The Harmonic Bobber

Jane is fishing in still water off of the old dock. She is using a cylindrical bobber as shown. The bobber has a cross sectional area of $A$, a length of $H$, a mean density of $\rho = \rho_w/2$ (recall $\rho_w = 1$ gram/cm$^3$), and is balanced so that it remains vertical. When it is floating at equilibrium (supporting the weight of the hook and worm dangling underneath) $2H/3$ of its length is submerged in the water. (Hint: What is the combined mass of the bobber, hook and worm from this data?) A fish gives a tug on the worm and pulls the bobber straight down a distance $y_0 < H/3$. At $t = 0$ it lets go.

a) What is the net restoring force on the bobber as a function of $y$? (Hint: The word net means that you don't need to worry about the weight of the worm or bobber explicitly.) Use this force and the calculated mass to write Newton's 2nd Law for the motion of the bobber up and down (in $y$).

b) Neglecting the damping effects of the water, write an equation for the displacement of the bobber from its equilibrium depth as a function of time, $y(t)$. With what frequency does the bobber bob?