Bulk modulus Formula

When a force is applied on a body in all directions and results in a deformation of the whole volume, the elastic coefficient is called the Bulk modulus. Is ratio of the change in pressure to the fractional volume compression:

Bulk modulus = (change in pressure stress)/(fractional volume) = (change in pressure) / (change in volume / original volume)

The equation is

B = ΔP /(ΔV/V)

We have:

B: Bulk modulus

ΔP: change of the pressure or force applied per unit area on the material

ΔV: change of the volume of the material due to the compression

V: Initial volume of the material

Bulk modulus equation Questions:

1) What is the bulk modulus of a body that experienced a change of pressure of 5*10⁴ N/m² and a its volume goes from 4 cm³ to 3.9 cm³?.

Answer: The bulk modulus is calculated using the formula,

B = ΔP /(ΔV/V)

B = (5*10⁴ N/m²)/((4 cm³ - 3.9 cm³)/4 cm³) = 0.125 *10⁴ N/m²

B = 1.25 *10⁴ N/m²

2) A sphere of radius 10 mm is stretched from its original volume to a half, using a force of 100 N. What is the bulk modulus of the system?

Answer: The bulk modulus is found from the equation:

B = ΔP /(ΔV/V)

The volume is calculated using V = 4/3 π (r)³, where r is 10 mm, then V is

A = 4/3 π (10 mm)³ = 4/3 π (0.01 m)³ = 4.2*10^(-6) m³

substituting the value of volume in

ΔV = Vf - Vi = 4.2*10^(-6) m³ - 1.1*10^(-6) m³ = 1.1*10^(-6) m³

Dividing by V, the fractional volume is,

ΔV/V = 1.1*10^(-6) m³ / 4.2*10^(-6) m³ = 0.26.

To find the pressure we use the formula, ΔP=F/A, where A is the area of the sphere A = 4 π r² = 4 π (0.01 m)² = 1.26*10^(-3) m²

Then ΔP = 100 N/1.26*10^(-3) m² = 79300 N/m².

Finally, the bulk modulus is,

B = (79300 N/m²) / (0.26) = 305000 N/m²