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William Kingdon Clifford.

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ever zero, and varies continuously, it is always zero. If s is multiplied
by P in every second, and = a for t = then it is = aP 1 whenever t is
commensurable. P is function of p which changes to P n vthenp changes
to np. Let e be value of P when p = l, then s = ae"' when p and t are
commensurable ; even when pt is commensurable. Extension of definition -
ae r ' is to mean result of making a grow at intrinsic rate p for t seconds.
Expansion in series. Prove that e x e y = e x+y . Extension of s ps to case



ABSTRACT OF DYNAMIC. 99

where s is vector in a plane, logarithmic spiral. Case of s J_ s ; circle ;
e 19 = cos 6 + 1 sin 6, construction of e and e 1 . Series of one sign,
convergent if sum of n terms never exceeds fixed quantity M. In this
case a limit exists, independent of order of terms. series, the four

cases ; if both divergent, the sum is arbitrary. Series for , con-

1 x

vergent for x 2 < 1 ; series 1 + \ + ^+... divergent. Exponential series
always convergent. Fluxion of sum of series = sum of series formed by
fluxions of its terms, except when convergence is infinitely slow.

XIII. Central Orbits.

1. Moment of velocity. Moment of resultant = sum of moments
of components. Hence rate of change of moment of velocity = moment
of acceleration. This is also h if h = twice rate of describing areas.
Proof by second fluxion of re 16 that r x transverse acceleration = h.
Newton's proof that if moment of acceleration = 0, then areas are pro-
portional to time. In central orbit r 2 6 = h, therefore angular velocity is
inversely as square of distance.

2. Belated curves. Inverse : cut radius vector at equal angles.
Description by Peaucellier's cell and Hart's cell. Pedal ; construction
of tangent by circle. Eeciprocal. Reciprocal of circle is conic.

3. Hence motion in conic with acceleration tending to focus has
circular hodograph. Acceleration proportional to angular velocity, .'. to

. Compare with Kepler's laws. Two constant parts of velocity.

4. Equation of Energy d, (%v*) = d t (v) =Pd,r, &*= - - in elliptic

motion. Velocity from infinity at any point. Case of parabola.

p

5. Any orbit, p=p+S^p in hodograph gives j 2 = (1 + S0) hu in

TT iP ft Jl^'D

orbit. So p = -7 , = P , vp = h gives P = -~ . Apply to r m =
p p r p A rr

a m coam6, here a m p = r ro + 1 , P = (m + 1) T^a 2 " 1 / 1 " 3 " 2 " 1 . These curves got
from | + itj = (x + iy) m : Examples.

XIV. Motion of Plane on Plane. General Theorems. Any finite
change of position may be produced by rotation. Case of translation
belongs to infinitely distant centre. Direct and inverse triangles. Com-
position of two finite rotations by inverse triangles. Hamilton's theorems
about inverse polygons. Effect of any number of successive rotations
represented by rolling of polygons. Limiting case; any motion of plane
can be represented by rolling of curves.

Theorem of the instantaneous centre. System of velocities ; their dis-
tribution. p=Iap with rotational velocity a. Composition of such systems:

72



100 DYNAMIC.

rotational velocity a, 6, c... at points A, B,C... have resultant a + b + c...
at point G such that (a + b + c + ...) OG a.OA + b. OB + c . OC + ...
Two equal and opposite rotations give translation. Composition of
translation with rotation.

In rolling, angular velocity = sum of curvatures
x velocity of point of contact. Velocity and accelera-
tion of every point in plane, locus of points of zero
normal or tangential acceleration is a circle. Curva-
ture of roulette. [Velocity of P I . AP (<j> + ^). Hence
acceleration = I. AP (0 + ^) + I. AP ($ + $) but AP
=P-A = I.AP((j> + \f<)-A, .-. acceleration P= - AP (j> + \j>) z - IA (<j> + fy
+ 1 . AP (0 + i/'). Curvature of path of
_ normal acceleration _ - AP(<f> + \f/) 2 + A cos 6 (<f> + \j/)
P 2 AP*(<j> + W

1 cos rr'




XV. Special Cases. Combination of circular translation with ro-
tation = rolling of circle on circle. Case of internal rolling, radii as 1:2.
Double generation of circular roulettes: envelop of diameter. Tricusp,
cardioid, quadricusp. General theorem for any three curves.

Circle on straight line. Curvature of cycloid. Theory of involute and
evolute. Arc of cycloid and epicycloid. Three-bar motion. Inverted
parallelogram ; rolling of equal conies. Locus of point rigidly attached
to bar is inverse of conic. Kite.

XVI. Motion of sphere on sphere. Analogues of Theorems in XIV
and XV. Eotational velocities about axes which meet combined by
addition of vectors. Theorem of moments.

XVII. General motion of Kigid Body. Any change of position may
be produced by a twist. Erroneous proof that it may be produced by
rotation. Two congruent figures ABC...K, A'B'C'...K, if Oa = AA',
Ob = BB' ...Ok = CC' will be directly congruent if abc...k has even dimen-
sions, inversely if it has odd.

Instantaneous motion is twist-velocity. If two systems of velocities
are consistent with rigidity, so is resultant system. Any number of twists
compound into single twist. Composition of two twists : cylindroid. Ee-
solved part of motion of any point of line along line = k sin -p cos 0.
Complex of screw, plane belonging to each point, and vice versa. Eesolu-
tion of twist into two rotations, axis of one given. Two rotations to meet
two given lines of the complex : viz. the tractors of the two lines and their
co-axes. Sum of motions of pair of axes along their lengths due to unit
rotation about any line = k sin 6 -p cos 6, whichever the two axes are.
Hence sum of such motions due to twist about second screw = k sin



CONTENTS. 101

- (P + l] cos & H thi 8 vanishes, a pair of axes of one screw are lines of
complex of the other, and the screws may be said to meet. Otherwise the
quantity is the moment of the screws.

XVIII. Strains. 1. String or rod: elongation or shortening;
ratio = a, amount = a - 1. Homogeneous stretch.

2. Membrane ; homogeneous strain may always be produced by two
longitudinal strains at right angles. Ellipse of strain. Function of

vector, ix +jy becomes ax + fty. Shear ; amount = a - - or nearly = 2X
if a = 1 + X.

3. Solid ; homogeneous strain produced by three longitudinal strains
at right angles. Strain- ellipsoid. Vector function (p). Resolution of
strain into elongation, shear, and uniform dilatation.



(C.)

BOOK III. STRAINS.

CHAPTER I. STEAIN-STEPS.

Strains in straight line. Strains in plane. Displacement conic.
Linear Function of a Vector. Shear. Composition of Strains. Eesultant.
Product. General strain of Solid. Displacement-quadric. Composition
of uniform strains. Non-uniform strain in terms of line-flux of displace-
ment.

CHAPTER H. STRAIN-FLUX.

Strain-flux due to given velocity-system. Instantaneous spin. Irrota-
tional velocity-system. Lines of flow and orthogonal surfaces. Velocity-
system consistent with constant volume.



BOOK IV. FORCES.

CHAPTER I. THE LAWS OF MOTION.

Mass. Acceleration due to strain of adjacent body. Gravitation.
Attraction and repulsion of electrified bodies, and of magnets. Law of
composition. Resultant mass-acceleration of particle. Attwood's Machine.
Law of Action and Reaction. D'Alembert's Principle. Effect of forces
on rigid body. Parallelogram of forces. Analogy to Spins. Wrench.
Work and Energy. Work of Twist against Wrench.



102 DYNAMIC.

CHAPTER II. THE CONDITIONS OF EQUILIBRIUM OF
A RIGID BODY.

Two forces. Three forces. Application to problems. Four or five
forces. Tractors. Six forces. Lines in involution. Screw determined
by five lines. General conditions. In passing through equilibrium no
work is being done.

CHAPTER m. THE COMPOSITION OF FORCES.

SECT. 1.

The link-polygon. Couples. Relation of link-polygons with different
poles. Reciprocal Diagrams. Parallel forces. Construction of Moment.
Centre of parallel forces.

SECT. 2.

Centre of inertia: triangle, quadrilateral, tetrahedron, pyramidal
frustum, circular arc, sector, segment, parabolic frustum, paraboloid,
hemispherical surface. Second Moment : neutral axis and plane, swing-
conic and swing-quadric. Core of parallelogram, triangle, ellipse, tetra-
hedron, ellipsoid.

SECT. 3.

Attraction and Repulsion inversely as square of distance. Spherical
Shell. Potential and level surfaces. Analogy with motion of liquid.
Sources and sinks. Lines of force. Theorems of Stokes and Chasles.
Electric images. Centrobaric bodies.

CHAPTER IV. MOTION OF A RIGID BODY.

Momentum and energy of rigid body. Momental ellipsoid. Pendulum.
Solution by elliptic functions. Motion under no forces. Poinsot's
representation. Euler's equations. Solution by elliptic functions. Sylves-
ter's theorem. Motion of Top. Spherical pendulum. Motion of Hoop.
Precession and Nutation.



BOOK V. STRESSES.

CHAPTER I. SOLIDS.

Elastic string in straight line. Flexible inextensible string. Caten-
aries. Wire. Kirchhoff's theorem. Elastic curve. Spiral Spring. Horizon-
tal rod slightly bent. Continuous girder. Plate strained in its own
plane. Stress-conic. Plate bent. Solid strained in any way. Stress-
quadric. Relation between stress and strain for isotropic solid. Crystals.
Energy of strain. Variation of stress under forces. Expression in terms
of displacement.



CONTENTS.



103



CHAPTEB H. FLUIDS.

Equilibrium of Fluids. Level surfaces. Floating bodies. Metacentre.
Molecular theory of Fluids. Theorems of Maxwell and Boltzmann.
Elasticity and specific heat. Surface tension. Elastic curve. Motion
of solid in a frictionless liquid. Vortex-lines and filaments.

CHAPTEB III. WAVES AND VTBEATIONS.

Wave-transmission along flexible string. Straight tube of air. Shear-
wave on rod. Tones of strings and pipes. Fundamental vibrations of a
system.



CHAP.
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.



XII.
XIII.
XIV.
XV.
XVI.
XVII.



XVIII.
XLX.
XX.



(D.)

ELEMENTS OF DYNAMIC.

BOOK I.
KINEMATIC OF TRANSLATIONS.

Divisions of the Subject.

Translations and their composition.

Harmonic Motion.

Parabolic Motion.

Velocity of Bectilinear Motion.

Velocity in general. Hodograph.

Logarithmic Motion.

Curvature.

The Inverse Method.

Elliptic Motion.

Central Orbits.

BOOK II.
KINEMATIC OF RIGID BODIES.

Motion of a Plane. General Theorems.

Circular Boulettes.

Three-bar Motion ; the two cases.

Solid with one point fixed. Composition of Botations.

Twist Motion.

Composition of Twists. Degrees of Freedom.

BOOK III.

STRAINS.

Classification and Measure of Strains : Solid, Plate, Wire.
Linear Function of a Vector.



104 DYNAMIC.

BOOK IV.

FORCES.

CHAP.

XXI. Mass. Attwood's Machine.

XXII. The Laws of Motion. Law of Energy for a Particle. Work.

XXIII. D'Alembert's Principle.

XXIV. The Conditions of Equilibrium.

XXV. Composition of Forces : the Link-Polygon.

XXVI. Centre of Inertia.

XXVII. Second Moment.

XXVIII. Attractions.

XXIX. Wrenches.

XXX. Momentum and Energy of Eigid Body. Bestatement of laws

of Motion and Energy.
XXXI. Motion of Rigid Body under no forces.

BOOK V.

STRESSES.

Classification of Stresses. Stress-Conic or Quadric.

Eelation of Stress to Strain in Isotropic Body. Energy of

Strain and Stress.

General relation of Stress to Strain.
Variation of Stress under forces.
Equilibrium of Fluids.
Floating Bodies. Metacentre.

BOOK VI.

HEAT.

Pressure of a gas. Boyle's Law.

Temperature. Law of Conduction.

Specific Heat and Elasticity. Properties of a Substance.

BOOK VII.

WAVES AND VIBRATIONS.

Eate of transmission of disturbance.

Strings and Pipes.

Fundamental Vibrations of a System. Fourier's Theorem.



APPENDIX III.



EXERCISES.

BOOK I. CHAPTEE I.

1. DEFINE a rigid body, and a movement of translation. Explain how
translations are compounded together.

Find the locus of a point P which moves so that the length of the
resultant of the translations PA, PB, PC is constant the points A,B,C
being fixed.

2. A leech crawls by alternately lengthening and shortening itself,
holding fast by its head when it shortens, and by its tail when it lengthens.
Describe this motion in kinematical language, analyzing it into its con-
stituent parts.

3. What is meant by compounding translations together? Show
from your definition that a change in the order of composition makes no
difference in the result.

A, B, G, D, E, F are the vertices of a regular hexagon, and is any
seventh point. Find the resultant of the translations AO, OB, 00, OD,
OE, OF.

4. Explain the equation of uniform rectilinear motion p = a + /St.
Two points are moving uniformly in straight lines AB and CD, and

in the same second they get from A to B, and from C to D respectively.
Find by construction the positions in which they are nearest together.

5. Define a simple harmonic motion, its period, amplitude, and
epoch. Prove that the resultant of any number of simple harmonic mo-
tions of the same period is motion in an ellipse.

The motion of a point is compounded of two simple harmonic motions
at right angles to one another which are very nearly equal in period, but
whose amplitudes are slowly diminishing at a uniform rate; find the
general shape of the curve which the point will describe.



106 DYNAMIC.

6. Define a "simple harmonic motion," its "amplitude" and its
"period," and show that the resultant of two simple harmonic motions
of the same period is in general an elliptic harmonic motion. What
special cases are included in this description?

7. Draw some of the figures produced hy compounding two simple
harmonic motions in directions at right angles to one another, the periods
being (a) as 2 to 1, (0) as 3 to 2.

8. State the experimental laws of motion, and define force. What
two kinds of force are there ? and which do you think more likely to be
explained as a case of the other ?

What is the difference between pressure due to contact with a
strained body, and "action at a distance"? Mention any hypotheses by
which it has been attempted to describe either of these as a case of the
other.

9. Define a motion of translation of a rigid body, and explain what
is meant by the composition of translations.

Three translations are represented by lines parallel and equal to the
sides of a triangle; discuss the different possible values of their resultant.

10. Any number of simple harmonic motions in one plane and of one
period compound into harmonic motion in an ellipse.

To two points A, B in the ceiling are fastened the ends of a string
slightly longer than AB; and from the middle of the string a ball is
hung by another string reaching nearly to the floor ; if the ball be set in
motion, what will be the nature of its path ?

11. Define a simple harmonic motion ; and show that the resultant
of two such motions, having the same period, in the same straight line,
is a simple harmonic motion of that period.

If the two motions differ by a quarter phase, prove that the squared
amplitude of the resultant is equal to the sum of the squared amplitudes
of the components.

Two simple harmonic motions take place 140 times and 150 times a
second respectively ; find how many times a second the amplitude of the
resultant goes through all its changes.

12. A pencil moving with S.H.M. on a generating line of a cylinder
which revolves uniformly in the same period will trace out an ellipse
upon the cylinder.

Explain how curves so drawn upon a cylinder may be used to repre-
sent the composition of harmonic motions at right angles to one another ;
and describe the curves produced when the cylinder makes (1) half a
revolution, (2) two revolutions in the period of the S.H.M.



EXERCISES. 107

13. Define a simple harmonic motion ; and draw a curve of velocities
for the compound of such a motion and its octave, the latter starting at
phase | when the former starts at phase 0.



CHAPTEB II.

1. A point moves uniformly round a circle while the centre of the
circle moves uniformly with less velocity along a straight line in its
plane ; find the nodes of the curve which the point describes.

2. Assuming the law of composition of velocities, and some rule for
drawing a tangent to a parabola, find the velocity at any instant of a
point moving with the horizontal component of its motion uniform in a
parabola whose axis is vertical.

3. Find the velocity at any time in the parabolic motion

p - a. + pt + yt 2 ,

and hence show that there is a point where the tangent to the curve is
perpendicular to y.

The equation p = (a + it) 2 a, where i is the operation of turning coun-
ter-clockwise through a right angle, represents a parabolic motion with
the origin for focus.

Hence, or in any other way, prove that the velocity in a parabolic
motion is that due to a fall from the directrix.

4. If p = a + pt n , prove that p =n^t n ~ l when n is a positive integer.

5. Explain what is meant by the "hodograph" of a given motion,
and find the hodograph in the case

p = at + j3 2 + yt 3 .

6. Find the normal and tangential accelerations of a moving point.
Find the curvature at any point of a parabola or of a cardioid.

7. Explain the equation e 1 ' 9 = cos 6 + i sin 0, and represent the series
for e, where a = 1 + %i, by a geometrical construction.

From the equation p = re, determine the radial and transversal ac-
celerations of a point moving in a plane curve.

8. Prove that the acceleration of a moving point consists of a
tangential part, which is the rate of change of the magnitude of the
velocity, and of a normal part, which is the square of the velocity
multiplied by the curvature of the path.

A train goes at 30 miles an hour round a curve of half-a-mile radius ;
find the deflection of a plumb-line hung in one of the carriages.



108 DYNAMIC.

9. If the acceleration of a moving point is always directed towards a
fixed point and inversely as the square of the distance from it, prove that
the hodograph is a circle.

What is the form of the orbit when this circle is at a distance from
the centre of acceleration very large compared with its radius ?

Conversely, if the hodograph is a circle, and the acceleration is
directed to a fixed centre, prove that it must vary as the inverse square of
the distance.

10. Prove that uniform velocities are compounded according to the
same rule as translations.

11. What are the dimensions of a velocity?

A molecule of air moves at the rate of twenty miles a minute ; and
light travels at the rate of 333 million kilometres per second ; express
each of these velocities in terms of the units here used to express the
other, assuming that

8 kilometres = 5 miles.

12. What sort of motion has a velocity at a given instant ? and how
is that velocity defined? Describe the "curve of velocities," and show
how it represents (1) the acceleration, (2) the space passed over.

13. Prove that the motion represented by the equation



is one of uniform acceleration, and that the path described is a parabola.
Prove in any other way that the path described under uniform accelera-
tion is a parabola.

14. If a point move in a parabola whose axis is vertical with uniform
horizontal velocity, show that the vertical component of its velocity
increases uniformly with the time.

Hence, or in any other way, show that if in rectilinear motion the
distance passed over is proportional to the square of the time, the
velocity will be proportional to the time simply.

15. In the elliptic harmonic motion



show that the acceleration p = - n?p.

A small bullet is fastened to the end A of a stiff elastic rod AB
without mass, and it is observed that when the end B of the rod is held
horizontally, the bullet weighs down the end A an inch and a half. The
whole is then placed on a smooth table, and the end B held tight. Prove
that the bullet will oscillate horizontally 8/?r times in a second, the
acceleration of gravity being 32 feet a second per second, and the
acceleration of the bullet due to the elasticity of the rod being supposed
proportional to the distance from its mean position.



EXERCISES. 109

16. A point moves in a parabola so that its distance from the axis
increases uniformly from zero ; show that its distance from the tangent
at the vertex varies as the square of the time.

The points A and B move in two different paraholas according to this
law, and the line AB is continually trisected in C and D; prove that
the points C and D move in two other parabolas, and find the relation
between the axes of the four curves.

17. State and prove the rule for composition of velocities.

A, B, C are points on an ellipse whose centre is 0, and P is any point
in its plane. Wherever P is, velocities represented by AP, BP, CP have
always a resultant in the direction of OP. Prove that EC is parallel to
the tangent at A, and that the area of the triangle ABC bears a constant
ratio to the area of the ellipse.

18. Define Acceleration, Mass, Force. State what is the approximate
relation between mass-acceleration and position in the following cases:

(1) A body falling freely in the neighbourhood of the earth's surface.

(2) A ball on a smooth table attached to the floor by an elastic string
which passes through a hole in the table and is just unstretched when
the ball is directly over the hole.

(3) The Moon.

19. If a particle moves in an ellipse under a force directed to one
focus, prove that the hodograph is a circle and the acceleration inversely
as the square of the distance.

20. Prove, by twice differentiating the vector of a moving point, that

v^ dv

its total acceleration is compounded of accelerations and -r- along the

r at

principal normal and the tangent to its path respectively.

21. Assuming that, for values of x between - IT and ;r, any function
/ (x) of x which is such that the curve yf(x) has an area between
those limits may be expanded in a series of the form



+ a a sin x + a. 2

prove that x times any one of the coefficients a, b is a projection of that
area on a diametral plane of a cylinder round which it can be wrapped
a certain number of times.

If / (x) = sin x from to \ , and = cos x from ^ to -IT, expand it in a
2 &

series of sines which shall be true between and IT.



110 DYNAMIC.

22. Define the velocity and the acceleration of a particle moving in a
straight line. If a curve be constructed whose ordinates represent the
velocities at times represented by the abscissae, in what way does the
figure indicate the acceleration and the space described ?

If s = atf + bcos(nt-a),

find the velocity and acceleration at any time.

23. How are velocities compounded ? Define the total acceleration of
a moving particle and the hodograph of its path. If a particle move
uniformly in a circle, show that the hodograph is also a circle uniformly
described; and find the magnitude and direction of the acceleration at
any time.

24. State the second law of motion ; and deduce from it that the path
of a projectile is nearly a parabola. If two particles are projected in the
same horizontal direction from two points not in a vertical line, find the
relation between their velocities of projection that the paths may touch,
and the times of arriving at the point of contact.

25. In what ways does the mass-acceleration of a body depend on its
position relatively to other bodies?

If the mass-acceleration of a particle is directly proportional to its
distance from a fixed point and directed towards the fixed point, show that
its motion will be compounded of two simple harmonic motions in lines
at right angles to one another.



BOOK II. CHAPTEB I.

1. A rigid body has a given twist-velocity about a given screw. Find
the velocity of any point in the body.

What is meant by the composition of two twist-velocities ?

Eotations about the axes A, B, G can be compounded into a rotation
about the axis D. Show that, when looked at from any point of D, the
lines A, B, C will appear to meet in a point.

2. Prove that any change in the position of a plane figure in its
plane may be produced by rotation about some point. What happens
when the point is at an infinite distance ?

Four rods AB, BC, CD, DA are jointed together, the length AB being
equal to CD, and BC to DA. The two longer rods cross one another.
If AB be fixed, prove that the motion of CD may be produced by the
symmetrical rolling of a conic upon an equal fixed conic.



EXERCISES. Ill

3. A plane slides on a fixed plane so that a fixed point of each lies
on a fixed line of the other, find the locus of the instantaneous centre on
the fixed and moving planes, and the locus of points which at any
instant have no tangential acceleration. The motion of one plane on
another is determined when two given curves on the moving plane touch


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Online LibraryWilliam Kingdon CliffordElements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) → online text (page 8 of 9)