Strain: The Unitless Property Explained

In physics and engineering, understanding the units of various physical properties is crucial. Some properties have specific units of measurement, while others are dimensionless, meaning they are "unitless." This solution will explore the concept of unitless properties and specifically analyze the given options to identify the one that fits this description.

Understanding Unitless Properties

A unitless property, also known as a dimensionless quantity, is a physical quantity that has no physical unit associated with it. This occurs when the quantity is defined as a ratio of two quantities with the same units, causing their units to cancel out.

Analyzing Material Properties and Their Units

Let's examine each of the listed properties to determine their respective units.

• Surface Tension
  Surface tension ($\gamma$ or T) is a contractive tendency of the surface of a liquid that allows it to resist an external force. It is defined as the force per unit length acting perpendicular to a line in the surface.
  Mathematically, it is expressed as:
  $$ \text{Surface Tension} = \frac{\text{Force}}{\text{Length}} $$ The standard International System of Units (SI units) for surface tension are newtons per meter ($\text{N/m}$) or joules per square meter ($\text{J/m}^2$). Since it has units, surface tension is not a unitless property.
• Dynamic Viscosity
  Dynamic viscosity ($\mu$) is a measure of a fluid's resistance to flow. It describes the internal friction of a fluid when it is in motion. It is defined by Newton's law of viscosity, relating shear stress to the rate of shear strain.
  The formula can be derived from the relationship:
  $$ \text{Shear Stress} (\tau) = \text{Dynamic Viscosity} (\mu) \times \text{Rate of Shear Strain} \left(\frac{du}{dy}\right) $$ Rearranging for dynamic viscosity:
  $$ \mu = \frac{\tau}{\frac{du}{dy}} = \frac{\text{Force/Area}}{\text{Velocity/Length}} = \frac{\text{N/m}^2}{\text{(m/s)/m}} = \frac{\text{N/m}^2}{\text{1/s}} = \text{N} \cdot \text{s/m}^2 $$ The SI unit for dynamic viscosity is pascal-second ($\text{Pa} \cdot \text{s}$), which is equivalent to newton-second per square meter ($\text{N} \cdot \text{s/m}^2$). Another common unit is Poise (P), where 1 Pa·s = 10 P. Since it has units, dynamic viscosity is not a unitless property.
• Shear Stress
  Shear stress ($\tau$) is the component of stress coplanar with the material cross-section. It arises from the shear force, which is applied parallel to a surface.
  It is calculated as:
  $$ \text{Shear Stress} = \frac{\text{Shear Force}}{\text{Area}} $$ The SI unit for shear stress is the pascal ($\text{Pa}$), which is equivalent to newtons per square meter ($\text{N/m}^2$). Since it has units, shear stress is not a unitless property.
• Strain
  Strain ($\varepsilon$) is a measure of the deformation of a material due to an applied force. It is defined as the ratio of the change in dimension to the original dimension.
  For example, linear strain (or normal strain) is given by:
  $$ \text{Linear Strain} = \frac{\text{Change in Length}}{\text{Original Length}} = \frac{\Delta L}{L_0} $$ Since both the change in length ($\Delta L$) and the original length ($L_0$) are measured in units of length (e.g., meters, millimeters), their ratio results in the units canceling out.
  For instance, if $\Delta L$ is in meters and $L_0$ is in meters, then:
  $$ \text{Strain} = \frac{\text{meters}}{\text{meters}} = \text{no units} $$ Therefore, strain is a dimensionless quantity, making it a unitless property.

Summary of Property Units

To provide a clear comparison, the table below summarizes each property and its corresponding SI unit:

Conclusion: Identifying the Unitless Property

Based on the detailed analysis of each physical property and their respective units, it is evident that Strain is the only property among the given options that is unitless. This is because strain is defined as a ratio of two quantities having the same physical dimensions, leading to the cancellation of units.