Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 7 / b

Table of contents
- Solution b

Solution

The quadratic Lagrangian is diagonal in

x, y, θ

, so the Euler-Lagrange equations give two pure translations and one pure rotation. Their angular frequencies are

ω_{x}^{2} = ω_{y}^{2} = \frac{3 k}{2 m} (1 + \frac{l - l _{0}}{l}), ω_{θ}^{2} = \frac{6 k ( l - l _{0} ) ( a + l )}{ma l} .

A horizontal translation has

(x, y, θ) = (A cos ω_{x} t, 0, 0)

; a vertical translation has

(0, A cos ω_{y} t, 0)

; and the rotational mode has

(0, 0, A cos ω_{θ} t)

. Any fixed-direction translation is another choice in the same two-dimensional degenerate normal mode space. The rotation keeps the centre fixed.

Figure 1.
Two degenerate translations and a separate rotation of a disk on symmetric radial springs
. Original schematic of the three normal modes; arrows indicate the independent motions, and the dashed disk marks the equilibrium position. Details.

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