Dimension of Bulk Modulus

The bulk modulus has been defined as the relative change in volume of a body caused by a unit compressive or tensile stress that is uniformly applied throughout the stress. The bulk modulus describes how well a substance reacts after being uniformly flattened. Whenever the external forces are perpendicular to the surface, the bulk modulus is distributed uniformly over the object’s surface. This could also happen when an object is immersed in a fluid and experiences volume changes without changing shape.

In hydraulic systems, the bulk modulus is an important consideration. The two most important factors influencing a fluid’s bulk modulus are trapped gas and temperature. Because a fluid’s bulk modulus decreases with increasing temperature, the lower the temperature of a fluid, the more difficult it is to compress. Trapped gas, particularly air, has a significant impact on the bulk modulus of fluids. The lesser the bulk modulus of a fluid, the more entrained or trapped gas there is. The only exception would be when the fluid’s bulk modulus is less than that of air; in that case, the opposite is true.

The actual value of a fluid’s bulk modulus can affect system performance in terms of power, stability, position, and response time. The two most significant factors in controlling bulk modulus are entrained air content and fluid temperature.

A liquid’s bulk modulus is indeed an estimate of its compressibility. It must be calculated as the amount of pressure required to cause a unit change in volume. In brief, given today’s demands for higher power and faster response times, it is more important than ever to pay attention to bulk modulus.

Dimensional Formula of Bulk Modulus

The dimensional formula of bulk modulus can be represented as:

[M¹ L^-1 T^-2]

Here,

M = Mass

L = Length

T = Time

Derivation of Dimension of Bulk Modulus:

We have, Bulk modulus (k) = Bulk Stress × [Bulk strain]^-1 . . . . (1)

As because Bulk stress = Force × [Area]^-1 . . . (2)

Then, the dimensional formula of,

Force = [M¹ L¹ T^-2] . . . . (3)

Area = [M⁰ L² T⁰] . . . . (4)

Now, when substituting equations (3) and (4) in equation (2) we get,

Bulk stress = [M¹ L¹ T^-2] × [M⁰ L² T⁰]^-1

Thus, the dimensions of Bulk stress = [M¹ L^-1 T^-2] . . . . . (5)

Also, Bulk strain = Change in Volume × Volume^-1

= ΔV/V = [M⁰ L⁰ T⁰] = Dimensionless Quantity . . . (6)

When substituting equations (5) and (6) in equation (1) we get,

Bulk modulus = Bulk Stress × [Bulk strain]^-1

That is, k = [M¹ L^-1 T^-2] × [M⁰ L⁰ T⁰]^-1 = [M¹ L^-1 T^-2]

Thus, the dimension of bulk modulus has been represented as [M¹ L^-1 T^-2].

Significance of Dimension of Bulk Modulus in IIT JEE exam

It is critical to take a holistic approach to every facet of a subject’s chapter. It will not only adequately prepare you for the exam, but will also clarify your understanding of each topic. It will help you in JEE preparation and answer conceptual problems in the exam. The number of questions from the chapter unit and dimensions would be one or two, with a weightage of roughly four marks.

FAQs

Why do liquids have a bulk modulus?

Fluids seem to be liquids and gases that have a bulk modulus but no shear modulus. Because there is no shear stress, even a negligible shear force can cause a fluid to flow. A liquid's bulk modulus is proportional to its compressibility and is defined as the pressure required to change the volume of a liquid by one unit. Because liquids are practically incompressible, significant volume changes require extremely high pressures.

What is the dimensional formula of bulk modulus?

The bulk modulus has been dimensionally represented as M¹ L^-1 T^-2.

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