Archimedes Principle - Example 7

Example 7:
A wooden sphere of density 0.9 g/cm³ and mass 180 g, is anchored by a string to a lead weight at the bottom of a vessel containing water. If the wooden sphere is completely immersed in water, find the tension in the string.

Answer:
Let's draw the diagram that illustrate the situation:

We need to determine the volume of the displaced water to find the upthrust.
Let the volume of the wooden sphere = V
$$V=\frac{m}{\rho }=\frac{180}{0.9}=200c{m}^3$$
Let,
Tension = T
Weight = W
Upthrust = F

All the 3 forces are in equilibrium, hence
$$\begin{array}{l}T+W=F\\ T=F-W\\ T={\rho }_{1}Vg-{\rho }_{2}Vg\\ T=({\rho }_1-{\rho }_2)Vg\\ =(1000-800)(0.0002)(10)=0.4N\end{array}$$