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
A. more than half-filled if `rho_(c)` is less than 0.5
B. more than half-filled if `rho_(c)` is more than 1.0
C. half-filled if `rho_(c)` is more than 0.5
D. less than half-filled if `rho_(c)` is less than 0.5

Answer 1

Correct Answer - D
Let r, l be the radius and length of the cylindrical shell. If `rho` is the density of the material of the shell, then mass of the shell, `m=pi r^(2)l rho`. The various force acting on the shell are shown in Fig. 7(CF).41. As the shell is floating vertically in water, therefore,

`mg+pi r^(2)h rho_(w)g=pip r^(2)(l//2)rho_(w) g`
or `h=(pi r^(2)(l//2)rho_(w)g)(-mg)/(pi r^(2) rho_(w)g) =(1)/(2) -(m)/(pi r^(2) rho_(w))`
`(1)/(2) -(pi r^(2)l rho)/(pi r^(2) rho_(w)) =(1)/(2)[1-2 rho_(c)]` `(.: rho_(c) =(rho)/(rho_(w)))`
From above we note that if `rho_(c) lt 0.5`, then shell is less than half filled. Thhus option (d) is true.