# Stability of a Beam

#### jchan79

Homework Statement
If a beam with square cross-section and very low density is placed in water, it will turn one pair of its long opposite faces horizontal. This orientation, however, becomes unstable as we increase its density. Find the critical density when this transition occurs. The density of water is $\rho_w = 1000 kg/m^3$.
Homework Equations
Balancing vertical forces: $\rho_w = \frac{\rho_{block}l^2}{xh}$ where l is the side length of the square cross-section and h is the depth of the beam that is submerged in the water.
The hint says that "The cross-section of the underwater part of the beam could be represented as a superposition of a rectangle and two symmetrically positioned narrow triangles (one of them of negative mass)." How do I find the torques from gravity and buoyancy on these figures?

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#### haruspex

Homework Helper
Gold Member
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Homework Statement: If a beam with square cross-section and very low density is placed in water, it will turn one pair of its long opposite faces horizontal. This orientation, however, becomes unstable as we increase its density. Find the critical density when this transition occurs. The density of water is $\rho_w = 1000 kg/m^3$.
Homework Equations: Balancing vertical forces: $\rho_w = \frac{\rho_{block}l^2}{xh}$ where l is the side length of the square cross-section and h is the depth of the beam that is submerged in the water.

The hint says that "The cross-section of the underwater part of the beam could be represented as a superposition of a rectangle and two symmetrically positioned narrow triangles (one of them of negative mass)." How do I find the torques from gravity and buoyancy on these figures?
Please post or describe the diagram you have drawn using the hint.

"Stability of a Beam"

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