Question 439: Properties of Matter

Question

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating
vertically in water in half-submerged state. If ${\rho _c}$ is the relative density of the material of the shell
with respect to water, then the correct statement is that the shell is

Accepted Answer

Consider a thin, uniform cylindrical shell that is closed at both ends and partially filled with water. This cylindrical shell is floating vertically in water, with exactly half of its height submerged. The relative density of the shell's material compared to water is denoted by ${\rho_c}$.
The inner radius of the cylindrical shell is $r_1$, the outer radius is $r_2$, the height is $h$, and the material density is $\rho$. The shell is filled with water, which has a density of $\rho_w$, up to a height of $x$. Given ${\rho_c = \frac{\rho}{\rho_w}}$, we know that the cylinder is in a half-submerged state, meaning $h/2$ of the cylinder's height is submerged in water.
In equilibrium, the weight of the cylindrical shell, including the water inside it, must equal the buoyant force (upthrust) acting on it. This relationship can be expressed as follows:
$ \left(\pi r_2^2-\pi r_1^2\right) h \rho g + \pi r_1^2 x \rho_w g = \pi r_2^2\left(\frac{h}{2}\right)\rho_w g $
By simplifying this equation, we get:
$ x = h\left[\frac{r_2^2}{r_1^2}\left(0.5-\rho_c\right)+\rho_c\right] = h\left[0.5+\frac{r_2^2-r_1^2}{r_1^2}\left(0.5-\rho_c\right)\right] ...........(i) $
From equation (i), we can determine the condition under which $x$ is greater than $0.5h$. This occurs if $\rho_c  r_1\right)$.

Answer

more than half-filled if ${\rho _c}$ is less than 0.5.

Answer

more than half-filled if ${\rho _c}$ is more than 1.0.

Answer

half-filled if ${\rho _c}$ is more than 0.5.

Answer

less than half-filled if ${\rho _c}$ is less than 0.5.

Properties of Matter

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If {\rho _c} is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

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