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Properties of Matter question

2021 · 17 Mar · Shift 2 · Q60
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Properties of Matter question

2021 · 17 Mar · Shift 2 · Q60

JEE MainPhysicsProperties of MatterMCQ+4 / −1
An object is located at 2 km beneath the surface of the water. If the fractional compression ΔVV{{\Delta V} \over V}VΔV​ is 1.36%, the ratio of hydraulic stress to the corresponding hydraulic strain will be ‾\underline{\hspace{2cm}}​. [Given : density of water is 1000 kgm −-− 3 and g = 9.8 ms −-− 2]
  1. A
    1.44 ×\times× 107 Nm −-− 2
  2. B
    1.44 ×\times× 109 Nm −-− 2
  3. C
    1.96 ×\times× 107 Nm −-− 2
  4. D
    2.26 ×\times× 109 Nm −-− 2
View written solutionFree

Correct answer: B

  1. Concept used

The ratio of hydraulic stress to hydraulic strain is the bulk modulus:

B=hydraulic stresshydraulic strainB = \frac{\text{hydraulic stress}}{\text{hydraulic strain}}B=hydraulic strainhydraulic stress​

Here,

  • Hydraulic stress =P=ρgh= P = \rho g h=P=ρgh
  • Hydraulic strain =ΔVV= \dfrac{\Delta V}{V}=VΔV​

So,

B=ρghΔV/VB = \frac{\rho g h}{\Delta V/V}B=ΔV/Vρgh​
  1. Given data
  • Depth, h=2 km=2000 mh = 2\ \text{km} = 2000\ \text{m}h=2 km=2000 m
  • Density of water, ρ=1000 kg m−3\rho = 1000\ \text{kg m}^{-3}ρ=1000 kg m−3
  • Acceleration due to gravity, g=9.8 m s−2g = 9.8\ \text{m s}^{-2}g=9.8 m s−2
  • Fractional compression,
ΔVV=1.36%=1.36100=0.0136\frac{\Delta V}{V} = 1.36\% = \frac{1.36}{100} = 0.0136VΔV​=1.36%=1001.36​=0.0136
  1. Calculate hydraulic stress
P=ρgh=1000×9.8×2000P = \rho g h = 1000 \times 9.8 \times 2000P=ρgh=1000×9.8×2000 P=1.96×107 N m−2P = 1.96 \times 10^7\ \text{N m}^{-2}P=1.96×107 N m−2
  1. Calculate bulk modulus
B=1.96×1070.0136B = \frac{1.96 \times 10^7}{0.0136}B=0.01361.96×107​ B=1.441176×109 N m−2B = 1.441176\times 10^9\ \text{N m}^{-2}B=1.441176×109 N m−2 B≈1.44×109 N m−2B \approx 1.44 \times 10^9\ \text{N m}^{-2}B≈1.44×109 N m−2
  1. Match with options

This corresponds to:

Option B: 1.44×109 N m−21.44 \times 10^9\ \text{N m}^{-2}1.44×109 N m−2

Final Answer

1.44×109 N m−2\boxed{1.44 \times 10^9\ \text{N m}^{-2}}1.44×109 N m−2​

So, the correct option is B.

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