Solution
= Solution
Unsteady Bernoulli gives the surface <pressure> from $p+\rho\partial_t\phi+\rho|\nabla\phi|^2/2=C(t)$. The velocity-squared contribution integrates to zero net force, while the unsteady term gives $F=-m_a\dot V$ with <added mass> $m_a=\tfrac12(4\pi\rho a^3/3)=2\pi\rho a^3/3$. Hence $(m+m_a)a=G$, so $m^*=m+2\pi\rho a^3/3$: accelerating the bubble also accelerates surrounding fluid.