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## Homework Statement

A cylindrical wooden log is loaded with lead at one end so that it floats upright in water. The length of the submerged portion is L = 2.56m. The log is set into vertical oscillation.

(a) Show that the oscillation is simple harmonic

(b) Find the period of oscillation

Neglect dampening effect by water

## Homework Equations

## The Attempt at a Solution

So I drew three pictures, one of the log in equilibrium, one with the log slightly raised, and one with the log slightly more submerged.

Case 2: Slightly Raised

[tex]F_b < W[/tex]

[tex]F_net = F_b - W = -m*a[/tex]

[tex]\rho_w * (A*L - A*\Delta y) * g - m*g = -m*\frac{dy^2}{dt^s}[/tex]

and, if this is correct, i'm not given the cross sectional area A or the density of the log, so not sure where to go.

Case 3: Slightly Pushed Down

[tex]F_b > W[/tex]

[tex]F_net = F_b - W = m*a[/tex]

[tex]\rho_w * (A*L + A*\Delta y) * g - m*g = m*\frac{dy^2}{dt^s}[/tex]

Same problem as case 2.

Some insight on where to go next would be nice :D

I'm thinking maybe trying to find the pressure differences at each delta y, but I'm sure there is a simpler way than that?

Thanks