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On a Problem in Mechanics and the Number of its Solutions

Published online by Cambridge University Press:  03 November 2016

F S. Macaulay
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Extract

To find the position of equilibrium of a particle which is acted on by forces of given magnitudes directed towards given points.

This problem was suggested by Mr. P A. Hillhouse, and arose in the first instance from the consideration of forces acting in a crane. The following note does not offer a solution of the problem, but contains a proof by Mr. A. L. Jones that there is at most one position of equilibrium when the forces are all of one sign, i.e. either all tensions or all thrusts. This is followed by an investigation of the total number of solutions, real and imaginary, for all combinations of thrusts and tensions, when the magnitude only of each force, and not its sign, is given. This number is surprisingly large. It would of course be of more practical interest to find the number of real solutions for each particular combination of thrusts and tensions, but I have only succeeded in doing this when the given points are in a straight line, and in one or two other special cases. The results for these are stated without proof at the end of the note.

Type
Research Article
Information
The Mathematical Gazette , Volume 3 , Issue 59 , October 1906 , pp. 365 - 373
Copyright
Copyright © Mathematical Association 1906

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References

Page 367 of note * The function t 1 r 1 + t 2 r 2+ ... +t n r n is the potential of the system of forces, and has a stationary value at a position of equilibrium P The curve of the family

t 1 r 1+t 2 r 2+ ... + t n r n=constant,

which passes through P, has a node at P; and a node is a position of equilibrium. If two positions of equilibrium move up to coincidence the node becomes a cusp.

Page 369 of note * This diminution to half the normal number of solutions is accounted for by the fact that the other half are at infinity. The infinite solutions have been previously considered separately.