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What are the conditions that would cause said sphere to sink or float?

What if the sphere was full of ice instead?

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up vote 5 down vote accepted

Simply if the average density $\rho_\text{avg}$ of the sphere + helium (or your horse, for that matter) is less than the density of water $\rho_w$.

This is because the weight is \begin{align} mg = \rho_\text{avg} V_\text{object} g \end{align} while the buoyancy force is \begin{align} F = \rho_w V_\text{displaced} g, \end{align} where $V_\text{object}$ is the volume of the object and $V_\text{displaced}$ is the volume of water displaced i.e. the volume of the object under water. Of course, $V_\text{object} \geq V_\text{displaced}$, because you can only displace as much volume of water as the volume you take up anyway.

If $\rho_\text{avg} \leq \rho_w$ then there exists some $V_\text{displaced}$ which is $ \leq V_\text{object}$ that satisfies $F = mg$, so the object floats, with a bit of it sticking out of the water, and if $\rho_\text{avg} > \rho_w$ then it just sinks and sinks and sinks...

cheers

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