$2\mathrm{.}9$ As shown in Figure $2$P$\mathrm{.}9,$ a sphere of mass $\backslash ($ m$\_\{\backslash$text $\{$sphere $\}\}=10\backslash$mathrm$\{$~kg$\}\backslash )$ is suspended from a massless rod attached to a frictionless pivot bearing. The support rod has a length $\backslash ($ L$=10\backslash$mathrm$\{$~m$\}\backslash )$ measured from the center of the sphere to the center of the pivot bearing. In the initial state of the sphere, it is held motionless by a support string in a position such that the support rod is horizontal. There is a pure translational spring of spring constant $\backslash ($ k $\backslash )$ located with its axis horizontal at a position $10$ m below the center of the pivot bearing. In its initial state, the spring is at its free length. The support string is cut and the sphere is allowed to descend. The free end of the spring is located so that the spring and the sphere make contact when the support rod is vertical.

$($a$)$ Describe in words the history of the motion of the spring and the sphere after the support string is cut.

$($b$)$ For the sphere as a system, determine the work transfer for the process by which the spring moves from its initial state to the state where it just begins to make contact with the spring.

$($c$)$ Determine the value of the spring constant $\backslash ($ k $\backslash )$ such that the maximum deflection of the spring is $1$ cm $.$ Assume that the radius of the sphere is much larger than $1$ cm $.$