# 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

## 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

Any clues?

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

At the equilibrium point, Fb = W. What is the expression for Fb 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.

• Emaquima
Astronuc said:
One knows W = mg, where m is the mass of the log.

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

At equilibrium, we have Fb = W = Weight of displaced water = ρw * g * V = ρw * 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?

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