Archimedes' Principle — Question 3

Question 3

Freeboard sets a stricter limit than merely staying afloat

A rectangular raft has horizontal area $A=2.0\,\mathrm{m^2}$, thickness $H=0.20\,\mathrm{m}$ and uniform density $600\,\mathrm{kg/m^3}$. It floats level in fresh water of density $1000\,\mathrm{kg/m^3}$. Cargo is centered, adds negligible submerged volume and must leave at least $0.050\,\mathrm{m}$ freeboard.

1. Find the unloaded draft and the maximum cargo mass allowed by the freeboard constraint.

2. How much larger would the cargo limit be if the only requirement were that the top just avoid submergence? Explain the distinction.

Question 3 - Solution

Freeboard sets a stricter limit than merely staying afloat

(a) Derivation and calculation

$$m_r=\rho_rAH=(600)(2.0)(.200)=240\,\mathrm{kg}.$$$$\boxed{d_0=m_r/(\rho_wA)=.120\,\mathrm m}.$$$$d_{\max}=H-f_{\min}=.200-.050=.150\,\mathrm m.$$$$\boxed{m_{c,\max}=\rho_wAd_{\max}-m_r=1000(2.0)(.150)-240=60\,\mathrm{kg}}.$$

(b) Result and interpretation

At zero freeboard,$$m_{c,0}=\rho_wAH-m_r=400-240=160\,\mathrm{kg}.$$$$\boxed{m_{c,0}-m_{c,\max}=100\,\mathrm{kg}}.$$The zero-freeboard limit can balance forces but fails the required reserve height.

Physical check

The unloaded freeboard is $.200-.120=.080\,\mathrm m$, greater than the required $.050\,\mathrm m$.