Calculating Young's Modulus for a Deformed Rod

The question asks us to find the formula for Young's modulus (E) of a material when a rod of uniform cross-section A and length L is subjected to a normal force P and deforms by $\delta$. Young's modulus is a fundamental property of an elastic material that describes its stiffness or resistance to elastic deformation under load. It is defined as the ratio of stress to strain.

Understanding Stress and Strain in a Material

Let's first define stress and strain in the context of this problem:

• Stress ($\sigma$): Stress is the internal force per unit area within a material that arises from external forces. In this case, the normal force P is applied over the cross-sectional area A of the rod.
  The formula for stress is: $\sigma = \frac{\text{Force}}{\text{Area}} = \frac{P}{A}$
• Strain ($\epsilon$): Strain is the measure of deformation of a material relative to its original size. For a rod subjected to a normal force causing it to lengthen or shorten, the strain is the change in length divided by the original length. Here, the change in length is $\delta$, and the original length is L.
  The formula for strain is: $\epsilon = \frac{\text{Change in Length}}{\text{Original Length}} = \frac{\delta}{L}$

Deriving the Young's Modulus Formula

Young's modulus (E) is defined as the ratio of normal stress to longitudinal strain, provided the material is within its elastic limit and the deformation is elastic.

The definition is: $E = \frac{\text{Stress}}{\text{Strain}}$

Substitute the formulas for stress ($\sigma = \frac{P}{A}$) and strain ($\epsilon = \frac{\delta}{L}$) into the definition of Young's modulus:

$E = \frac{\sigma}{\epsilon} = \frac{\frac{P}{A}}{\frac{\delta}{L}}$

To simplify this complex fraction, we can multiply the numerator by the reciprocal of the denominator:

$E = \frac{P}{A} \times \frac{L}{\delta}$

This gives the formula for Young's modulus in terms of the given parameters:

$E = \frac{P \times L}{A \times \delta}$

Comparing with Options for Young's Modulus

Now let's compare the derived formula with the given options:

• Option 1: $\frac{P\times \delta}{A\times L}$ - Does not match our derived formula.
• Option 2: $\frac{A\times \delta}{P\times L}$ - Does not match our derived formula.
• Option 3: $\frac{P\times L }{A\times \delta}$ - Matches our derived formula.
• Option 4: $\frac{P\times A }{L\times \delta}$ - Does not match our derived formula.

The formula $E = \frac{P \times L}{A \times \delta}$ correctly represents Young's modulus based on the given parameters.

Revision Table: Material Properties Formulas

Additional Information on Elasticity and Hooke's Law

The relationship between stress and strain defined by Young's modulus is part of a broader concept called elasticity.

• Elasticity: This is the property of a material that allows it to return to its original shape and size after the deforming force is removed. The deformation discussed in the question ($\delta$) is assumed to be elastic deformation.
• Hooke's Law: For many materials, within their elastic limit, stress is directly proportional to strain. Hooke's Law can be expressed as $\sigma \propto \epsilon$. The constant of proportionality in this relationship for uniaxial stress and strain is Young's modulus. So, $\sigma = E \epsilon$. Rearranging this gives the definition $E = \frac{\sigma}{\epsilon}$, which we used.
• Elastic Limit: If the applied stress exceeds the material's elastic limit, the deformation becomes permanent (plastic deformation), and the material will not fully return to its original shape after the load is removed. The formula for Young's modulus is only applicable for elastic deformation.