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Stability of Equilibrium
Published online by Cambridge University Press: 03 November 2016
Extract
This article contains a discussion of the problem considered by Routh and other writers under the general heading “Rocking Stones”, but is restricted to cases which can be treated as problems in two dimensions. Thus Routh writes: “A perfectly rough heavy body rests in equilibrium on a fixed surface; it is required to determine whether the equilibrium is stable or unstable. We shall suppose the body to be displaced in a plane of symmetry so that the problem may be considered to be one in two dimensions.”
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- Copyright © Mathematical Association 1945
References
page 104 note * E. J. Routh, Treatise on Analytical Statics, 2nd ed. (1896 and 1909), Vol. I, pp. 175-180.
page 104 note † G. M. Minchin, Treatise on Statics, 4th ed. (1889), Vol. II, pp. 134-139; 5th ed. (1915), pp. 86-91.
page 104 note ‡ Ibid., pp. 178-180.
page 104 note § “Critical case” is the term used by Minchin, while Larmor writes “On critical τ ‘apparently neutral’ equilibrium”.
page 105 note * This example is given by Minchin, 4th ed., p. 139; or 5th ed., p. 91, Ex. 4. H
page 106 note * If the order of the critical cases is regarded as unimportant, these results may be obtained readily by noting that the values of ρ are respectively h + 5aψ 4 and h + 5a sin4 ψ cos ψ, so that ρ 0 is a minimum or maximum according as a is positive or negative.
page 106 note † This “one-sided stability” seems to be actually instability. See § 5.
page 107 note * Note that u and m are now initial values.
page 108 note * These include all the results given in Routh’s Statics, p. 179.
page 108 note † Ibid., 4th ed., p. 139; or 5th ed., p. 91, Ex. 5.