 Solution

ID: past-exam-of-the-mathematical-mechanical-biology-course-of-the-university-of-oxford/2023/3/a/i/solution

Before contact with both walls, the elastic rod can grow freely, so its elastic stretch ratio is

α = 1

and its length is its stress-free grown length,

ℓ (t) = L_{0} γ (t) = L_{0} (1 + c t) .

The first contact occurs when

ℓ (t_{c}) = L_{1}

. For the intended growth case

c > 0

t_{c} = \frac{L _{1} / L _{0} - 1}{c} = \frac{L _{1} - L _{0}}{c L _{0}} .

The condition

L_{1} > L_{0}

ensures

t_{c} > 0

. If

c = 0

, contact never occurs; negative

c

describes shrinkage rather than the stated continuing growth.

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