In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity. In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit. In the case of a central orbit described under the law of the inverse square we have v = h/SY = h. SZ/b2, where S is the centre of force, SY is the perpendicular to the tangent at P, and Z is the point where YS meets the auxiliary circle again. Hence the hodograph is similar and similarly situated to the locus of Z (the auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case. The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation. The pole O of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola. In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the “reciprocal polar” of the orbit with respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole, described with constant velocity.
In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity. In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit. In the case of a central orbit described under the law of the inverse square we have v = h/SY = h. SZ/b2, where S is the centre of force, SY is the perpendicular to the tangent at P, and Z is the point where YS meets the auxiliary circle again. Hence the hodograph is similar and similarly situated to the locus of Z (the auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case. The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation. The pole O of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola. In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the “reciprocal polar” of the orbit with respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole, described with constant velocity.
§ 15.Kinetics of a System of Discrete Particles.—The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (§ 8). Thus taking any point O as base, we have first alinear momentumwhose components referred to rectangular axes through O are
Σ(mẋ), Σ(mẏ), Σ(mż);
(1)
its representative vector is the same whatever point O be chosen. Secondly, we have anangular momentumwhose components are
Σ {m (yż − zẏ) }, Σ {m (zẋ − xż) }, Σ {m (xẏ − yẋ) },
(2)
these being the sums of the moments of the momenta of the several particles about the respective axes. This is subject to the same relations as a couple in statics; it may be represented by a vector which will, however, in general vary with the position of O.
The linear momentum is the same as if the whole mass were concentrated at the centre of mass G, and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t + δt. Thus
(3)
Analytically we have
(4)
with two similar formulae.
Again, if the instantaneous position of G be taken as base, the angular momentum of the absolute motion is the same as the angular momentum of the motion relative to G. For the velocity of a particle m at P may be replaced by two components one of which (v) is identical in magnitude and direction with the velocity of G, whilst the other (v) is the velocity relative to G. The aggregate of the components mvof momentum is equivalent to a single localized vector Σ(m)·vin a line through G, and has therefore zero moment about any axis through G; hence in taking moments about such an axis we need only regard the velocities relative to G. In symbols, we have
(5)
since Σ(mξ) = 0, Σ(mξ̇) = 0, and so on, the notation being as in § 11. This expresses that the moment of momentum about any fixed axis (e.g.Ox) is equal to the moment of momentum of the motion relative to G about a parallel axis through G, together with the moment of momentum of the whole mass supposed concentrated at G and moving with this point. If in (5) we make O coincide with the instantaneous position of G, we havex,z, z = 0, and the theorem follows.
Finally, the rates of change of the components of the angular momentum of the motion relative to G referred to G as a moving base, are equal to the rates of change of the corresponding components of angular momentum relative to a fixed base coincident with the instantaneous position of G. For let G′ be a consecutive position of G. At the instant t + δt the momenta of the system are equivalent to a linear momentum represented by a localized vector Σ(m)·(v+ δv) in a line through G′ tangential to the path of G′, together with a certain angular momentum. Now the moment of this localized vector with respect to any axis through G is zero, to the first order of δt, since the perpendicular distance of G from the tangent line at G′ is of the order (δt)2. Analytically we have from (5),
(6)
If we putx,y,z= 0, the theorem is proved as regards axes parallel to Ox.
Next consider the kinetic energy of the system. If from a fixed point O we draw vectorsOV1>,OV2>to represent the velocities of the several particles m1, m2..., and if we construct the vector
(7)
this will represent the velocity of the mass-centre, by (3). We find, exactly as in the proof of Lagrange’s First Theorem (§ 11), that
1⁄2Σ (m·OV2) =1⁄2Σ (m)·OK2+1⁄2Σ (m·KV2);
(8)
i.e.the total kinetic energy is equal to the kinetic energy of the whole mass supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G. The latter may be called theinternal kinetic energyof the system. Analytically we have
(9)
There is also an analogue to Lagrange’s Second Theorem, viz.
(10)
which expresses the internal kinetic energy in terms of the relative velocities of the several pairs of particles. This formula is due to Möbius.
The preceding theorems are purely kinematical. We have now to consider the effect of the forces acting on the particles. These may be divided into two categories; we have first, theextraneous forcesexerted on the various particles from without, and, secondly, the mutual orinternal forcesbetween the various pairs of particles. It is assumed that these latter are subject to the law of equality of action and reaction. If the equations of motion of each particle be formed separately, each such internal force will appear twice over, with opposite signs for its components, viz. as affecting the motion of each of the two particles between which it acts. The full working out is in general difficult, the comparatively simple problem of “three bodies,” for instance, in gravitational astronomy being still unsolved, but some general theorems can be formulated.
The first of these may be called thePrinciple of Linear Momentum. If there are no extraneous forces, the resultant linear momentum is constant in every respect. For consider any two particles at P and Q, acting on one another with equal and opposite forces in the line PQ. In the time δt a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant linear momentum of the system is unaltered. If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric addition of the total impulse of the extraneous forces. It follows, by the preceding kinematic theory, that the mass-centre G of the system will move exactly as if the whole mass were concentrated there and were acted on by the extraneous forces applied parallel to their original directions. For example, the mass-centre of a system free from extraneous force will describe a straight line with constant velocity. Again, the mass-centre of a chain ofparticles connected by strings, projected anyhow under gravity, will describe a parabola.
The second general result is thePrinciple of Angular Momentum. If there are no extraneous forces, the moment of momentum about any fixed axis is constant. For in time δt the mutual action between two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have equal and opposite moments about the fixed axis. If extraneous forces act, the total angular momentum about any fixed axis is in time δt increased by the total extraneous impulse about that axis. The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time δt we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G. If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect. A plane through G perpendicular to this vector has a fixed direction in space, and is called theinvariable plane; it may sometimes be conveniently used as a plane of reference.
For example, if we have two particles connected by a string, the invariable plane passes through the string, and if ω be the angular velocity in this plane, the angular momentum relative to G ism1ω1r1·r1+ m2ωr2·r2= (m1r12+ m2r22) ω,where r1, r2are the distances of m1, m2from their mass-centre G. Hence if the extraneous forces (e.g.gravity) have zero moment about G, ω will be constant. Again, the tension R of the string is given byR = m1ω2r1=m1m2ω2a,m1+ m2where a = r1+ r2. Also by (10) the internal kinetic energy is1⁄2m1m2ω2a2m1+ m2
For example, if we have two particles connected by a string, the invariable plane passes through the string, and if ω be the angular velocity in this plane, the angular momentum relative to G is
m1ω1r1·r1+ m2ωr2·r2= (m1r12+ m2r22) ω,
where r1, r2are the distances of m1, m2from their mass-centre G. Hence if the extraneous forces (e.g.gravity) have zero moment about G, ω will be constant. Again, the tension R of the string is given by
where a = r1+ r2. Also by (10) the internal kinetic energy is
The increase of the kinetic energy of the system in any interval of time will of course be equal to the total work done by all the forces acting on the particles. In many questions relating to systems of discrete particles the internal force Rpq(which we will reckon positive when attractive) between any two particles mp, mqis a function only of the distance rpqbetween them. In this case the work done by the internal forces will be represented by
−Σ∫Rpgdrpq,
when the summation includes every pair of particles, and each integral is to be taken between the proper limits. If we write
V = Σ∫Rpqdrpq,
(11)
when rpqranges from its value in some standard configuration A of the system to its value in any other configuration P, it is plain that V represents the work which would have to be done in order to bring the system from rest in the configuration A to rest in the configuration P. Hence V is a definite function of the configuration P; it is called theinternal potential energy. If T denote the kinetic energy, we may say then that the sum T + V is in any interval of time increased by an amount equal to the work done by the extraneous forces. In particular, if there are no extraneous forces T + V is constant. Again, if some of the extraneous forces are due to a conservative field of force, the work which they do may be reckoned as a diminution of the potential energy relative to the field as in § 13.
§ 16.Kinetics of a Rigid Body. Fundamental Principles.—When we pass from the consideration of discrete particles to that of continuous distributions of matter, we require some physical postulate over and above what is contained in the Laws of Motion, in their original formulation. This additional postulate may be introduced under various forms. One plan is to assume that any body whatever may be treated as if it were composed of material particles,i.e.mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction. In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as in § 15.
An alternative procedure is to adopt the principle first formally enunciated by J. Le R. d’Alembert and since known by his name. If x, y, z be the rectangular co-ordinates of a mass-element m, the expressions mẍ, mÿ, mz̈ must be equal to the components of the total force on m, these forces being partly extraneous and partly forces exerted on m by other mass-elements of the system. Hence (mẍ, mÿ, mz̈) is called the actual oreffectiveforce on m. According to d’Alembert’s formulation, the extraneous forces together with theeffective forces reversedfulfil the statical conditions of equilibrium. In other words, the whole assemblage of effective forces is statically equivalent to the extraneous forces. This leads, by the principles of § 8, to the equations
Σ(mẍ) = X, Σ(mÿ) = Y, Σ(mz̈) = Z,Σ {m (yz̈ − zÿ) } = L, Σ {m (zẍ − xz̈) } = M, Σ{m (xÿ − yẍ) } = N,
(1)
where (X, Y, Z) and (L, M, N) are the force—and couple—constituents of the system of extraneous forces, referred to O as base, and the summations extend over all the mass-elements of the system. These equations may be written
(2)
and so express that the rate of change of the linear momentum in any fixed direction (e.g.that of Ox) is equal to the total extraneous force in that direction, and that the rate of change of the angular momentum about any fixed axis is equal to the moment of the extraneous forces about that axis. If we integrate with respect to t between fixed limits, we obtain the principles of linear and angular momentum in the form previously given. Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies; they are assumed to hold for all material systems whatever. The peculiar status of rigid bodies is that the principles in question are in most cases sufficient for the complete determination of the motion, the dynamical equations (1 or 2) being equal in number to the degrees of freedom (six) of a rigid solid, whereas in cases where the freedom is greater we have to invoke the aid of other supplementary physical hypotheses (cf.Elasticity;Hydromechanics).
The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done by the extraneous forces acting on the body. This is an immediate consequence of the fundamental postulate, in either of the forms above stated, since the internal forces do on the whole no work. The statement may be extended to a system of rigid bodies, provided the mutual reactions consist of the stresses in inextensible links, or the pressures between smooth surfaces, or the reactions at rolling contacts (§ 9).
§ 17.Two-dimensional Problems.—In the case of rotation about a fixed axis, the principles take a very simple form. The position of the body is specified by a single co-ordinate, viz. the angle θ through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then dθ/dt, = ω say, is theangular velocityof the body. The angular momentum of a particle m at a distance r from the axis is mωr·r, and the total angular momentum is Σ(mr2)·ω, or Iω, if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the extraneous forces about the axis, we have
(1)
This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation. If N = 0, ω is constant.
As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig. 72). Neglecting frictional resistance we have, if R be the tension of the string,Iω̇ = Rb, mu̇ = mg − R,whencebω̇ =mb2g.1 + mb2(2)This gives the acceleration of m as modified by the inertia of the wheel.A “compound pendulum” is a body of any form which is free to rotate about a fixed horizontal axis, the only extraneous force (other than the pressures of the axis) being that of gravity. If M be the total mass, k the radius of gyration (§ 11) about the axis, we haved(Mk2dθ)= −Mgh sin θ,dtdt(3)where θ is the angle which the plane containing the axis and the centre of gravity G makes with the vertical, and h is the distance of G from the axis. This coincides with the equation of motion of a simple pendulum [§ 13 (15)] of length l, provided l = k2/h. The plane of the diagram (fig. 73) is supposed to be a plane through G perpendicular to the axis, which it meets in O. If we produce OG to P, making OP = l, the point P is called thecentre of oscillation; the bob of a simple pendulum of length OP suspended from O will keep step with the motion of P, if properly started. If κ be the radius of gyration about a parallel axis through G, we have k2= κ2+ h2by § 11 (16), and therefore l = h + κ2/h, whenceGO · GP = κ2.(4)This shows that if the body were swung from a parallel axis through P the new centre of oscillation would be at O. For different parallel axes, the period of a small oscillation varies as √l, or √(GO + OP); this is least, subject to the condition (4), when GO = GP = κ. The reciprocal relation between the centres of suspension and oscillation is the basis of Kater’s method of determining g experimentally. A pendulum is constructed with two parallel knife-edges as nearly as possible in the same plane with G, the position of one of them being adjustable. If it could be arranged that the period of a small oscillation should be exactly the same about either edge, the two knife-edges would in general occupy the positions of conjugate centres of suspension and oscillation; and the distances between them would be the length l of the equivalent simple pendulum. For if h1+ κ2/h1= h2+ κ2/h2, then unless h1= h2, we must have κ2= h1h2, l = h1+ h2. Exact equality of the two observed periods (τ1, τ2, say) cannot of course be secured in practice, and a modification is necessary. If we write l1= h1+ κ2/h1, l2= h2+ κ2/h2, we find, on elimination of κ,1⁄2l1+ l2+1⁄2l1− l2= 1,h1+ h2h1− h2whence4π2=1⁄2(τ12+ τ22)+1⁄2(τ12− τ22).gh1+ h2h1− h2(5)The distance h1+ h2, which occurs in the first term on the right hand can be measured directly. For the second term we require the values of h1, h2separately, but if τ1, τ2are nearly equal whilst h1, h2are distinctly unequal this term will be relatively small, so that an approximate knowledge of h1, h2is sufficient.As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position. The equation of motion is of the typeI θ̈ = −Kθ,(6)and the period is therefore τ = 2π√(I/K). If by the attachment of another body of known moment of inertia I′, the period is altered from τ to τ′, we have τ′ = 2π√{ (I + I′)/K }. We are thus enabled to determine both I and K, viz.I / I′ = τ2/ (τ′2− τ2), K = 4π2τ2I / (τ′2− τ2).(7)The couple may be due to the earth’s magnetism, or to the torsion of a suspending wire, or to a “bifilar” suspension. In the latter case, the body hangs by two vertical threads of equal length l in a plane through G. The motion being assumed to be small, the tensions of the two strings may be taken to have their statical values Mgb/(a + b), Mga/(a + b), where a, b are the distances of G from the two threads. When the body is twisted through an angle θ the threads make angles aθ/l, bθ/l with the vertical, and the moment of the tensions about the vertical through G is accordingly −Kθ, where K = M gab/l.
As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig. 72). Neglecting frictional resistance we have, if R be the tension of the string,
Iω̇ = Rb, mu̇ = mg − R,
whence
(2)
This gives the acceleration of m as modified by the inertia of the wheel.
A “compound pendulum” is a body of any form which is free to rotate about a fixed horizontal axis, the only extraneous force (other than the pressures of the axis) being that of gravity. If M be the total mass, k the radius of gyration (§ 11) about the axis, we have
(3)
where θ is the angle which the plane containing the axis and the centre of gravity G makes with the vertical, and h is the distance of G from the axis. This coincides with the equation of motion of a simple pendulum [§ 13 (15)] of length l, provided l = k2/h. The plane of the diagram (fig. 73) is supposed to be a plane through G perpendicular to the axis, which it meets in O. If we produce OG to P, making OP = l, the point P is called thecentre of oscillation; the bob of a simple pendulum of length OP suspended from O will keep step with the motion of P, if properly started. If κ be the radius of gyration about a parallel axis through G, we have k2= κ2+ h2by § 11 (16), and therefore l = h + κ2/h, whence
GO · GP = κ2.
(4)
This shows that if the body were swung from a parallel axis through P the new centre of oscillation would be at O. For different parallel axes, the period of a small oscillation varies as √l, or √(GO + OP); this is least, subject to the condition (4), when GO = GP = κ. The reciprocal relation between the centres of suspension and oscillation is the basis of Kater’s method of determining g experimentally. A pendulum is constructed with two parallel knife-edges as nearly as possible in the same plane with G, the position of one of them being adjustable. If it could be arranged that the period of a small oscillation should be exactly the same about either edge, the two knife-edges would in general occupy the positions of conjugate centres of suspension and oscillation; and the distances between them would be the length l of the equivalent simple pendulum. For if h1+ κ2/h1= h2+ κ2/h2, then unless h1= h2, we must have κ2= h1h2, l = h1+ h2. Exact equality of the two observed periods (τ1, τ2, say) cannot of course be secured in practice, and a modification is necessary. If we write l1= h1+ κ2/h1, l2= h2+ κ2/h2, we find, on elimination of κ,
whence
(5)
The distance h1+ h2, which occurs in the first term on the right hand can be measured directly. For the second term we require the values of h1, h2separately, but if τ1, τ2are nearly equal whilst h1, h2are distinctly unequal this term will be relatively small, so that an approximate knowledge of h1, h2is sufficient.
As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position. The equation of motion is of the type
I θ̈ = −Kθ,
(6)
and the period is therefore τ = 2π√(I/K). If by the attachment of another body of known moment of inertia I′, the period is altered from τ to τ′, we have τ′ = 2π√{ (I + I′)/K }. We are thus enabled to determine both I and K, viz.
I / I′ = τ2/ (τ′2− τ2), K = 4π2τ2I / (τ′2− τ2).
(7)
The couple may be due to the earth’s magnetism, or to the torsion of a suspending wire, or to a “bifilar” suspension. In the latter case, the body hangs by two vertical threads of equal length l in a plane through G. The motion being assumed to be small, the tensions of the two strings may be taken to have their statical values Mgb/(a + b), Mga/(a + b), where a, b are the distances of G from the two threads. When the body is twisted through an angle θ the threads make angles aθ/l, bθ/l with the vertical, and the moment of the tensions about the vertical through G is accordingly −Kθ, where K = M gab/l.
For the determination of the motion it has only been necessary to use one of the dynamical equations. The remaining equations serve to determine the reactions of the rotating body on its bearings. Suppose, for example, that there are no extraneous forces. Take rectangular axes, of which Oz coincides with the axis of rotation. The angular velocity being constant, the effective force on a particle m at a distance r from Oz is mω2r towards this axis, and its components are accordingly −ω2mx, −ω2my, O. Since the reactions on the bearings must be statically equivalent to the whole system of effective forces, they will reduce to a force (X Y Z) at O and a couple (L M N) given by
X = −ω2Σ(mx) = −ω2Σ(m)x, Y = −ω2Σ(my) = −ω2Σ(m)y, Z = 0,L = ω2Σ(myz), M = −ω2Σ(mzx), N = 0,
(8)
wherex,yrefer to the mass-centre G. The reactions do not therefore reduce to a single force at O unless Σ(myz) = 0, Σ(msx) = 0,i.e.unless the axis of rotation be a principal axis of inertia (§ 11) at O. In order that the force may vanish we must also havex,y= 0,i.e.the mass-centre must lie in the axis of rotation. These considerations are important in the “balancing” of machinery. We note further that if a body be free to turn about a fixed point O, there are three mutually perpendicular lines through this point about which it can rotate steadily, without further constraint. The theory of principal or “permanent” axes was first investigated from this point of view by J. A. Segner (1755). The origin of the name “deviation moment” sometimes applied to a product of inertia is also now apparent.
Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle θ through which the body has turned from some standard position. The components of linear momentum are then Mẋ, Mẏ, and the angular momentum relative to G as base is Iθ̇, where M is the mass and I the moment of inertia about G. If the extraneous forces be reduced to a force (X, Y) at G and a couple N, we have
Mẍ = X, Mÿ = Y, Iθ̈ = N.
(9)
If the extraneous forces have zero moment about G the angular velocity θ̇ is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant angular velocity, whilst its centre describes a parabola.
We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal on a plane of inclination α. If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we haveMẍ = Mg sin α − F, 0 = Mg cos α − R, Mκ2θ̈ = Fa,(10)where κ is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig. 74. We have also the kinematical relation ẋ = aθ̇. Henceẍ =a2g sin α, R = Mg cos α, F =κ2Mg sin α.κ2+ a2κ2+ a2(11)The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(κ2+ a2). For a homogeneous sphere this ratio is5⁄7, for a uniform circular cylinder or disk2⁄3, for a circular hoop or a thin cylindrical shell1⁄2.
We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal on a plane of inclination α. If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have
Mẍ = Mg sin α − F, 0 = Mg cos α − R, Mκ2θ̈ = Fa,
(10)
where κ is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig. 74. We have also the kinematical relation ẋ = aθ̇. Hence
(11)
The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(κ2+ a2). For a homogeneous sphere this ratio is5⁄7, for a uniform circular cylinder or disk2⁄3, for a circular hoop or a thin cylindrical shell1⁄2.
The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions.It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have
M (ẋẍ + ẏÿ) + lθ̇θ̈ + Xẋ + Yẏ + Nθ̇,
(12)
whence, integrating with respect to t,
1⁄2M (ẋ2+ ẏ2) +1⁄2Iθ̇2=∫(X dx + Y dy + N dθ) + const.
(13)
The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (§ 15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.
The formula (13) may be easily verified in the case of the compound pendulum, or of the solid rolling down an incline. As another example, suppose we have a circular cylinder whose mass-centre is at an excentric point, rolling on a horizontal plane. This includes the case of a compound pendulum in which the knife-edge is replaced by a cylindrical pin. If α be the radius of the cylinder, h the distance of G from its axis (O), κ the radius of gyration about a longitudinal axis through G, and θ the inclination of OG to the vertical, the kinetic energy is1⁄2Mκ2θ̇2+1⁄2M·CG2·thetȧ2, by § 3, since the body is turning about the line of contact (C) as instantaneous axis, and the potential energy is −Mgh cos θ. The equation of energy is therefore1⁄2M (κ2+ α2+ h2− 2 ah cos θ) θ̇2− Mgh cos θ − const.(14)
The formula (13) may be easily verified in the case of the compound pendulum, or of the solid rolling down an incline. As another example, suppose we have a circular cylinder whose mass-centre is at an excentric point, rolling on a horizontal plane. This includes the case of a compound pendulum in which the knife-edge is replaced by a cylindrical pin. If α be the radius of the cylinder, h the distance of G from its axis (O), κ the radius of gyration about a longitudinal axis through G, and θ the inclination of OG to the vertical, the kinetic energy is1⁄2Mκ2θ̇2+1⁄2M·CG2·thetȧ2, by § 3, since the body is turning about the line of contact (C) as instantaneous axis, and the potential energy is −Mgh cos θ. The equation of energy is therefore
1⁄2M (κ2+ α2+ h2− 2 ah cos θ) θ̇2− Mgh cos θ − const.
(14)
Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete determination of the motion. If q be any variable co-ordinate defining the position or (in the case of a system of bodies) the configuration, the velocity of each particle at any instant will be proportional to q̇, and the total kinetic energy may be expressed in the form1⁄2Aq̇2, where A is in general a function of q [cf. equation (14)]. This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the co-ordinate q.
Thus in the case of a railway truck travelling with velocity u the kinetic energy is1⁄2(M + mκ2/α2)u2, where M is the total mass, α the radius and κ the radius of gyration of each wheel, and m is the sum of the masses of the wheels; the reduced inertia is therefore M + mκ2/α2. Again, take the system composed of the flywheel, connecting rod, and piston of a steam-engine. We have here a limiting case of three-bar motion (§ 3), and the instantaneous centre J of the connecting-rod PQ will have the position shown in the figure. The velocities of P and Q will be in the ratio of JP to JQ, or OR to OQ; the velocity of the piston is therefore yθ̇, where y = OR. Hence if, for simplicity, we neglect the inertia of the connecting-rod, the kinetic energy will be1⁄2(I + My2)thetȧ2, where I is the moment of inertia of the flywheel, and M is the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent moment of inertia of the flywheel by the variable amount My2. If, on the other hand, we take OP (= x) as our variable, the kinetic energy is1⁄2(M + I/y2)ẋ2. We may also say, therefore, that the effect of the flywheel is to increase the apparent mass of the piston by the amount I/y2; this becomes infinite at the “dead-points” where the crank is in line with the connecting-rod.
Thus in the case of a railway truck travelling with velocity u the kinetic energy is1⁄2(M + mκ2/α2)u2, where M is the total mass, α the radius and κ the radius of gyration of each wheel, and m is the sum of the masses of the wheels; the reduced inertia is therefore M + mκ2/α2. Again, take the system composed of the flywheel, connecting rod, and piston of a steam-engine. We have here a limiting case of three-bar motion (§ 3), and the instantaneous centre J of the connecting-rod PQ will have the position shown in the figure. The velocities of P and Q will be in the ratio of JP to JQ, or OR to OQ; the velocity of the piston is therefore yθ̇, where y = OR. Hence if, for simplicity, we neglect the inertia of the connecting-rod, the kinetic energy will be1⁄2(I + My2)thetȧ2, where I is the moment of inertia of the flywheel, and M is the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent moment of inertia of the flywheel by the variable amount My2. If, on the other hand, we take OP (= x) as our variable, the kinetic energy is1⁄2(M + I/y2)ẋ2. We may also say, therefore, that the effect of the flywheel is to increase the apparent mass of the piston by the amount I/y2; this becomes infinite at the “dead-points” where the crank is in line with the connecting-rod.
If the system be “conservative,” we have
1⁄2Aq2+ V = const.,
(15)
where V is the potential energy. If we differentiate this with respect to t, and divide out by q̇, we obtain
(16)
as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this must be satisfied by q̇ = O; this requires that dV/dq = 0,i.e.the potential energy must be “stationary.” To examine the effect of a small disturbance from equilibrium we put V = ƒ(q), and write q = q0+ η, where q0is a root of ƒ′ (q0) = 0 and η is small. Neglecting terms of the second order in η we have dV/dq = ƒ′(q) = ƒ″(q0)·η, and the equation (16) reduces to
Aη̈ + ƒ″ (q0)η = 0,
(17)
where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if ƒ″ (q0) > 0,i.e.if V is a minimum in the configuration of equilibrium, the variation of η is simple-harmonic, and the period is 2π √{A/ƒ″(q0) }. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If ƒ″ (q0) < 0, the solution of (17) will involve real exponentials, and η will in general increase until the neglect of the terms of the second order is no longer justified. The configuration q = q0, is then unstable.
As an example of the method, we may take the problem to which equation (14) relates. If we differentiate, and divide by θ, and retain only the terms of the first order in θ, we obtain{x2+ (h − α)2} θ̈ + ghθ = 0,(18)as the equation of small oscillations about the position θ = 0. The length of the equivalent simple pendulum is {κ2+ (h − α)2}/h.
As an example of the method, we may take the problem to which equation (14) relates. If we differentiate, and divide by θ, and retain only the terms of the first order in θ, we obtain
{x2+ (h − α)2} θ̈ + ghθ = 0,
(18)
as the equation of small oscillations about the position θ = 0. The length of the equivalent simple pendulum is {κ2+ (h − α)2}/h.
The equations which express the change of motion (in two dimensions) due to an instantaneous impulse are of the forms
M (u′ − u) = ξ, M (ν′ − ν) = η, I (ω′ − ω) = ν.
(19)
Here u′, ν′ are the values of the component velocities of G just before, and u, ν their values just after, the impulse, whilst ω′, ω denote the corresponding angular velocities. Further, ξ, η are the time-integrals of the forces parallel to the co-ordinate axes, and ν is the time-integral of their moment about G. Suppose, for example, that a rigid lamina at rest, but free to move, is struck by an instantaneous impulse F in a given line. Evidently G will begin to move parallel to the line of F; let its initial velocity be u′, and let ω′ be the initial angular velocity. Then Mu′ = F, Iω′ = F·GP, where GP is the perpendicular from G to the line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be
u′ − ω′·GC = (F/M) · (I − GC·CP/κ2),
where κ2is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided GC·GP = κ2. If this condition be satisfied there would be no impulsive reaction at C even if this point were fixed. The point P is therefore called thecentre of percussionfor the axis at C. It will be noted that the relation between C and P is the same as that which connects the centres of suspension and oscillation in the compound pendulum.
§ 18.Equations of Motion in Three Dimensions.—It was proved in § 7 that a body moving about a fixed point O can be brought from its position at time t to its position at time t + δt by an infinitesimal rotation ε about some axis through O; and the limiting position of this axis, when δt is infinitely small, was called the “instantaneous axis.” The limiting value of the ratio ε/δt is called theangular velocityof the body; we denote it by ω. If ξ, η, ζ are the components of ε about rectangular co-ordinate axes through O, the limiting values of ξ/δt, η/δt, ζ/δt are called thecomponent angular velocities; we denote them by p, q, r. If l, m, n be the direction-cosines of the instantaneous axis we have
p = lω, q = mω, r = nω,