JEE Advanced 2020 · Paper 2 · Physics Q7

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PhysicsMultiple correct+4 / -2Question 7 of 54

A beaker of radius $r$ is filled with water $\left(\text{refractive index }\frac43\right)$ 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 $h\ (h \ll H,\ h \ll r)$ , 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)

Left: a beaker on a table filled with water to height H. Right: the same beaker rotating about its vertical axis with angular speed ω; the water surface is concave, lower at the centre than at the wall by h.

R = \dfrac{h^2 + r^2}{2h}

R = \dfrac{3r^2}{2h}

Apparent depth of the bottom of the beaker is close to

\dfrac{3H}{2}\left(1 + \dfrac{\omega^2H}{2g}\right)^{-1}

Apparent depth of the bottom of the beaker is close to

\dfrac{3H}{4}\left(1 + \dfrac{\omega^2H}{4g}\right)^{-1}

Answer and worked solution

Official answer: A, D

(A)/(B) Radius of curvature. The surface dips by $h$ over a chord of half-width $r$ :
$(R-h)^2+r^2=R^2\implies R=\frac{h^2+r^2}{2h}\approx\frac{r^2}{2h}.$
(A) is true and (B) is false.

The rotating surface is a paraboloid with $h=\dfrac{\omega^2r^2}{2g}$ , so $R\approx\dfrac{r^2}{2h}=\dfrac g{\omega^2}$ .

(C)/(D) Apparent depth. Light goes up from the bottom (in water, $\mu_1=\frac43$ ) into air ( $\mu_2=1$ ) through a surface whose centre of curvature lies above it, on the air side, so $R>0$ . With $u\approx-H$ (since $h\ll H$ ):
$\frac1v-\frac{4/3}{-H}=\frac{1-\frac43}R\implies-\frac1v=\frac4{3H}+\frac1{3R}.$
So the apparent depth is
$\frac{3H}{4+H/R}=\frac{3H}4\left(1+\frac{\omega^2H}{4g}\right)^{-1}.$
(D) is true and (C) is false.

(A), (D)

Topics:Refraction at spherical surfaces Pseudo forces and non-inertial frames

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JEE Advanced 2020 · Paper 2, Physics question 7