Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 2 / c

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- Solution c

Solution

Force and torque freedom of the cell--helix pair give

(A + A_{0}) U + B (Ω + ω) = 0,

B U + (D + D_{0}) Ω + D ω = 0.

Δ = (A + A_{0}) (D + D_{0}) - B^{2},

solution of this linear system yields

U = - \frac{B D _{0}}{Δ} ω, Ω = \frac{B ^{2} - ( A + A _{0} ) D}{Δ} ω .

The body counter-rotates relative to the motor, and the swimming direction reverses with the helix handedness through

B

.

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