A solid cylinder of mass m and volume V is su | vedclass

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Question

(A) Let the cylinder have length $L$. The volume $V = AL$. When the cylinder is submerged up to half its length, the buoyant force is $F_B = (A \cdot

Vedclass · Accepted Answer

$\frac{M_0 g}{k + A\rho g}$

Vedclass · Answer

(A) Let the cylinder have length $L$. The volume $V = AL$. When the cylinder is submerged up to half its length, the buoyant force is $F_B = (A \cdot L/2) \rho g = (V/2) \rho g$.At equilibrium, the spring force $k x_0 = (m + M_0)g - F_B$, where $x_0$ is the initial extension.When the cylinder is displaced by a small distance $x$ downwards, the additional buoyant force is $F_{B}' = A x \rho g$.The net restoring force is $F_{net} = -kx - A x \rho g = -(k + A \rho g)x$.This is the equation of simple harmonic motion with an effective spring constant $k_{eff} = k + A \rho g$.The amplitude of oscillation $A_{osc}$ is given by the displacement from the new equilibrium position. When the mass $M_0$ is added, the equilibrium position shifts by $\Delta x = \frac{M_0 g}{k_{eff}}$.Thus, the amplitude of oscillation is $A_{osc} = \frac{M_0 g}{k + A \rho g}$.

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