A solid cylinder of height h and mass m is floating in a liquid of density ρ as shown in the figure. Find the acceleration of the vessel (in$$ $$m/s2) containing liquid for which the relative downward acceleration of the completely immersed cylinder w.r.t. vessel becomes equal to $$one-third$$ of that of the vessel. (Take$$ g $$=$$ $$10$$ $$m/s2)

Question

A solid cylinder of height h and mass m is floating in a liquid of density ρ as shown in the figure. Find the acceleration of the vessel (in$$ $$m/s2) containing liquid for which the relative downward acceleration of the completely immersed cylinder w.r.t. vessel becomes equal to $$one-third$$ of that of the vessel. (Take$$ g $$=$$ $$10$$ $$m/s2)

Infinity Learn · Accepted Answer

Let the volume of the cylinder be V When the cylinder is floating, upthrust weight. Hence,

$ρ (\frac{3}{4} V) g = mg \Rightarrow V = \frac{4 m}{3 ρ}$

Let the acceleration of the particle vessel be A (upwards). In the reference frame of the vessel, the acceleration of the cylinder is A.3.

$\begin{array}{l} ∴ mg + mA - uptrust = m (\frac{A}{3}) \\ mg + mA - {ρVg}^{'} = m (\frac{A}{3}) \end{array}$

where g' = g + A = effective value of g for upthrust.

$\begin{array}{l} ∴ mg + mA - ρV (g + A) = m (\frac{A}{3}) \\ \Rightarrow mg - \frac{4}{3} m (g + A) = - m \frac{2}{3} A \\ A = (- \frac{g}{2}) upwards \end{array}$

The acceleration of the vessel should be $\frac{g}{2} = \frac{10}{2} = 5 {ms}^{- 2}$ (downwards).

Pressure in a fluid-mechanical properties of fluids

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