Simple harmonic motion and buoyant force

[csnsc14320]

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

$$F_b < W$$

$$F_net = F_b - W = -m*a$$

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

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

$$F_b > W$$

$$F_net = F_b - W = m*a$$

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

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

 

[csnsc14320]

Any clues?

 

[Astronuc]

One knows W = mg, where m is the mass of the log.

At the equilibrium point, F_b = W. What is the expression for F_b in terms of ρ, where ρ = density of water, and L.

Let $\xi$ be the displacement from equilibrium, so one must be concerned about $$\ddot{\xi}$$.

Think about the restoring force per unit length.

 

[csnsc14320]

One knows W = mg, where m is the mass of the log.

At the equilibrium point, F_b = W. What is the expression for F_b in terms of ρ, where ρ = density of water, and L.

At equilibrium, we have F_b = W = Weight of displaced water = ρ_w * g * V = ρ_w * g * L * A

Where L * A is the length times the cross sectional area - but I am not given A, but that is a volume in terms of L?