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Physics · Geometrical Optics

JEE Advanced 2020 — Paper 2 — Question 52

A beaker of radius r is filled with water Refractive index of water=43\text{Refractive index of water} = \frac{4}{3} 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 . This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is h h≪H,h≪rh \ll H,\quad 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)

Question figure
  1. Option A:
    (A)R=h2+r22h\text{(A)}\quad R = \frac{h^{2} + r^{2}}{2h}
    Correct
  2. Option B:
    (B)R=3r22h\text{(B)}\quad R = \frac{3r^{2}}{2h}
  3. Option C:

    Apparent depth of the bottom of the beaker is close to 3H2(1+ω2H2g)−1\frac{3H}{2} \left( 1 + \frac{\omega^{2} H}{2g} \right)^{-1}

  4. Option D:

    Apparent depth of the bottom of the beaker is close to 3H4(1+ω2H4g)−1\frac{3H}{4} \left( 1 + \frac{\omega^{2} H}{4g} \right)^{-1}

    Correct

Answer: A, D

Step-by-step solution

tan⁡θ=dydx=ω2xg\tan\theta = \frac{dy}{dx} = \frac{\omega^{2} x}{g} ⇒y=ω2x22g⇒h=ω2r22g\Rightarrow y = \frac{\omega^{2} x^{2}}{2g} \Rightarrow h = \frac{\omega^{2} r^{2}}{2g} Pythagoras: r2+(R−h)2=R2\text{Pythagoras: } r^{2} + (R - h)^{2} = R^{2} ⇒R=r2+h22h⇒A option is correct\Rightarrow R = \frac{r^{2} + h^{2}}{2h} \Rightarrow \text{A option is correct}

For apparent depth applying refraction formula

1v=43(H−h)=1−43R⇒1v=13R+13H(H≫h)\frac{1}{v} = \frac{4}{3(H - h)} = \frac{1 - \frac{4}{3}}{R} \Rightarrow \frac{1}{v} = \frac{1}{3R} + \frac{1}{3H} \quad (H \gg h) ⇒∣v∣=3H4(1+ω2H4g)⇒D option is correct\Rightarrow |v| = \frac{3H}{4} \left( 1 + \frac{\omega^{2} H}{4g} \right) \Rightarrow \text{D option is correct}
Solution figure

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2020
Paper
Paper 2
Subject
Physics
Chapter
Geometrical Optics
Topic
Apparent Depth, Glass Slab, Prism and Dispersion
A beaker of radius r is filled with water Refractive index of water =… | JEE Advanced 2020 PYQ with Solution · DhiX AI