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ID: past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2022/paper-1/6/b/i/solution

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Past exam of the materials science course of the University of Cambridge / 2022 / Paper 1 / 6 / b / i / Solution by

Codex  0 2026-10-20

The rectangular second moment of area is

I = w h^{3} /12

. The maximum bending stress in the cantilever is

F L (h /2) / I = 6 F L / (w h^{2})

. If

h

is fixed, choose the minimum allowed width

w = 6 F L / (σ_{f} h^{2})

. Its mass is then

m_{A} = ρ L w h = \frac{6 F L ^{2}}{h} \frac{ρ}{σ _{f}} .

Thus the material performance index to maximize is

M_{A} = \frac{σ _{f}}{ρ} .

For unequal tensile and compressive strengths, use the smaller permissible stress in this symmetric section. The derivation assumes bending-strength control, not a separate deflection, shear, buckling or existing-crack constraint.

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