Mechanical Properties of Solids

JEE Main level, numerical answer, medium

91 of 118

Question 91 of 118

If average depth of an ocean is 4000m4000m and the bulk modulus of water is 2×109Nm−22\times {10}^{9}N{m}^{-2}, then fractional compression ΔVV\dfrac{\Delta V}{V} of water at the bottom of ocean is α×10−2\alpha \times {10}^{-2}. The value of α\alpha is _______, (Given, g=10ms−2,ρ=1000kgm−3g=10m{s}^{-2},\rho =1000kg{m}^{-3})

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Answer: 2

The formula to calculate the Bulk modulus is given by
B=−∆P(∆VV)...(1)B=-\dfrac{∆P}{\left(\dfrac{∆V}{V}\right)}...\left(1\right)
From equation (1), the fractional compression in volume may be obtained as follows:
−(ΔVV)=∆PB=ρg∆hB...(2)\begin{matrix}-\left(\dfrac{\Delta V}{V}\right) & = & \dfrac{∆P}{B} \\ & = & \dfrac{\rho g∆h}{B}...\left(2\right)\end{matrix}
From equation (2) it follows that
∣−(∆VV)∣=1000×10×40002×109⇒(∆VV)=2×10−2\begin{matrix}\left|-\left(\dfrac{∆V}{V}\right)\right| & = & \dfrac{1000\times 10\times 4000}{2\times {10}^{9}} \\ & \Rightarrow & \left(\dfrac{∆V}{V}\right)=2\times {10}^{-2}\end{matrix}
Hence, α=2\alpha =2.

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