A tall cylinder partially filled with mercury (Hg) is rotated about its axis with a constant angular velocity ω 0 as shown in the figure. If ρ ( x ) is the radius of curvature at any coordinate x (a

Question

A tall cylinder partially filled with mercury (Hg) is rotated about its axis with a constant angular velocity $ω_{0}$ as shown in the figure. If $ρ (x)$ is the radius of curvature at any coordinate x (assuming the lowest point of the Hg surface at origin) and h(x) is the height of the free surface of Hg at coordinate x, then match List-I with List-II.

List-I	List-II
(I) $h (\frac{r _{0}}{2})$	(P) $\frac{ω _{0}^{2} r _{0}^{2}}{8 g}$
(II) $h (r_{0})$	(Q) $\frac{( 1 + \frac{ω _{0}^{4} r _{0}^{2}}{4 g ^{2}} ) ^{3/2}}{\frac{ω _{0}^{2}}{g}}$
(III) $ρ (\frac{r _{0}}{2})$	(R) $\frac{ω _{0}^{2} r _{0}^{2}}{2 g}$
(IV) $ρ (0)$	(S) $\frac{g}{ω _{0}^{2}}$
	(T) $\frac{2 g}{ω _{0}^{2}}$

TestHub · Accepted Answer

Rotating liquid surface: h ( x ) = 2 g ω 0 2 ​ x 2 ​ h ( r 0 ​ /2 ) = 8 g ω 0 2 ​ r 0 2 ​ ​ ( P ) ; h ( r 0 ​ ) = 2 g ω 0 2 ​ r 0 2 ​ ​ ( R ) Radius of curvature: ρ ( x ) = ω 0 2 ​ / g [ 1 + ( ω 0 2 ​ x / g ) 2 ] 3/2 ​ ρ ( r 0 ​ /2 ) matches (Q); ρ ( 0 ) = ω 0 2 ​ g ​ ( S ) . Option C.

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