= Solution
For the <normal modes of two spring-coupled equal pendulums>, a <normal mode> has every coordinate oscillating at the same frequency with a fixed displacement ratio. The linearized equations here are
$$
m\ddot x_1=-m\omega_0^2x_1-k(x_1-x_2),\qquad m\ddot x_2=-m\omega_0^2x_2-k(x_2-x_1).
$$
Their normalized displacement patterns and angular frequencies are
$$
\boxed{\mathbf e_+=\frac{(1,1)}{\sqrt2},\quad\omega_+=\omega_0;\qquad\mathbf e_-=\frac{(1,-1)}{\sqrt2},\quad\omega_-=\sqrt{\omega_0^2+2k/m}=1.1\omega_0.}
$$
The spring is unstretched in the in-phase mode, and stretched or compressed in the opposite-phase mode. The numerical ratio follows from $k/m=(21/200)\omega_0^2$.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-pendulum-modes.png]
{title=In-phase and opposite-phase displacement patterns of the two coupled pendulums}
{source=paper-1-pendulum-modes.py}
The initial displacement is the equal sum of the two mode patterns, and the initial velocities vanish. Therefore
$$
x_1(t)=\frac a2[\cos\omega_0t+\cos(1.1\omega_0t)],\qquad\boxed{x_2(t)=\frac a2[\cos\omega_0t-\cos(1.1\omega_0t)]=a\sin(1.05\omega_0t)\sin(0.05\omega_0t).}
$$
With the printed $\omega_0=2\pi\,\mathrm{rad\,s^{-1}}$ this is $x_2/a=\sin(2.1\pi t)\sin(0.1\pi t)$, with $t$ in seconds. The fast oscillation has frequency $1.05\,\mathrm{Hz}$ and the amplitude is bounded by $a|\sin(\pi t/10)|$. The first complete displacement transfer is at $t=5\,\mathrm s$, when $x_1=0$, $x_2=a$ and both velocities vanish; the original state returns at $10\,\mathrm s$. Distinguishing <absolute and signed beat envelopes>, the absolute beat envelope repeats every $10\,\mathrm s$, although its signed sine factor has period $20\,\mathrm s$.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-pendulum-beats.png]
{title=The second pendulum oscillates inside a ten-second beat envelope and first receives the full displacement at five seconds}
{source=paper-1-pendulum-beats.py}
Back to article page
