A hemispherical bowl just floats without sink | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The weight of the hemispherical bowl is given by $W = mg = V_{material} \rho g$,where $V_{material} = \frac{2}{3} \pi [ (D/2)^3 - (d/2)^3 ]$ is th

Vedclass · Accepted Answer

$0.98$

Vedclass · Answer

(C) The weight of the hemispherical bowl is given by $W = mg = V_{material} ho g$,where $V_{material} = \frac{2}{3} \pi [ (D/2)^3 - (d/2)^3 ]$ is the volume of the material of the bowl.Here,$D = 1 \ m$ is the outer diameter and $d$ is the inner diameter. The density of the bowl is $ho = 2 	imes 10^4 \ kg/m^3$.According to the principle of flotation,the weight of the bowl equals the weight of the liquid displaced by the submerged part of the bowl. Since the bowl is hemispherical and just floats,the volume of the liquid displaced is the volume of the outer hemisphere,$V_{displaced} = \frac{2}{3} \pi (D/2)^3$.The weight of the displaced liquid is $W_{liquid} = V_{displaced} \sigma g$,where $\sigma = 1.2 	imes 10^3 \ kg/m^3$ is the density of the liquid.Equating the two weights: $\frac{2}{3} \pi (D/2)^3 \sigma g = \frac{2}{3} \pi [ (D/2)^3 - (d/2)^3 ] ho g$.Canceling common terms: $(D/2)^3 \sigma = [ (D/2)^3 - (d/2)^3 ] ho$.Substituting the values: $(0.5)^3 	imes 1.2 	imes 10^3 = [ (0.5)^3 - (d/2)^3 ] 	imes 2 	imes 10^4$.$0.125 	imes 1.2 	imes 10^3 = [ 0.125 - (d/2)^3 ] 	imes 20 	imes 10^3$.$0.15 = [ 0.125 - (d/2)^3 ] 	imes 20$.$0.0075 = 0.125 - (d/2)^3$.$(d/2)^3 = 0.125 - 0.0075 = 0.1175$.$d/2 = (0.1175)^{1/3} \approx 0.49$. $d = 2 	imes 0.49 = 0.98 \ m$.

$A$ hemispherical bowl just floats without sinking in a liquid of density $1.2 \times 10^3 \ kg/m^3$. If the outer diameter and the density of the bowl are $1 \ m$ and $2 \times 10^4 \ kg/m^3$ respectively,then the inner diameter of the bowl will be........ $m$.

Similar Questions

$A$ $0.5 \,kg$ block of brass (density $=8 \times 10^3 \,kg \,m^{-3}$) is suspended from a string. What is the tension in the string if the block is completely immersed in water? $(g=10 \,ms^{-2})$

Why does a ship made of iron float while a piece of iron sinks in water?

An ice block floats in a liquid whose density is less than water. $A$ part of the block is outside the liquid. When the whole of the ice has melted,the liquid level will

$A$ body floats in a liquid contained in a beaker. If the whole system as shown in the figure falls freely under gravity, then the upthrust on the body due to the liquid is

$A$ container of liquid is in free fall. Without any liquid spilling out, does this container obey the principle of Archimedes?

Vedclass Test Series

Exam Paper Generator

Online Exam Module