$4.2.12$ Determine the force of the tension of the thread connecting two balls of volume 10 cm$^3$ each, if the upper ball floats half-immersed in water. The lower ball is three times heavier than the upper one.
Applying Newton's Second Law...
For upper ball $$m_{\uparrow}~g+T = F_{\uparrow}$$ $$m_{\uparrow}~g+T = \rho_0~g~\frac{V}{2} \;(1)$$ For lower ball $$m_{\downarrow}~g = T+F_{\downarrow}$$ $$m_{\downarrow}~g = T+\rho_0~g~V \;(2)$$ Separating the gravity's force term from $(1)$ $$m_{\uparrow}~g = \rho_0~g~\frac{V}{2}-T \;(3)$$ Dividing $(2)$ by $(3)$, and taking account that $m_{\downarrow}=3~m_{\uparrow}$ $$\frac{\rho_0~g~V+T}{\rho_0~g~\frac{V}{2}-T}=3$$ and separating $T$ $$T = \frac{\rho_0~g~V}{8}$$ Taking $\rho_0 = 1~\rm{g/cm^3} = 10^{-3}~\rm{kg/cm^3}$ (water's density) and $g=9.8~\rm{m/s^2}$ (acceleration due to gravity at Earth's surface), $$\boxed{T \simeq 1.2 \cdot 10^{-2}~\rm{N}}$$
BSc. Luis Daniel Fernández Quintana
Physics Department (FCNE)
Universidad de Oriente, Cuba