To answer the question about the validity of Hooke's Law, we need to understand some basic concepts of solid mechanics.
Hooke's Law: Hooke's Law states that, within the elastic limit of a material, the stress is directly proportional to the strain. Mathematically, it is expressed as:
\(\sigma = E \cdot \epsilon\)
Where:
\(\sigma\) is the stress applied to the material.
\(\epsilon\) is the strain experienced by the material.
E is the modulus of elasticity or Young's modulus, a constant for the material.
This law is valid only up to a certain point. Let's analyze the options:
Breaking point: This is the point at which the material breaks. Beyond the elastic limit, materials do not obey Hooke's Law, as they have undergone permanent deformation.
Yield point: The yield point is the stress level at which a material begins to deform plastically. Beyond this point, deformation is not completely recoverable, and hence, Hooke's Law does not hold.
Elastic limit: This is the maximum point up to which a material will return to its original shape and size upon the removal of the load. Beyond this point, permanent deformation occurs. Hooke's Law is valid up to the elastic limit.
Ultimate stress: This is the maximum stress that a material can withstand. This point is beyond the elastic limit, and hence, Hooke's Law is not applicable.
Thus, the correct answer is Elastic limit because Hooke's Law is valid only within this range. Beyond the elastic limit, materials enter the plastic deformation phase, and the linear relationship between stress and strain no longer holds.
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Approach Solution -2
Step 1: Understanding Hooke’s law.
Hooke’s law states that within certain limits, stress is directly proportional to strain. Mathematically, it is expressed as:
\[
\text{Stress} \propto \text{Strain}
\]
or
\[
\text{Stress} = E \times \text{Strain}
\]
where $E$ is the modulus of elasticity. Step 2: Meaning of elastic limit.
The elastic limit is the maximum stress a material can withstand without undergoing permanent deformation. Up to this point, the material returns to its original shape once the load is removed. Step 3: Relationship between Hooke’s law and elastic limit.
Hooke’s law is valid only in the region where stress and strain remain proportional. This proportional behavior exists only up to the elastic limit. Beyond this limit, the stress–strain relationship becomes non-linear. Step 4: Analysis of options. (A) Breaking point: Incorrect — At this point, the material fractures, and Hooke’s law no longer applies. (B) Yield point: Incorrect — Plastic deformation begins near the yield point, violating Hooke’s law. (C) Elastic limit: Correct — Hooke’s law holds true only up to the elastic limit. (D) Ultimate stress: Incorrect — This corresponds to maximum stress before failure, far beyond the elastic region. Step 5: Conclusion.
Since Hooke’s law requires proportionality between stress and strain, it is valid only up to the elastic limit.
