= Normal modes of two arc-spring-coupled beads on a hoop
{title2=$\omega_0=0,\quad\omega_1^2=2k(m_1+m_2)/(m_1m_2)$}
Two frictionless beads of positive <masses> $m_1,m_2$ on a circular hoop are joined by two identical <springs> of <spring constant> $k$, constrained to follow the two arcs. At their diametrically opposite equilibrium, each <spring> has natural length equal to half the circumference. For signed <arc length> displacements in the same rotational sense, the <potential energy> is $U=k(x_1-x_2)^2$. The <normal mode> of common <rotation> has <mode shape> $(1,1)$ and zero <angular frequency>, while the other <normal mode> has <mode shape> $(1,-m_1/m_2)$ and the positive <angular frequency> in the title.
Adding the equations of motion shows that $X=(m_1x_1+m_2x_2)/(m_1+m_2)$ satisfies $\ddot X=0$. Subtracting them gives $\ddot\delta+\omega_1^2\delta=0$ for $\delta=x_1-x_2$. Thus the complete <zero mode> is $X=C+Dt$, including uniform <rotation>, and the relative motion is a <harmonic oscillator>. In the limit $m_2/m_1\to\infty$ at fixed $m_1,k$, the oscillatory motion of the heavier bead vanishes for fixed lighter-bead <amplitude>, and $\omega_1\to\sqrt{2k/m_1}$. A common drift still survives if its initial <velocity> is held fixed.
Back to article page
