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A cube made of wood has density d and edge length ℓ. It floats in a liquid of density ρ with its top and bottom faces parallel to the liquid surface. When it is given a small downward displacement and then released, it executes SHM with period T. Which expression gives T?
- 2π√(ℓd/ρg)
- 2π√(ρℓ/dg)
- 2π√(ℓd/((ρ−d)g))
- 2π√(ℓρ/((ρ−d)g))
Correct answer: 2π√(ℓd/ρg)
Solution
The correct option is derived from the principles of buoyancy and simple harmonic motion (SHM). When the cube is displaced, the restoring force acting on it is proportional to the volume of the cube submerged in the liquid, which relates to its density and the density of the liquid, leading to the expression for the period of oscillation being dependent on the ratio of the cube's density to the liquid's density.
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