Question

Define bulk modulus of elasticity. Give its units and dimensions.

Answer 1

Hint- Here, we will proceed by defining the term bulk modulus of elasticity. Then, we will write down the formula given for bulk modulus of elasticity in terms of the applied pressure.

Complete step-by-step answer:
Bulk modulus of elasticity is simply the ratio of the pressure to the corresponding volumetric strain (or relative decrease in the volume of the substance). Mathematically, bulk modulus of elasticity is expressed as given under.

Bulk modulus of elasticity ${\text{K}} = \dfrac{{\text{P}}}{{{\text{Volumetric strain}}}} = \dfrac{{\text{P}}}{{ - \dfrac{{\Delta {\text{V}}}}{{\text{V}}}}}$ where P denotes the pressure applied to the body, $\Delta {\text{V}}$ denotes the change in the volume of the body due to the applied stress (or pressure) and V denotes the original volume of the body.

In the mathematical expression, negative sign represents that there occurs a decrease in volume due to increase in the value of the pressure applied.

Since, the SI units of pressure is Pascal (Pa) or Newton per square metre (N/${{\text{m}}^2}$) and volumetric strain (i.e., ratio of the change in the volume to the volume) has no unit. Clearly, according to the above mentioned formula, the units of the bulk modulus of elasticity will be the same as those of the pressure. So, the SI units of bulk modulus of elasticity will be Pascal (Pa) or Newton per square metre (N/${{\text{m}}^2}$). Other units of bulk modulus of elasticity include Megapascal (MPa), Kilopascal (Kpa), etc.

Since, the dimension of pressure is same as that of force per unit area (or stress) which is $\left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]$ and the volumetric strain is dimensionless. So, the dimension of the bulk modulus of elasticity will be equal to the dimension of the pressure. Therefore, the dimensions of bulk modulus of elasticity is also $\left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]$.

Additional Information:
Bulk modulus is known as the ratio of the volumetric stress to the volumetric strain of a physical quantity, while the material deformation is within the elastic limit. To put it in simpler terms, the bulk modulus is nothing more than a numerical constant used to calculate and define the elastic properties of a solid or fluid when applying pressure on all surfaces.
Bulk elasticity modulus is one measure of solids' mechanical properties. Certain elastic modules include Young's modulus of elasticity and Shear's modulus of elasticity. In any case, a material's bulk elastic properties are used to determine how much it can compress under a specified amount of external pressure.

Note- Some of the important characteristics of bulk modulus of elasticity includes that it exists in all three phases (i.e., solids, liquids and gases). It also determines the amount of the compression that will occur when a given external pressure is applied. The reciprocal of bulk modulus of elasticity is compressibility.