Bulk Modulus of Elasticity Definition Formula Questions

What is Bulk Modulus of Elasticity?

Bulk modulus of elasticity is defined as the proportion of volumetric stress linked to the volumetric strain of definite material, although the deformation of the material is within the elastic limit. Bulk modulus is basically a numerical constant that measures and defines the elastic properties of a solid or fluid when pressure is applied on all the surfaces.

Bulk Modulus Formulas

Mathematically we can express bulk modulus of elasticity as-

Bulk modulus of elasticity (K) =P/Volumetric strain = K = (P/ (- ΔV/V))

Where, P = pressure applied to the body,

ΔV represents the change in the volume of the body due to the applied stress and V is the original volume of the body.

In the mathematical expression, a negative sign indicates that due to increase in the value of the pressure applied, the volume decreases.

This is the same as, the change in pressure divided by the change in volume divided by initial volume:

Bulk Modulus (K) = (p₁ – p₀) / [(V₁ – V₀) / V₀]

Here, initial pressure and volume will be p₀ and V₀ and p₁ and V1 are the pressure and volume measured upon compression.

Hence, in terms of pressure and density bulk modulus elasticity may also be expressed as

K = (p₁ – p₀) / [(ρ₁ – ρ₀) / ρ₀]

Here, ρ₀ and ρ₁ are the initial and final density values.

SI Unit of Bulk Modulus of Elasticity

According to the above formula, the bulk modulus of elasticity units will be the same as the unit of pressure because the volumetric strain is unitless, (N/m²) is the SI units of pressure. Hence, Pascal (Pa) or Newton per square metre (N/m²) will be the SI units of bulk modulus of elasticity. Megapascal (MPa), Kilopascal (Kpa) are some other units of bulk modulus of elasticity too.

Dimensions of Bulk Modulus of Elasticity

Since, volumetric strain is dimensionless and the dimension of pressure is the same as that of force per unit area which is [ML⁻¹T⁻²]. Therefore, the dimension of the bulk modulus of elasticity will be equal to the dimension of the pressure that is [ML⁻1T⁻2].

Important Questions on Bulk Modulus of Elasticity

1) Bulk Modulus of elasticity is the ratio of?

a) Compressive stress to linear strain

b) Compressive stress to volumetric strain

c) Tensile stress to volumetric strain

d) Tensile stress to linear strain

Correct Option: (b)

Explanation: Bulk Modulus k is associated with the decrease in volume per unit volume and the compression of a liquid. Therefore, it is the ratio of compressive stress to the volumetric strain.

2) The exact relationship between compressibility (β) and Bulk Modulus (k) is which one among the following?

a) β = 1/k

b) β = k

c) β = 3k

d) β = k/4

Correct Option: (a)

Explanation: β = 1/k is the correct relation because the compressibility β of a liquid is defined as the ratio of volumetric strain to the compressive stress while Bulk Modulus is the ratio of compressive stress to volumetric strain.

3) For an incompressible fluid, what will be the value of Bulk Modulus of elasticity?

a) Infinity

b) Unity

c) Zero

d) very low

Correct Option (a)

Explanation: k = 1/β, where k is Bulk Modulus of elasticity and β is compressibility. Therefore, β=0, for an incompressible fluid, hence the value of k will be infinity.

4) Which among the following is the dimension of compressibility?

a) [M¹L¹T^-1].

b) [M¹L¹T^-2].

c) [M^-1L¹T²].

d) [M^-1L¹T^-2].

Correct Option: (c)

Explanation: Since, k = 1/β, where β= compressibility and k = Bulk Modulus of elasticity and the dimensions of k is [ML^-1T^-2]

5) Which among the following is the unit of compressibility?

a) m²/N

b) m/N

c) m³/N

d) Compressibility is unitless

Correct Option: (a)

Explanation: Since, k (Bulk Modulus of elasticity) = Compressive Stress/Volumetric Stress, therefore, the unit of bulk modulus is N/m² and hence, the unit of compressibility will be m²/N.

6) Which one of these won’t have the same unit as the others?

a) Bulk Modulus

b) Force

c) Pressure

d) Stress

Correct Option: (b)

Explanation: N/m² is the SI unit of Bulk Modulus, Stress and Pressure but the unit of Force is N.

7) The Bulk Moduli of 3 fluids, that is, a, b and c are p1, p2 and p3, respectively. If p1 > p2 > p3, which liquid will have the maximum compressibility?

a) liquid a

b) liquid b

c) liquid c

d) they’ll have equal compressibility

Correct Option: (c)

Explanation: p = 1=β, where p= Bulk Modulus of elasticity and β= compressibility. If p1 > p2 > p3, then β1 < β2 < β3. Thus, liquid c will have the highest compressibility.

8) Which one of the following is the correct relationship between young’s modulus (E), bulk modulus (K) and Poisson’s ratio (µ)?

a) E=3K(1-2µ)

b) E=2K(1-2µ)

c) E=2K(1-2µ)

d) E=2K(1-3µ)

Correct Option: (a)

Explanation: relationship between young’s modulus (E), bulk modulus (K) and Poisson’s ratio (µ) will be given as E=3K(1-2µ)

9) Factor of safety is the ratio of?

a) Ultimate stress / Permissible stress

b) Compressive stress / Ultimate stress

c) Tensile stress / Permissible stress

d) Ultimate stress / Shear stress

Correct Option: (a)

Explanation: Factor of safety is defined as the ratio of ultimate stress to the permissible stress, therefore the correct answer will be option a.

10) The bulk modulus of elasticity

a) Increases with pressure

b) Decreases with pressure

c) Is independent of temperature

d) Is independent of pressure and temperature

Correct Option: (a)

Explanation: Since, the bulk modulus of elasticity is directly proportional to the change of pressure, therefore as the pressure increases the bulk modulus of elasticity increases.

Practice Questions

1. Define bulk modulus of elasticity.
2. Differentiate between bulk modulus and density of a liquid.
3. Calculate the bulk modulus of a body that experiences a change in pressure of 6*10⁴N/m² and its volume goes from 5 cm³ to 4.9 cm³.
4. Define Poisson’s ratio.
5. Define young’s modulus of elasticity.

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