It may also be deduced from the principles of linear and angular
momentum as embodied in the equations (9). We have
whence, integrating with respect to t,
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.
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| Fig. 75. |
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 is 12Mκ2θ.2 + 12M·CG2·θ. 2, 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
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 form 12Aq̇2, 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.
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| Fig. 76. |
Thus in the case of a railway truck travelling with velocity u the kinetic energy is 12 (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 yθ., where y = OR. Hence if, for simplicity, we neglect the inertia of the connecting-rod, the kinetic energy will be 12 (I + My2)θ. 2, 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 is 12 (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
where V is the potential energy. If we differentiate this with respect to t, and divide out by q̇, we obtain
| Aq̣̈ + 12 | dA | q̇2 + | dV | = 0 |
| dq | dq |
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 q0 is 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
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
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
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| Fig. 77. |
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
where κ2 is 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 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 + 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 the angular velocity of 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 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
If we draw a vector OJ to represent the angular velocity, then J 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).
