# 4.6.1.5: Neutral frequency of Floating Bodies

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This case is similar to pendulum (or mass attached to spring). The governing equation for the pendulum is $ll\ddot{\beta} - g\beta = 0$ Where here $$ll$$ is length of the rode (or the line/wire) connecting the mass with the rotation point. Thus, the frequency of pendulum is $$\frac{1}{2\pi}\sqrt{\frac{g}{ll}}$$ which measured in $$Hz$$. The period of the cycle is $$2\pi \sqrt{ll/g}$$. Similar situation exists in the case of floating bodies. The basic differential equation is used to balance and is $I\ddot{\beta} - V \rho_{s} \overline{GM} \beta = 0$ In the same fashion the frequency of the floating body is $\frac{1}{2\pi}\sqrt{\frac{V\rho_{s}\overline{GM}}{I_{body}}}$ and the period time is $2\pi\sqrt{\frac{I_{body}}{V\rho_{s}\overline{GM}}}$ In general, the larger $$\overline{GM}$$ the more stable the floating body is. Increase in $$\overline{GM}$$ increases the frequency of the floating body. If the floating body is used to transport humans and/or other creatures or sensitive cargo it requires to reduce the $$\overline{GM}$$ so that the traveling will be smoother.