Question:
A cone with half the density of water is floating in water as shown. It is depressed down by a small distance \( \delta \ll H \) and released. The frequency of SHM of the cone is:
A cone with half the density of water is floating in water as shown. It is depressed down by a small distance \( \delta \ll H \) and released. The frequency of SHM of the cone is:
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Floating body SHM uses buoyancy-based restoring force and mass of displaced liquid.
Updated On: May 18, 2025
- \( \frac{1}{2\pi} \sqrt{\frac{6g}{H} \cdot \frac{1}{4\sqrt{3}}} \)
- \( \frac{1}{2\pi} \sqrt{\frac{3g}{H} \cdot \frac{1}{4\sqrt{3}}} \)
- \( \frac{6g}{2H} \)
- \( \frac{1}{2\pi} \sqrt{\frac{g}{H}} \)
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The Correct Option is A
Solution and Explanation
The restoring force is due to buoyancy and acts as effective restoring force in SHM. Using geometry and Archimedes’ principle, the effective spring constant \( k \) is proportional to the displaced volume gradient. After deriving and simplifying, the frequency of SHM comes out to be:
\[
f = \frac{1}{2\pi} \sqrt{\frac{6g}{H} \cdot \frac{1}{4\sqrt{3}}}
\]
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