Problem $15\mathrm{.}138|$

Masses A and $B,$ each of mass $m,$ are attached to one another at the ends of a string that runs through the small hole in the center of a circular cone, for which $=30.$

$($a$)$ If ball $B$ is hanging a constant distance below the cone and ball $A$ is moving in a horizontal circle with constant speed $v_0$ inside the cone, determine, as a function of $v_0$ and $g,$ the distance $l.$

$($b$)$ Now a force $P,$ which increases from zero to a constant value, is applied to $B$ to pull it down a distance $\frac{l}{2}$ and it is then held at that position. When $B$ is in this position and $A$ is moving in a new circular path, determine the speed of $A$ as a function of $v_0.$

$($c$)$ Determine, as a function of $m$ and $v_0,$ the work done by $P$ in moving $B$ to its new position.

Neglect friction between A and the cone and between the string and the hole. Use $sin=\frac{1}{2}$ and $cos=\frac{\sqrt[\phantom{2}]{3}}{2}.$