Consider the operation of the lava lamp. Polymer is heated at the base and due to buoyant forces, begins to rise through a colored liquid in the form of small spheres \(\left(r_{o}=1.5 \mathrm{~cm}\right)\). Assume that the colored liquid has a density of \(ho_{o}=910 \mathrm{~kg} / \mathrm{m}^{3}\) and a uniform temperature of \(T_{\infty}=325 \mathrm{~K}\). The polymer sphere has a temperature dependent density given by:

\[ho=ho_{o}-\beta\left(T-T_{\infty}\right) \quad ho_{o}=910 \mathrm{~kg} / \mathrm{m}^{3} \quad \beta=2 \mathrm{~kg} / \mathrm{m}^{3} \mathrm{~K}\]

If the polymer liquid begins its ascent at a temperature of \(350 \mathrm{~K}\), estimate how long before it begins to fall back to be reheated? You may assume the sphere remains a constant size and:

\[h=20 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K} \quad C_{p}(\text { sphere })=400 \mathrm{~J} / \mathrm{kg} \mathrm{K} \quad k(\text { sphere })=10 \mathrm{~W} / \mathrm{m} \mathrm{K}\]