The Force Law for Simple Harmonic Motion — Question 8

Question 8

A floating cylinder has a buoyant restoring stiffness

A vertical cylinder of constant cross-sectional area $S$ and mass $m$ floats at rest in liquid of density $\rho$. Displace it downward by a small distance $q$ while keeping it vertical and partially immersed. Ignore drag, added fluid inertia and surface tension.

1. Derive its small vertical motion equation and period.

2. Find the period for $S=.010\,\mathrm{m^2}$, $m=2.0\,\mathrm{kg}$, $\rho=1000\,\mathrm{kg/m^3}$ and $g=10\,\mathrm{m/s^2}$. State the immersion constraint on the amplitude.

Question 8 - Solution

A floating cylinder has a buoyant restoring stiffness

(a) Derivation and calculation

Let $d$ be the equilibrium draft. Equilibrium and the displaced-volume increment give$$mg=\rho gSd,\qquad\Delta V=Sq,\qquad\Delta F_B=\rho gSq.$$$$m\ddot q=mg-\rho gS(d+q)=-\rho gSq.$$$$\boxed{\omega^2=\frac{\rho gS}{m}},\qquad\boxed{T=2\pi\sqrt{\frac{m}{\rho gS}}}.$$

(b) Result and interpretation

$$k_{\mathrm{eff}}=(1000)(10)(0.010)=100\,\mathrm{N/m},$$$$T=2\pi\sqrt{2/100}=\boxed{0.88858\,\mathrm s},\qquad d=\frac{2}{1000(0.010)}=0.200\,\mathrm m.$$$$0<d-A,\quad d+A<H\Rightarrow\boxed{A<\min(d,H-d)}.$$Emergence or full submergence changes the buoyancy-displacement relation.

Physical check

The result excludes fluid added inertia and drag as specified; these would change the frequency or damping in an actual experiment.