The Oscillation Of Floating Bodies - Practice Questions & MCQ

A floating body is in a stable equilibrium. When it is displaced up and released, it accelerates down and when it is pushed down
and released, it accelerates up. It means a floating body experiences a net force towards its stable equilibrium position. Hence, a floating body oscillates when displaced up or down from its mean position.

Consider a solid cylinder of density $\sigma$ and height h , is floating in a liquid of density $\rho$ as shown below figure, And $(\sigma<\rho)$.

If l is the length of cylinder dipping in liquid as shown in the above figure.

If it is depressed slightly and allowed to oscillate vertically.  

Then the time period of the oscillation is given by

$
T=2 \pi \sqrt{\frac{l}{g}}
$

- The time period of the oscillation of the above SHM is also given in term of $h, \rho, \sigma$

$
\begin{aligned}
& \text { at mean position } \\
& F_{n e t}=0 \Rightarrow \text { Weight of solid }=\text { buoyant } \text { force } \Rightarrow m g=V \rho g \\
& \text { As } m=\sigma h A \\
& \Rightarrow \sigma h A g=\rho l A g \\
& \Rightarrow l=\frac{h \sigma}{\rho}
\end{aligned}
$

So time period of the oscillation is given by

$
T=2 \pi \sqrt{\frac{h \sigma}{g \rho}}
$