A uniform cylinder of length L and mass M having...

A uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended with its vertical length, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $d$ at equilibrium position. When released, it starts oscillating vertically with a small amplitude. If the force constant of the spring is $k$, the frequency of oscillation of the cylinder is $$ \frac{1}{2 \pi}\left(\frac{k-A d g}{M}\right)^{1 / 2} \quad \text { (b) } \frac{1}{2 \pi}\left(\frac{k+A d g}{M}\right)^{1 / 2} $$ (c) $\frac{1}{2 \pi}\left(\frac{k-d g L}{M}\right)^{1 / 2}$ (d) $\frac{1}{2 \pi}\left(\frac{k+A g L}{A d g}\right)^{1 / 2}$

A uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended with its vertical length, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $d$ at equilibrium position. When released, it starts oscillating vertically with a small amplitude. If the force constant of the spring is $k$, the frequency of oscillation of the cylinder is
$$
\frac{1}{2 \pi}\left(\frac{k-A d g}{M}\right)^{1 / 2} \quad \text { (b) } \frac{1}{2 \pi}\left(\frac{k+A d g}{M}\right)^{1 / 2}
$$
(c) $\frac{1}{2 \pi}\left(\frac{k-d g L}{M}\right)^{1 / 2}$
(d) $\frac{1}{2 \pi}\left(\frac{k+A g L}{A d g}\right)^{1 / 2}$

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Key Concepts

Simple Harmonic Motion

This concept describes the oscillatory behavior of a mass attached to a spring, where the restoring force is proportional to the displacement from equilibrium. In a mass-spring system, the oscillation frequency is given by the square root of the effective spring constant divided by the mass, demonstrating how the system’s parameters affect its periodic motion.

Archimedes’ Principle and Buoyancy

Archimedes’ principle states that a body submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. In problems involving bodies partially submerged, the buoyant force plays a critical role in the force balance and can modify the effective forces acting on the body, especially when analyzing small oscillations about an equilibrium position.

Effective Force Constant in Oscillatory Systems

When additional forces, such as buoyancy, contribute to the overall restoring force in a system, the effective force constant must account for both the spring’s inherent stiffness and the linearized buoyant force gradient. This adjustment is crucial to accurately determine the oscillation frequency, as it reflects the combined effect of mechanical and fluid dynamic restoring forces in the system.

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