17Author(s): William HooverSource: The American Mathematical Monthly, Vol. 2, No. 5 (May, 1895), p. 161Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2968141 .

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161

MECHANICS.

Condueted by B.F.FINKEL. Kidder, Missouri. All oontributions to this department should be sent to him.

SOLUTIONS OF PROBLEMS.

17. Proposed by WILLIAM HOOVER, A.M., Ph. D., Profssor of Mathematics and Astronomy, Ohio University, Athenss Ohio.

Find the law of density of strings collected into a heap at the edge of a table with the end of the string just over the edge, so that equal masses may always pass over in equal units of time.

II. Solution by the PROPOSER.

Let w,=the length of string which depends from the edge of the table at the end of the time t from the beginning of motion. k=the density of the stringf at a unit's distance from the end, and assume that the density varies as the 7uth power of the distance from the end. The mass of the depending length

k is then =JZkXndx= +1 xin+1 and it a= the mass passing over the edge in a

tinit of time, and y=the acceleration of gravity, we have for the equation of

motion,dd (dA; k f+ldx )=gat.. . . (1), dtn+1 dt

dx _ ~ ~~d k li dt n+1 dt =n+

11 dx ~ ~ ~ ~ ~ ( x o r ' dl 4+ 1 dd xn l . *.*(3)

Multiplying both members of (3) by 2 (x+' dt) and integrating,

( Idx\ 2q _r dx2 2g . V

l d$ )tl =

2n+3 ,*n+3... (4), or, d c 2 *(5)

Equation (5) gives c - (6), d22n+3-

whence - 9 t* dt -,i ('7)

From (7), gat=a(2n+3) dx- = dt k+ n+ dt .... (8).

Integrating (8) and dividing by x, k nndx a(2n+3).. .(9).

1 k _

Equation (9) gives t- .- xn +l=a(n+1)(2n+3)=a.... (10),

since the na8 of string passes over at the uniform rate a. Equation (10) gives 2n2+5n=-2, or n=-i, or n=-2.

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