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Properties of Matter question

2019 · 12 Jan · Shift 2 · Q46
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Properties of Matter question

2019 · 12 Jan · Shift 2 · Q46

JEE MainPhysicsProperties of MatterMCQ+4 / −1
A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the centre and the sides, in cm, will be :
  1. A
    2.0
  2. B
    1.2
  3. C
    0.1
  4. D
    0.4
View written solutionFree

Correct answer: A

  1. Surface shape of a rotating liquid

When a liquid in a cylindrical vessel rotates with angular speed ω\omegaω, its free surface becomes a paraboloid. The height difference between the liquid surface at radius rrr and at the centre is

Δh=ω2r22g\Delta h = \frac{\omega^2 r^2}{2g}Δh=2gω2r2​

Here, the required difference is between the centre and the side wall, so we take r=Rr=Rr=R.

  1. Given data
  • Radius of vessel: R=5 cm=0.05 mR = 5\text{ cm} = 0.05\text{ m}R=5 cm=0.05 m
  • Rotational speed: f=2 rotations/sf = 2\text{ rotations/s}f=2 rotations/s

Therefore, angular speed is

ω=2πf=2π×2=4π rad/s\omega = 2\pi f = 2\pi \times 2 = 4\pi\text{ rad/s}ω=2πf=2π×2=4π rad/s
  1. Substitute into the formula
Δh=ω2R22g=(4π)2(0.05)22×9.8\Delta h = \frac{\omega^2 R^2}{2g} = \frac{(4\pi)^2(0.05)^2}{2\times 9.8}Δh=2gω2R2​=2×9.8(4π)2(0.05)2​

Now,

(4π)2=16π2≈16×9.87≈157.9(4\pi)^2 = 16\pi^2 \approx 16 \times 9.87 \approx 157.9(4π)2=16π2≈16×9.87≈157.9

and

(0.05)2=0.0025(0.05)^2 = 0.0025(0.05)2=0.0025

So,

Δh=157.9×0.002519.6\Delta h = \frac{157.9 \times 0.0025}{19.6}Δh=19.6157.9×0.0025​ Δh=0.3947519.6≈0.0201 m\Delta h = \frac{0.39475}{19.6} \approx 0.0201\text{ m}Δh=19.60.39475​≈0.0201 m
  1. Convert to cm
0.0201 m=2.01 cm0.0201\text{ m} = 2.01\text{ cm}0.0201 m=2.01 cm

So the height difference is approximately

2.0 cm\boxed{2.0\text{ cm}}2.0 cm​
  1. Check options
  • A: 2.02.02.0 ✅
  • B: 1.21.21.2 ❌
  • C: 0.10.10.1 ❌
  • D: 0.40.40.4 ❌

Therefore, the correct option is A.

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