A solid sphere of radius R is floating in a liquid of density ρ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.
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Question

A solid sphere of radius R is floating in a liquid of density ρ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.

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G1 Physics · Accepted Answer

Half-submergence gives sphere mass m=(ρ/2)(4πR³/3). A small downward displacement x increases buoyant force by ρgπR²x, so kₑ=ρgπR². Thus ω²=kₑ/m=3g/(2R), and f=(1/2π)√(3g/2R). Correct answer: A.