Past exam of the mathematical methods II course of the University of Oxford / 2017 / 9 / a

For small horizontal displacement $x_i\simeq l{\theta }_i$, the gravitational restoring force is $-mg{\theta }_i\simeq -(mg/l)x_i$. A spring between adjacent bobs has extension equal to their displacement difference; Hooke's law pulls each bob towards its neighbour's displaced position. Thus comparison with the stated equations givesThe small-angle approximation makes the bob motion horizontal to first order; its vertical shift changes spring geometry only at higher order. Multiplying all force equations by minus one puts the positive restoring coefficients on the displayed right-hand side.