(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A cork with a density [tex]\rho[/tex]_{0}in the form of a cube of side length l floats on water with a density of [tex]\rho[/tex]_{w}. The pressure in water depends on depth h from the surface as P=[tex]\rho[/tex]_{w}*g*h.

A. Find the equilibrium depth of the bottom surface of the cube (how much length is below the surface)

Let the equilibrium position of the bottom surface be at y=0. At time t=o, the position of the bottom surface (below the equilibrium point) is at y=-A_{0}and the cube is moving upwards with a speed of V_{0}.

B. Find the behavior of y(t)

C. Find the earliest time it take to reach the origin and the next time it takes to reach it again. Find the maximum speed the cork can have.

D. Friction is now present with friction force -aV(y,t) where a is a contant and V(y,t) is the velocity. Assume the friction results in a very lightly damped motion. Find the Q value.

2. Relevant equations

All in next part.

3. The attempt at a solution

This problem should be easy enough but I am having some problems.

A. The depth under water is [tex]\frac{P0}{Pw}[/tex] *l

B. I did the force diagram on the block, with F_{g}in the negative y direction and F_{b}in the positive y direction.

F=ma

Fb -Fg = ma

where Fb=[tex]\rho[/tex]w*g*y*l^{2}

and Fg=m*g=[tex]\rho[/tex]o*g*l^{3}

this gives me

[tex]\rho[/tex]w*g*l^{2}*y - m*g=[tex]\rho[/tex]o*g*l^{3}= m*a

where m= ([tex]\rho[/tex]w+[tex]\rho[/tex])o*l^{3}

a= [tex]\frac{d2*x}{dt2}[/tex]

[tex]\rho[/tex]w*g*l^{2}*y - m*g=[tex]\rho[/tex]o*g*l^{3}= ([tex]\rho[/tex]w+[tex]\rho[/tex]o)*l^{3}* [tex]\frac{d2*x}{dt2}[/tex]

is this the right setup? my prof. said there was something wrong with this, but I have no idea what..

I am also not sure how to approach part B. The undamped, unforced harmonic osc. eq. is [tex]\frac{d2*x}{dt2}[/tex]+ w_{0}^{2}*x=0

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# Buoyancy and Harmonic Oscillation

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