Archimedes' Principle — Question 4

Question 4

Moving a dense stone from a boat to the water lowers the level

A floating boat carries a stone of mass $m=3.0\,\mathrm{kg}$ and density $\rho_s=3000\,\mathrm{kg/m^3}$. The boat floats in a straight-sided tank with horizontal area $A=0.50\,\mathrm{m^2}$, containing water of density $1000\,\mathrm{kg/m^3}$. The stone is gently transferred to the bottom, where it becomes fully submerged. No water spills; the boat remains afloat.

1. Compare the water volume excluded because of the stone before and after transfer.

2. Find the water-level change and its sign. Explain the limiting result if the stone density approached water density.

Question 4 - Solution

Moving a dense stone from a boat to the water lowers the level

(a) Derivation and calculation

While carried, the stone adds boat displacement$$\boxed{V_i=m/\rho_w=3.0/1000=.00300\,\mathrm{m^3}}.$$After transfer its own submerged volume is$$\boxed{V_f=m/\rho_s=3.0/3000=.00100\,\mathrm{m^3}}.$$$$\Delta V=V_f-V_i=-.00200\,\mathrm{m^3}.$$

(b) Result and interpretation

$$\boxed{\Delta h=\frac{m}{A}\left(\frac1{\rho_s}-\frac1{\rho_w}\right)=\frac{-.00200}{.500}=-.00400\,\mathrm m}.$$The level falls 4.00 mm.$$\rho_s\to\rho_w\Rightarrow V_f\to V_i\Rightarrow\Delta h\to0.$$

Physical check

Weight determines the extra floating displacement, whereas geometrical volume determines the bottom-resting exclusion.