Bulk Modulus And Compressibility

Question

A solid sphere of volume 0.5 m³ is taken deep into the ocean where the pressure increases by 2 	imes 10^7 Pa, causing its volume to decrease by 100 cm³; find the bulk modulus.

RankGuru · Accepted Answer

NCERT Class 11, Chapter 9 (Mechanical Properties of Solids) defines bulk modulus as the ratio of the applied normal pressure to the resulting fractional decrease in volume, $B = \dfrac{\Delta P}{(\Delta V / V)}$. This describes how strongly a material resists uniform compression on all sides. The volume change must be expressed in SI units: $\Delta V = 100\ 	ext{cm}^3 = 100 	imes 10^{-6}\ 	ext{m}^3 = 1 	imes 10^{-4}\ 	ext{m}^3$. The volumetric strain is $\dfrac{\Delta V}{V} = \dfrac{1 	imes 10^{-4}}{0.5} = 2 	imes 10^{-4}$. Therefore $B = \dfrac{2 	imes 10^7}{2 	imes 10^{-4}} = 1 	imes 10^{11}\ 	ext{Pa}$. The option $2 	imes 10^{11}$ wrongly omits the factor of 0.5 in the strain. The option $1 	imes 10^{10}$ results from a units slip leaving volume in cm³. The option $5 	imes 10^{10}$ comes from inverting the strain ratio. A useful habit is to first compute the dimensionless volumetric strain and only then divide the pressure by it, which keeps the calculation transparent. As a magnitude check, the value lies in the typical range for stiff solids, far higher than that of gases which compress easily, and the units of pressure divided by a dimensionless strain correctly give pascals. The very small fractional volume change for such a large pressure also signals that the material is highly incompressible, consistent with a large bulk modulus.

Bulk Modulus And Compressibility

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