GATE 2020 ME (ME2) – Question 47
A hollow spherical ball of radius 20 cm floats in still water with half its volume submerged. Water density is 1000 kg/m³ and g = 10 m/s². Find natural frequency of small oscillations normal to the surface (rad/s), to two decimals.
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Correct answer: 8.6 to 8.7
Explanation
At equilibrium mass is half the displaced full-sphere mass: m=ρ(2πr³/3). Small displacement changes buoyancy byρgπr²y, so stiffness isρgπr². Thus $\omega_n=\sqrt{3g/(2r)}=\sqrt{75}=\mathbf{8.66}$ rad/s. Added fluid mass and damping are neglected in the intended model.