It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have M(>¢55+3';V)+199+X2%+Y?+N9, (12)

whence, integrating with respect to t, gm (fe +y2> +§ 1é2 =f<Xdx+Ydy+Ndo> +¢<msf. (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 (§ 1 5); 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 ex centric point,

rolling on a horizontal plane.

This includes the case of a comound

pendulum in which the

knife-edge is replaced by a cylindrical pin. If a be the radius of

the cylinder, h the distance of G

from its axis (O), rc the radius of

gyration about a longitudinal

axis through G, and 0 the inclination of OG to the vertical,

the kinetic energy is éMx202+

§ lVl . CG' . 92, by § 3, since the

body is turning about the line of contact (C) as instantaneous axis, and the potential energy is-Mgh cos 0. The equation of energy is therefore

5M (nz-I-a'-l-h'-2 ah cos 0) 0”-Mgh cos 0-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 dehning 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 Q, and the total kinetic energy may be expressed in the form %Aq”, where A is in general a function of q [cf. equation (I4)]. This coefficient A is called the coqficient 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 is HM -l-mx'/a2)u2, where M is the total mass, a the radius and x the radius of gyration of each wheel, and -m is the sum of the masses of the wheels; the reduced inertia is therefore M 1)-mx'/a2. Again, take the system composed of the fiywheel, connecting rod, and piston of a steam-engine. We have here a limiting case of threebar motion (§ 3), and the

lite. 75

/1] instantaneous centre I of

" l the connecting-rod PQ will

I have the position shown in

1 5 the figure. The velocities

g, of P and Q will be in the

R;' Q, ' ratio of IP to ]Q, or OR to |

~ . OQ; the velocity of the

piston is therefore y0, where

" y=OR. Hence if, for

simplicity, we neglect the

inertia of the connecting rod,

the kinetic energy will

be § (H-My”)02, 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 iiywheel by the variable amount My? If, on the other hand, we take OP (=x) as our variable, the kinetic energy is é(M+I/y2)ot2. We may also say, therefore, that the effect of the flywheel is to increase the apparent mass of the piston by the amount I/yz; this becomes infinite at the “ dead points " where the crank is in line with the connecting-rod. a I

J -o—~....P

FIG. 76.

If the system be “ conservative, ” we have %Aq'+V=const., (15)

where V is the potential energy. If we differentiate this with respect to t, and divide out by q, we obtain Ar+s'§ , %r +4, };=o <16>

as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this must be satisfied by q=O; this requires that dV/dq=o, i.e. the potential energy must be “ stationary.” To examine the effect of a small disturbance from equilibrium we put V=]'(q), and write q=q0-I-17, where go is a root of f'(q0) =o and 11 is small. Neglecting terms of the second order in 'rp we have dV/dq =f'(q) = f"(q0).1;, and the equation (16) reduces to Ati +f”(q<»)f/ =0, (17)

where A may be supposed to be constant and to have the value corresponding to q=qu. Hence if f”(q@) >o, i.e. if V is a minimum in the configuration of equilibrium, the variation of 1) is simple-harmonic, and the period is 21r/{A/f”(q°)}. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If f”(q0)<O, the solution of (17) will involve real exponential, and 11 will in general increase until the neglect of the terms of the second order is no longer justified. The connguration 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 li, and retain only the terms of the first order in 0, we obtain lx”-|-(h -10219 -I-ghff =0, (IS)

as the equation of small oscillations about the position 0=o. The length of the equivalent simple pendulum is {:¢2-|-(h -a)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) =£, M(U'f1))=1], I(o.>'—w) =1f. (19) Here u', 11' are the values of the component velocities of G just before, and u, 1/ their values just after, the impulse, whilst co', <0 denote the corresponding angular velocities. Further, E, 71 are the time-integrals of the forces parallel to the co-ordinate axes, and v 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 co' be the initial angular velocity. Then M1/=F, Iw' =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'-at GC = (F/M).(1 -GC.CP/K2),

where K2 is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided GC. GP=l<2. 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 the centre of percussion for 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-l-6t by an infinitesimal rotation eabout some axis through O; and the limiting position of this axis, when 5t is infinitely small, was called the “ instantaneous axis.” The limiting value of the ratio e/6t is called the angular velocity of the body; we denote it by co. If 5, 11, fare the components of e about rectangular co-ordinate axes through O, the limiting values of E/5t, 17/Bt, § '/5t are called the component angular velocities; we denote them by p, q, r. If l, m, n be the direction-cosines of the instantaneous axis we have

F

Fig. 77.

p=lw, q=rnw, r=nw, (I)

P2+a*+r2 =w°'. (2)

If we draw a vector O] to represent the angular velocity, then I traces out a certain curve in the body, called the polhode, and a certain curve in space, called the herpolhode. The cones generated by the instantaneous axis in the body and in space are called the polhode and 'herpolhode cones, respectively; in

the actual motion the former cone rolls on the latter (§ 7).