Physics

Physics

Problem

A solid sphere of radius 0.15 m0.15\,\text{m} and density 1.10×103 kg m−31.10\times10^{3}\,\text{kg}\,\text{m}^{-3} is completely submerged in liquid water of density 1.00×103 kg m−31.00\times10^{3}\,\text{kg}\,\text{m}^{-3}. Consider placing the water-filled container holding the sphere on the surfaces of three different celestial bodies: • Earth: g=9.80 m s−2g=9.80\,\text{m}\,\text{s}^{-2} • Moon: g=1.62 m s−2g=1.62\,\text{m}\,\text{s}^{-2} • Jupiter: g=24.8 m s−2g=24.8\,\text{m}\,\text{s}^{-2} (a) For each environment, calculate the magnitude of the buoyant force on the sphere. (b) For each environment, compute the apparent weight of the sphere (the difference between its true weight and the buoyant force). (c) Rank the environments from the one in which the sphere is easiest to lift vertically through the water to the one in which it is hardest, and justify your ranking quantitatively. (d) Derive an algebraic expression for the apparent weight of any fully submerged object in terms of its density ρobj\rho_{\text{obj}}, the fluid density ρfluid\rho_{\text{fluid}}, its volume VV, and the local gravitational field strength gg. Explain how this expression supports the numerical results obtained in parts (a) and (b).
Intuitive explanation

Buoyancy depends on the weight of fluid displaced, which scales with local gravity—so the net upward or downward force on the sphere changes as gg changes, even though the sphere's density relative to water stays constant. The key insight is that buoyancy is not a fixed force; it's fundamentally tied to gravity through Fb=ρfluidVgF_b = \rho_{\text{fluid}} V g. On each world, you'll recalculate whether the sphere sinks, floats, or hovers based on how gravity reshapes the balance between the sphere's weight and the buoyant force it experiences.

Intuitive explanation

Buoyancy depends on the weight of fluid displaced, which scales with local gravity—so the net upward or downward force on the sphere changes as gg changes, even though the sphere's density relative to water stays constant. The key insight is that buoyancy is not a fixed force; it's fundamentally tied to gravity through Fb=ρfluidVgF_b = \rho_{\text{fluid}} V g. On each world, you'll recalculate whether the sphere sinks, floats, or hovers based on how gravity reshapes the balance between the sphere's weight and the buoyant force it experiences.

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