Physics

Physics

Problem

A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency ω\omega . If the radius of the bottle is 2.5 cm then ω\omega is close to: (density of water = 103 kg/m310^3 \text{ kg/m}^3 ) a. 3.75 rad s−13.75 \text{ rad s}^{-1} b. 1.25 rad s−11.25 \text{ rad s}^{-1} c. 2.50 rad s−12.50 \text{ rad s}^{-1} d. 5.00 rad s−15.00 \text{ rad s}^{-1}
Intuitive explanation

When you push a floating bottle down and release it, the restoring force comes from the change in buoyant force—the water pushes back harder as the bottle sinks deeper. The key is recognizing that this restoring force is proportional to displacement, which is the hallmark of SHM. Use the relationship between the cross-sectional area, the mass of water, and the effective "spring constant" of the fluid to find ω=k/m\omega = \sqrt{k/m}, where the restoring force depends on how much extra water is displaced per unit depth.

Intuitive explanation

When you push a floating bottle down and release it, the restoring force comes from the change in buoyant force—the water pushes back harder as the bottle sinks deeper. The key is recognizing that this restoring force is proportional to displacement, which is the hallmark of SHM. Use the relationship between the cross-sectional area, the mass of water, and the effective "spring constant" of the fluid to find ω=k/m\omega = \sqrt{k/m}, where the restoring force depends on how much extra water is displaced per unit depth.

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