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Standard XII

Physics

Area and Volume Expansion

A solid cylin...

Question

A solid cylinder of height h and mass m floats in a liquid of density $ρ$ as shown in Fig. Now the cylinder is released inside a liquid of density $\frac{ρ}{4}$ , contained in a downward accelerated vessel. Determine the magnitude of acceleration of vessel(A), for which cylinder sinks with relative acceleration $\frac{A}{3}$ with respect to vessel.
Neglect any dissipative force:

$\frac{1}{3} g$

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$\frac{3}{4} g$

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$\frac{4}{3} g$

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$\frac{2}{3} g$

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Solution

The correct option is D
$\frac{2}{3} g$

Volume of cylinder = V

$ρ (\frac{3}{4} V) g = m g$

$V = \frac{4 m}{3 ρ}$

In reference frame of vessel

$m g - \frac{ρ}{4} V g_{e f f} - m A = \frac{m A}{3} g_{e f f} = g - A$

$m g - \frac{ρ}{4} (\frac{4 m}{3 ρ}) (g - A) - m A = \frac{m A}{3}$

or $A = \frac{2 g}{3}$

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The correct option is D 23g Volume of cylinder = V ρ(34V)g=mg V=4m3ρ In reference frame of vessel mg−ρ4Vgeff−mA=mA3geff=g−A mg−ρ4(4m3ρ)(g−A)−mA=mA3 or A=2g3