William Kingdon Clifford.

# Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) online

. (page 6 of 9)
Font size with the brain-disturbances the mental facts of a sensation
of effort to push the object, and a sensation of pressure on
the hand and of resistance to the effort. Now in the case
of a bullet in contact with a strained spring, there is no-
thing to correspond, either with the nervous and cerebral
mechanism, or with the sensations of effort and resistance.
The only fact common to the two events is that the flux
of momentum of a body A stands in a definite numerical
relation with the strained condition of an adjacent body B.
In one case A is the bullet and B the spring. In the
other case A is the object pressed and B the surface
of my hand.

The scientific meaning of the word force relates only
to this common fact. The various literary and conver-
sational meanings imply a reference, direct or by metaphor,
to the complex structure of an organism, and the mental
facts which accompany it.

70 DYNAMIC.

GENERAL STATEMENT OF THE LAWS OF MOTION.

We may now put together the inductions from ex-
perience which we have lately described.

Definition of Mass. The mass of a body is the number
of grams of platinum in a lump of platinum which under
the same circumstances (contact with the same strained
body in the same state of strain) has the same acceleration
as the given body.

Law of Force. The product of the mass and accelera-
tion of a body which does not rotate depends partly upon
the strain of adjacent bodies and partly upon the position
and state of distant bodies; but not directly upon the
velocity of any of them.

Considered as so depending, this product of mass and
acceleration is called Force. Force is the mass-accelera-
tion of a body expressed in terms of the strain and
position of other bodies.

Law of Composition. The actual mass-acceleration of
a body which does not rotate is the resultant, or vector-sum,
of the mass-accelerations which it might have separately in
virtue of the strains of the several adjacent bodies and the
positions of the several distant bodies.

We may also state this as follows. Forces in a
particle, or in a body which does not rotate, are com-
pounded according to the law of addition of vectors, or
of rotors passing through the same point.

Law of Reciprocity. The mass-acceleration of a body
A, due to the strain or position of another body B, is equal
and opposite to the mass-acceleration of B due to the strain
or position of A.

The mass-acceleration of A due to B will be called
the force in A due to B. When we wish to mention the
equal and opposite forces together, we shall speak of the
forces between A and B.

D'ALEMBEET'S PRINCIPLE. 71

D'ALEMBERT'S PRINCIPLE.

In all that precedes we have considered the change of
momentum only of a body which either does not rotate,
or is so small that its rotation may be neglected. We
now go .on to treat the case of rotating or straining
bodies, by means of a theorem first explicitly stated by
D'Alembert.

Consider a body A, in contact with various strained
bodies B, C, D. Any small por-
tion a of the body A will have
a force (mass-acceleration) which
is partly due to the strain of the
adjacent parts of A, partly to the
position of the distant parts, such
as b, and partly to the position
of distant bodies. If the small
portion be in contact with a
strained adjacent body, as at c, &

it will of course have a force
due to that strain.

The forces in a which are due to the other parts of the
body A, whether adjacent or distant, are called internal
forces. Those which are due to other bodies, whether
adjacent or distant, are called external forces.

Now if a and b are two parts so small that their
rotation may be neglected, we know that the force in a
due to b is equal and opposite to the force in b due to a ;
and this is true whether the two parts are adjacent or
distant. Each of these forces then may be represented
by the same length measured on the line ab, but measured
in opposite senses.

The moment of a force (the mass-acceleration of a par-
ticle) about a point, is the mass of the particle multiplied
by the moment of its acceleration. Suppose that the forces
in a and b due to b and a respectively are represented by
ab and ba ; their moments about any point o will be the
doubles of oab and o&a, these areas being regarded as
vectors. Hence the sum of the moments about any point
o of the force in a due to b and the force in b due to a is
zero.

72 DYNAMIC.

We know also that the moment about o of the whole
force in a is the vector-sum of the moments of the several
forces of which it is the resultant. That is to say

moment of whole force in a = moment of internal forces
+ moment of external forces

and the same thing is true for every point of the body.
Hence, adding all these equations together, we find

sum of moments of forces^

in all parts of A )

_ (sum. of moments') (sum of moments
~ (of internal forces/ (of external forces.

Now when we add together the moments of the in-
ternal forces of all the particles, they disappear in pairs,
because the moment of the force in a due to b is equal
and opposite to the moment of the force in 6 due to a.
Hence the sum of the moments of all the internal forces
is zero. Consequently we find that, about any point
o whatever

sum of moments of forces') _ ( sum of moments
in all parts of A ) ~ (of external forces.

It must be carefully observed that in the demonstra-
the nature of the body A ; it may be rigid or elastic, solid,
liquid, or gaseous. Only the rotation of very small parts
of it has been neglected. This neglect will be subse-
quently remedied. It is implied in this that no part of
the body has a force due to another part like that in a
magnetic north pole due to an element of current, which
is not in the line joining them ; because the law of recipro-
city cannot be understood in such a case without taking-
account of rotation.

FORCE. 73

(C.)

Now the doctrine that Force is at the bottom of things
is essentially an attempt to answer the question " why ? "
For Force is defined to be any cause which changes or
tends to change a body's state of rest or motion. To the
question, " why does the body move in such a way ? "
would be answered "because it is acted on by such and
such a force" force being now* measured by the quantity
of motion which it produces in unit of time.

We are not to conclude, however, that the doctrine of
Force is a mere attempt to answer the question why. If
the common definition included all the meaning of the
word, the above question and answer might be expressed
thus : why does this body's motion change ? because it is
acted on by that which changes it and so be either tauto-
logical or absolutely unmeaning, according to the doubt
expressed above. But as a matter of fact the word has a
connotation going much beyond this definition ; it links to-
gether in itself two meanings, and this linking is at once
the eftect and the expression of belief in a great physical
have replied "other things". Aristotle, for instance, would
no doubt have given this answer. After the step which
substitutes for " other things " some virtue or property of
these things their attraction, repulsion, resistance there
still remains the idea that this property or virtue depends
in some way upon the position, relative to the moving
thing, of the bodies to which it belongs ; that is to say,
that there are some rules (whether we know them all or
not) by which when the relative position of bodies is
known, their force may be calculated ; or, in mathematical
phraseology, that force is a function of situation. The
expression "force exercised by a body" comprehends as
part of its meaning the notion of something that depends
on the situation of the body relative to the thing moved
by its force. When therefore it is held further that the

* Thomson and Tait, Nat. Phil. Art. 220.

74 DYNAMIC.

effect of force is not motion but the acceleration of motion,
to which, therefore it is proportional; the binding up of
these two ideas in the same word is equivalent to a state-
ment of the following law, which for convenience may be
called the law of acceleration by place :

There are rules by which when the position of a body
in relation to surrounding things is known, its acceleration
may be calculated ; and this acceleration is the same, with
whatever velocity the body passes through that position.

(D.)

THE ROTATION OF A RIGID BODY.

When a body has a fixed point o, the angular velocity
of the body may be represented by a vector 6 in
the direction of the instantaneous axis, of length
equal to the rotational velocity. It is so drawn
that when we look back on it the rotation ap-
pears counter-clockwise. Thus a rotation re-
presented by oq will cause p to move away from
the eye.

The resulting velocity of p is the vector product o/oq, op,
or it is N6p if p = op. For p moves in a circle of
radius pm with angular velocity oq; therefore
p = oq.pm perpendicular to the plane qop. But
this is V6p. Or if 0=ip+jq+kr, p=ix+jy+kz,
then p = i (qz ry] + j (rx pz) + k (py qx).

The moment about o of the momentum ofp
is mVpp, or mVpV6p since p = Vdp. For in
general the moment of momentum of p is twice the area
opv multiplied by the mass of p. But this is twice the
area uop multiplied by m, or mVpp if p = op, p =pv = ou.
Expressing this in terms of the components, we find
Vpp = i (py* + pz* rzx qxy) + j (qz* + qx 2 ryz pxy}
+ k (rx* + ry 1 ' pzx qyz).

The moment of momentum of the whole body about o is

MOVING AXES. 75

clearly fm Vpp ; or, putting a = fm (y* + z 2 ), b = fm (z* + # 2 ),
C == jw (x 2 + ?/ 2 )> f= \$myz, g =fmzas, h = fmxy
it is /a = i (ap hq gr) +j ( hp + bq fr)
4- k ( gp fq + cr) = <f> (0) say. Hence
there is an ellipsoid with centre o such that
if 6 (= oq) is a vector to its surface, qt the
tangent plane at q, ot the perpendicular upon it, then p is
parallel to ot and of magnitude ot' 1 . This is called the
momental ellipsoid. Its axes are a set of lines at right-
angles through o such that o =Jmyz = \$mzx = Jmxy ; that
is, they are the same as the axes of the swing-ellipsoid, or
principal axes at o. If ijk coincide with these at the
moment in question, then 0=ip +jq +kr, fj,=iap+jbq+kcr.

By D'Alembert's principle the flux of the moment of
momentum is equal to the moment of the external forces.
Let this be OT, then /i = BT. The problem is, from this
equation to find 0, and thence the position of the body, in
terms of the time. But as (i involves not only the flux of
B but also those of abcfgh, the equation in this form is un-
manageable.

If however no forces act on the body, /i- = r = o, or //, is
constant. Hence ot is fixed in magnitude and direction,
and therefore the tangent plane qt is fixed in space. Since
oq is the instantaneous axis of rotation, q has no velocity ;
therefore the momental ellipsoid rolls on a fixed plane.
This representation of the motion is due to Poinsot.

MOVING AXES.

Let a, /3, 7 be three unit-vectors along the principal
axes of the body, and consider the flux of any vector
p = ax + fty + yz. This vector changes not only on account
of the change in its components x, y, z, but also on account
of the change in the unit- vectors a, /?, 7. Let us first suppose
that p moves with the body, so that x, y, z are constant ;
and that the body has an angular velocity 9 = ap + @q + yr.
Then p = V0p, or

dx + fiy + yz = a. (qz ry) + ft (rx pz) + 7 (py qx) ;
whence d = fir <yq, etc. Next let p move relatively to the

76 DYNAMIC.

body, and let (p) be the relative velocity, or (P)=OLX -

Then the whole flux of p is the resultant of these two

motions, or p = (p) + V6p.

Applying this to the flux of the moment of momentum,
we find -EJ = p = (/JL) + V6p. Or, since the axes are prin-
cipal axes and therefore p = aap + ftbq + jcr, where a, b, c
are constant,

al + @m + yn =

oLp+/3q + yr + a.(c b)qr + /3(a-c)rp + <y(ba) pq.

Here I, m, n are the components of ts. The three equations
here involved are called Euler's equations,

KINETIC ENERGY.

The kinetic energy of a particle is half the scalar
product of rotation and moment of momentum.
For moment of momentum = p, perpendicu-
lar to op, of length =m.op. pm . oq. Hence
its scalar product with oq is m . pm* . oq 3
or mv*, which is twice the kinetic energy.
Therefore, the kinetic energy T of the whole body is

- S0fji = \ (ap* + bq* + cr* - 2fqr - 2grp - 2hpq)
which when the axes are principal axes reduces to
i (ap 2 + b(f + cr 2 ).

If no forces act on the body T is a constant ; this is
equivalent to saying that the extremity of 6 is always on
a fixed ellipsoid with centre o. [Sd/j, + 2T = 0].

For solution of Euler's equation when CT = o, see Kirch-
hoff, 1 . 66. We have <f>=am (nt+e), p= Pcos<, <?=Qsin<,
; where

D2 - n , -

* = - , U = -y - f - , Ji =

a. a c' b.b-c ' c.a c '

a b

^

abc ' ~

where T is the kinetic energy and M the magnitude of
moment of momentum.

TOP.

77

TOP.

Let the axis of the top make an angle t with the vertical,
and let the vertical plane through it make an angle u with
a fixed vertical plane. Let 7 be the vertical through the
tip, a the projection of the axis; then the rotation of these
axes = = 7%. Rotation of top

= a sin t . p + J3i + 7 (p cos t + u).
Momentum = acp sin t + fiat + 7 (cp cos t au).
Also

a = fiu, fi = a.u, 7 = 0, w = figl sin t = a (cpi cos afai)
+ fi (ai + cpw sin t) + 7 (cm cpi sin ).

Whence cp cos = aw, and

cV
ai = ( cpw + gl) sin = (^ *- cos t) sin t.

CL

Momentum = cp (7 cos t + a. sin t).

Rotation of top

= p (7 cos t + a. sin t) + u sin t (7 sin t - a. cos
Momentum

= cp (7 cos t + a. sin t) + au sin t (7 sin a cos

p = a. (cp sin t au sin < cos t) + fiat + Ey,

78 DYNAMIC.

where

E= cp cos t + au sin 2 which is constant.
Now w = /i or,
ftgh sin t = a (cp/ cos t au sin cos 2 att cos 2t aid}

+ /3 (at + cpu sin i au* sin cos t)
+ 2T= [terms erased.]
H - 2gh cos t = cp* + a (u* sin 2 1 + i*)

[side work.] c (p + u cos t) a, + au sin t ( a. - ^- 1

V cos 1 1

. z . ,
u z sm 2 1 =

.

' sm' *

_ , a a- A

+ 2a/i cos i + c a - H = 0,

2j

a sin t

or if cos = x, i sin = x

aa?+(E- cpxf + (2ghx + cp*-H)a(l- a') = 0,
-H)a- c*p*} a? + 2

From T+ T', p. 445,

a = inclination of spiral,
t = twist of wire,

- = curvature ; .'. components of bending stress in

CL

plane perpendicular to axis 9 in plane through axis are

cos 2 a , D cos 2 a . . , *._._

B - cos a and ti - sm a. Also components of twist-
a a

ing stress in same planes are At sin a and At cos a.
Hence, for equilibrium

n B cos 2 a .

tr= - cosa + Atsma,
a

* [This line is in pencil ; a second line also in pencil, is omitted.]

TOP. 79

D B cos 2 a .

ia= sin a At cos or.
a

TVT i n A T- 1-5 cos 3 a J.

Making G = 0. we have t = -; , R =

a A sin a

a cos a

For rotation,

6 = t (i cos a -f k sin a) + it cos a (& cos a i sin a)
yu. = At (i cos a + k sin a) + Bu cos a (& cos a.i sin a)

cos a = or = / = .M cos a w cos a sn a,

or gh = Atu Bu 2 sin a, w =

CL

G = coefficient of k in //, = J. sin a + Bu cos 2 a.

i = o, u o, and gh sin i = cpu sin au* sin i cos i
or cpu =gh + au* cos t.

Suppose now that the top has a rotation p about a
the axis of it, and a rotation u about 7 the vertical;
/3 = Vay, and the whole rotation p + u cos t about the axis
is constant ; say p + u cos t = u. Then whole rotation
= px + i/3 + uy, and momentum

= cpy. + ai/3 + (c cos" t + a sin 2 1) uy.

cos a cos 2 a

w = - , p = T, curvature = u cos a = - .

80 DYNAMIC.

E=cpcost + ausm*t = cp sin a + au cos 2 a

cos 3 a
= cp sin a + a .

cos a cos 2 a .
.R = cpu au cos = cp a ., sin a.

The two equations are cpw = gh+ aw 2 cos ,
and E = cp cos t + au sin 2 .

(E.)

If cro-j be two different velocities, imagined as belong-
ing to a particle w, the scalar product of either into the
momentum of the other is the same, namely ra\$<70- r

When this quantity vanishes, the energy of the
resultant motion is equal to the sum of the energies
of the components, and the motions are called conjugate.

In general, the energy of the resultant is less than
the sum of the energies of the components by twice the
scalar product of either velocity into the momentum of
the other.

Let now QR be two velocity systems, imagined as
belonging to a body m, in the same situation. The
momenta belonging to them may be called </>Q, <j>R. If
we calculate for each particle the scalar product of its
velocity in either system by its momentum in the other,
and add all the results, we shall obtain a quantity which
may be called the scalar product of the velocity of one
motion into the momentum of the other, and denoted
by 8 . Q(f)R or S . R<f>Q. It is clear from the case of the
particle that these expressions are always identical.

When this quantity vanishes, the energy of the
resultant of the two motions is equal to the sum of the
energies of the components, and the motions are called
conjugate.

MOMENTUM. 81

In general, the energy of the resultant of any number
of motions is equal to the sum of their energies less
twice the scalar products of their momenta and velocities,
two and two.

If the body have n degrees of freedom, its situation
may be determined by n quantities q, r,... Let Q be that
velocity which the body has when q 1 and all the other
variables are constant, R that which it has when r = 1
and the rest are constant, etc. Then d r Q = d q R. For if
p be the position-vector of a particle, its velocity in the
system Q is d q p, and in the system R it is d r p. The part
of d r Q belonging to this particle is d g d r p, which is also
the part of d q R belonging to it. Similarly for all particles.

The energy T of any motion U is given by the
equation ^T^^S.Q^Rqr, where for q, r are put
independently all the variables. For the motion U is
made up of qQ, rR,... and its energy is consequently
calculated by the rule stated above.

Hence - S q T=2S. Q<f>R . r = S.Q2(j>R .r=S. Q^U
where <f>U is the whole momentum.

The motions Q and U depend on the variables q and t.
Hence d q U=Q. Now since

-2T=S.U<f>U,-<2d q T=S.d q U<j>U+S.U<l>d q U=2S.d q U<l>U;
therefore d q T=S.Q<f>U.

(F.)

Momentum of particle = mass x velocity.

But mass being in a definite place, this is a localized
vector or rotor.

Momenta of two particles to be compounded by rules
for rotors. This is convention, to be justified by results.

Hence resultant momentum of body having trans-
lation velocity passes through mass-centre.

Def.: moment of momentum, moment of resultant
= sum of moments of components.

c. 6

82 DYNAMIC.

Momentum of particle m at end of p, having velocity
<r, is equivalent to momentum ma- through origin + moment
of momentum mVpa: This is a vector, not localized at
all. Hence any system of momenta is equivalent to a
momentum along a certain axis together with moment
of momentum in plane perpendicular to that axis.

To determine momentum due to rotation about axis
<w through the origin. Velocity of end of p is Vwp ;
hence momentum = jVwpdm through origin + moment
of momentum JV. p Vcopdm. Now fVcopdm = V. cofpdm,
which is the momentum of the resultant mass. Hence
if axis passes through mass-centre, momentum is pure
vector. Now we know that whole momentum = momentum
of translation velocity equal to that of mass-centre -4-
momentum of rotation about mass-centre, hence the
former is the rotor part of the whole momentum.

Moment of momentum of rotation about axis through
mass-centre is the same for every point, we therefore
now consider moment of momentum in regard to mass-
centre of rotation about principal axis oz. Velocity of
isc + jy is iy + jx, moment = V(ix +jy + kz) ( iy + jx).
For whole body this is = kj(x* + y*)dm ijxzdm jfyzdm,
= Ak, and therefore parallel to axis of rotation when that
is a principal axis.

Hence for rotation 6, moment of momentum = \$ (6\
where (p = (AQO). The ellipsoid of this function is called

050

OOC

the momental ellipsoid; it is reciprocal to the ellipsoid
of gyration.

If therefore mass-centre has velocity <r, and there is
rotation &> about axis through it, the momentum is rotor
ma- together with vector <&>.

Let </>p = exi 4- fyj + gzk,

Vp<j>p = f- g.yzi + g- e.zxj + e-f.xyk.
Of course J<ppdm = m^p.

MOMENTUM. 83

r r
Next, ll^.r*dldr = J/mMl; thus, in a squirt, if we

take a small cone, vertex at the source, the momentum
is uniformly distributed along its axis.

Energy of translation = ^mv 2 .
Energy of rotation

Vwp = (qz ry) i + (rx pz) j + (py qx) k,
-}f0>dm=\${p*Stf+f)dm+fj(f+at)dm + r' 2 \$(x? + f)dm}

= \ (Ap* + Bq* + CV 2 ) = -
for two rotations CD,

-S.Vo>pV0p

= (qz - ry} (q^z - r\$) + (rx - pz) (r^ - p^z)

for if /( Vcop) 2 dm = Sw^w,

then /( Vwp + VepYdm = S(o> + 0)<p(o> + 0),

whence 2/fif . Vwp Vdpdm = Sco(f)0 + S0<f>to = 2So><j>0.

Two motions in which the velocity of a particle 8m
is cr and <r l respectively are called conjugate when
/Sacrum = 0, or when the scalar product of the velocity
of one by the momentum of the other vanishes. In
that case the energy of their resultant is equal to the
sum of their energies.

Two rotations about axes which meet are conjugate
when the axes are conjugate in regard to the momental
ellipsoid at their point of intersection.

Consider motions ma- 1 and <,, wcr 2 and o> 2 ; their scalar
product is mScr^^ + w^oa^ and

fSa Vapdm = jSawpdm = S . Vo-o)\$pdm

which vanishes for mass-centre. Therefore translation
is conjugate to spin about mass-centre.

G 2

84 DYNAMIC.

Momentum of a. + &>/3 is ra/3 + axfxx, mass-centre being
origin. If the screws coincide

a 4- 60/3 = x (mft + co(f>oL), . '. a. xm{3, ft = x\$ (xm
therefore /8 coincides with a principal axis, say 1 =
then pitch = - = \M- This gives us six principal
screws of the body.

APPENDIX II.

(A.)

SYLLABUS OF LECTURES ON MOTION.

Division of the Subject.

The science which teaches how to describe motion accurately, and
how to compound different motions together, without considering the
circumstances under which motions take place, is called Kinematic
(wrHM, motion).

The simplest kind of motion is that in which a body without changing
its size or shape moves so that all straight lines in the body remain
parallel to their original positions ; this motion is called a Translation.
As all parts of the body move alike, we may confine our attention to any
one of them, however small; for this reason that part of Kinematic
which treats of translations is often called the Kinematic of Particles.

A body which does not change its size or shape during the time
considered is called a rigid body. That part of Kinematic which treats of
motions in which there is no change of size or shape is called the
Kinematic of Bigid Bodies.

A change of size or shape, considered without reference to change of
position, is called a strain. The Kinematic of Strains teaches how to
describe strains accurately, and how to compound them together. Bodies
which change their size or shape are called elastic; and the corresponding
branch of Kinematic is called the Kinematic of Elastic Bodies.

The science which teaches under what circumstances particular
motions take place is called by one or other of two different names
according to the view that is taken of it. If it is regarded as mainly
based upon the Law of Force, and if its results are expressed in
terms of force, it is called Dynamic (5wa/j.is, force); but if it is regarded
as mainly based upon the Law of Energy, and if its results are expressed
in terms of energy, it is called Energetic (tvepyeia). In either case it is
divided into two parts; Static, which treats of those circumstances
under which rest or null motion is possible, and Kinetic, which treats
of circumstances under which actual motion takes place. Properly
speaking, Static is a particular case of Kinetic which it is convenient
to consider separately.

86 DYNAMIC.

We may also make divisions between the Static and Kinetic of
particles, rigid bodies, and elastic bodies; but the Static of particles
and of rigid bodies is generally treated as one subject, while the Kinematic