Chapter 6

${}^{†}$ A uniform rod $AB$ of mass $m$ and length $2a$ is suspended by two light inextensible strings $PA$ and $QB$, each of length $2a$. The fixed points $P$ and $Q$ are at the same level, distance $2a$ apart; and in equilibrium, $PA$ and $QB$ are vertical. The system performs small oscillations about its equilibrium configuration. Determine the normal frequencies and describe the normal modes.

$†$ A particle of mass $m$ is attached by identical light elastic strings of natural length $a$ and modulus of elasticity $\lambda$ to four points $A$, $B,C$, and $D$, which lie at the corners of a square of side $2a$ in a horizontal plane. In equilibrium, the particle hangs under gravity at a distance $a\sqrt{2}$ below $ABCD$. Show that $mg=\lambda \sqrt{8}$. Find the normal frequencies for small oscillations about equilibrium and describe the normal modes.

${}^{†}$ A particle $A$ of mass $m$ is suspended from a fixed point $O$ by a light string of length $3a$. A second particle $B$ of mass $m$ is suspended from $A$ by a second light string of length $2b$ and a third particle $C$ of mass $m$ is suspended from $B$ by a third light string of length $c$. The system can move in a vertical plane through $O$. Let $x,y,z$ denote the horizontal displacements of $A,B,C$ from their equilibrium positions. Show that an approximate Lagrangian

${}^{†}\mathrm{A}$ double pendulum consists of a bob of mass $m$ suspended under gravity from a fixed point by a light string of length $a$, and a second bob, also of mass $m$, suspended from the first by a further light string, also of length $a$. Show that if the motion is near equilibrium and is confined to a vertical plane through the fixed point then the normal frequencies are $\sqrt{(2\pm \sqrt{2})g/a}$. Describe the corresponding normal modes. Without doing further calculation, describe the normal modes when the motion is not confined to a vertical plane.