Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 329 / 2 / d

Table of contents
- Solution d

Solution

The zero-couple equations in the two coupled horizontal blocks are

L Ω_{y} = - \frac{3}{13} U_{x}, L Ω_{x} = - \frac{3}{13} U_{y},

and

Ω_{z} = 0

. The force equations then become

k_{x} = 5 U_{x} + L Ω_{y} = \frac{62}{13} U_{x}, k_{y} = \frac{62}{13} U_{y}, k_{z} = 5 U_{z} .

Therefore

α = \frac{13}{62}, γ = \frac{3}{62},

and

U = (\frac{13}{62} k_{x}, \frac{13}{62} k_{y}, \frac{1}{5} k_{z}), L Ω = - \frac{3}{62} (k_{y}, k_{x}, 0) .

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