
and
퐵
at the ends of the
top bar. This requires adding two couples represented by the
푃
’
and
푄
’
force pairs,
which go directly into the supports at
퐹
and
푓
.
The relocated forces
푃
and
푄
are
assumed to be carried entirely by the top bar, and summing moments about F gives
푃
·
푑
=
푄
·
푒
,
(
7
)
where the
lengths
푑
and
푒
of the moment arms
are
18
푑
=
퐴퐹
̅
̅
̅
̅
·
푠푖푛
(
훼
−
훽
)
;
푒
=
퐹퐵
̅
̅
̅
̅
·
푠푖푛
(
훼
−
훾
)
.
(
8
)
Substitution into Equation 7 gives the generalized balance condition
푃
푄
=
퐹퐵
̅
̅
̅
̅
·
푠푖푛
(
훼
−
훾
)
퐴퐹
̅
̅
̅
̅
·
푠푖푛
(
훼
−
훽
)
,
(
9
)
which depends on the angles of the forces relative to that of
the top bar.
A final remark here concerns Poinsot’s assumption that the top bar carries all of
the load
s
from
푃
and
푄
.
This is one of the equilibrium solutions for this
statically
indeterminant problem, and since the balance condition is the same
for all
solutions, the correct balance condition is obtained. If the top bar is made
infinitely rigid and the stiffness of the bottom bar is finite (i.e., it is flexible), then
Poinsot’s assumed solution is the correct one. However, it is doubtful that P
oinsot
thought through the problem in this manner.
3.
7
Free
-
body method
John Henry
Pratt (1809
-
1871)
was a
mathematician and clergyman of the Church
of England. In 1836, the first edition of his
The Mathematical Principles of
Mechanical Philosophy and their Application to the Theory of Universal
Gravitation
(
13
) was published.
One section
is on Roberval’s balance, which Pratt
terms
remarkable. He describes the peculiarity of the
scale
as: if two weights
푃
and
푄
are in balance, their positions can be shifted along the platforms, and
equilibrium is maintained. He also mentions that in a balanced state, if o
ne arm is
pushed down and the other consequently rises, the whole will remain at rest in the
position it was left. Pratt
explain
s these features using the principle of virtual
velocities, but he says this does not remove the paradox.
Pratt addresses the paradox later in his book in a chapter o
f
problems. His method
of solution is to write equilibrium equations by summing forces and moments to
zero and then solving the equations. For a system consisting of multiple bodies, he
considers each body separately; thus, he thinks in terms of modern fr
ee
-
body
analysis even though he does not actually draw the free
-
body diagrams.
Instead,