Question

A beaker of radius r is filled with water (refractive index $\left(\frac{4}{3}\right)$) up to a height H as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of water level at the center and at the circumference of the beaker is h ( $h < < H , h < < r$ ), as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature $R$. Which of the following is/are correct? ($g$ is the acceleration due to gravity)

• A. $R =\frac{ h ^{2}+ r ^{2}}{2 h }$
• B. $R =\frac{3 r ^{2}}{2 h }$
• C. Apparent depth of the bottom of the beaker is close to $\frac{3 H }{2}\left(1+\frac{\omega^{2} H }{2 g }\right)^{-1}$
• D. Apparent depth of the bottom of the beaker is close to $\frac{3 H }{4}\left(1+\frac{\omega^{2} H }{4 g }\right)^{-1}$

Answer 1

Answer: (A), (D)

$R =\frac{ h ^{2}+ r ^{2}}{2 h }$

Answer 2

(A) $R =\frac{ h ^{2}+ r ^{2}}{2 h }$
(D) Apparent depth of the bottom of the beaker is close to $\frac{3 H }{4}\left(1+\frac{\omega^{2} H }{4 g }\right)^{-1}$