A solid paraboloid of base radius $R_{0}$ and height $h_{0}$ is having uniform volume mass density of $ρ_{s}$ is inverted and placed just above the surface as shown, of a liquid having volume mass density of $ρ_{L}$ . When paraboloid is released from rest it has been observed that when it becomes completely submerged in the liquid the paraboloid comes to the rest again. There exist uniform gravity of g. Assume that the liquid body is very large so that liquid level is not changing as the paraboloid enters into the liquid

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The phenomenon is true for all paraboloids irrespective of their dimensions as long as density ratio is maintained.

The phenomenon is not true for all paraboloids irrespective of their dimensions as long as density ratio is maintained.

The ratio of $\frac{ρ_{L}}{ρ_{s}}$ is 2.

The ratio of $\frac{ρ_{L}}{ρ_{s}}$ is 3.

answer is A, D.

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Detailed Solution

Let ${Y=ax}^{2}$ (A parabola)
$Y= \frac{H}{R^{2}} x^{2}$ (from given data)
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$v= \frac{d y}{d t} {mg-ρ}_{L} . \frac{1}{2} {πx}^{2} y.g=m. \frac{dv}{dt} \frac{dv}{dt} =g- \frac{{πx}^{2} {ygρ}_{L}}{{2ρ}_{s} \frac{{πR}^{2} H}{2}} g \Rightarrow g- \frac{ρ_{L}}{ρ_{s}} \frac{x^{2}}{R^{2}} × \frac{yg}{H} v. \frac{dv}{dy} =g- \frac{ρ_{L}}{ρ_{s}} {gy}^{2} \int_{0}^{0} vdv= \int_{0}^{H} (g- \frac{ρ_{L}}{ρ_{s}} {gy}^{2}) dy o=gH- \frac{{gρ}_{L}}{{3ρ}_{s}} H \frac{ρ_{L}}{ρ_{s}} =3$