- #1

green-beans

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## Homework Statement

A mass m is suspended by a light elastic string. When the mass remains at rest it is at a point 0, which is a distance a + b below the point from which the string is suspended from the ceiling, where a is the natural length of the string. The mass is pulled down a distance h below 0 and released from rest. The moving mass is subject to air resistance of magnitude cv per unit mass, where c is a positive constant and v is the speed of the mass.

Show that if bc

^{2}< 4g then the largest value of h such that the string never becomes slack is h = be

^{(πµ/ν)}, where µ = c/2 and ν = 1/2√{(4g/b) − c

^{2}}.

## Homework Equations

I deduced the differential equation of motion in terms of the displacement x of the mass below 0 (until the string goes slack) to be x'' + cx' + (g/b)x = 0.

## The Attempt at a Solution

bc

^{2}< 4g occurs when the characteristic equation has 2 complex roots λ = (-c/2) ± i{√|c

^{2}- 4g/b|}/2

So, the solution would be

x=e

^{(-c/2)t}*(Asin{1/2*√|c

^{2}-4g/b|}t + Bcos{1/2*√|c

^{2}-4g/b|}t)

Then I tried differentiating the expression and setting first derivative to zero to find the maximum value of x but it did not work as I could not even solve the equation. Also I do not quite understand where π in the answer came from.

I would appreciate if someone could give me some direction on how to do this question.

Thanks!