A wooden cylinder floats in water. At equilibrium the cylinder floats with a depth 10 cm submerged. When the cylinder is pushed downward a small distance and then released, it is observed that it bobs up and down periodically. Assume viscosity is negligible. The period of oscillation of the cylinder is ; A) 0.63 s. B) 0.32 s. C) 0.10 s. D) 1.58 s

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A wooden cylinder floats in water. At equilibrium the cylinder floats with a depth 10 cm submerged. When the cylinder is pushed downward a small distance and then released, it is observed that it bobs up and down periodically. Assume viscosity is negligible. The period of oscillation of the cylinder is ; A) 0.63 s. B) 0.32 s. C) 0.10 s. D) 1.58 s

Physics
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Have you had differential equations?
I know simple differential equations taught in calculus II ...
I'm thinking about how to approach this. There isn't much given. We need to be able to come up with an expression of the buoyancy force in terms of distance from equilibrium position, and we need the mass of the cylinder. Then we can use\[f = \sqrt{k \over m}\]where \(f\) is the frequency of the oscillations in \(\bf rad \over sec\), \(k\) is the spring constant in \(N*m\) (Here \(k\) is related to the buoyancy force as \(F_B = k*d\) where \(d\) is the depth of the cylinder from the equilibrium condition.), and \(m\) is the mass of the cylinder. Then the period of oscillation can be found to be \[T = {2 \pi \over f}\]

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I was thinking about using T=\[T=2\pi \sqrt{m/k}\] and then instead of k we can use \[k=F_b/d\] and use \[m=\rho.V\] instead of the mass and k , but still we don't know how much the cylinder is dicplaced. I don't know if we can cancel it out with the length in volume or not ? because it's very smal we might be able to cancel it out with the length but I'm not sure :/
I agree with your methodology. However you ran into the same problem I did, we don't seem to have enough information given in the problem.

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