Hooke's Law Explained

Hooke's law is a fundamental principle in physics and mechanics that describes the elastic behavior of materials. It states that within the elastic limit of a material, the stress applied is directly proportional to the strain produced.

Mathematically, Hooke's law for a simple case like tension or compression can be written as:

\(\sigma = E \epsilon\)

Where:

• \(\sigma\) is the stress (force per unit area)
• \(\epsilon\) is the strain (relative deformation)
• \(E\) is the Young's modulus of elasticity, a material property that relates stress and strain in tension or compression.

For shear deformation, Hooke's law has a similar form:

\(\tau = G \gamma\)

Where:

• \(\tau\) is the shear stress
• \(\gamma\) is the shear strain
• \(G\) is the shear modulus of elasticity, relating shear stress and shear strain.

Hooke's Law Application to Different Stresses

Hooke's law applies to various types of stress and strain, provided the material behaves elastically and the deformation is within the proportional limit. Let's look at the options:

• Tension: When a material is pulled, it experiences tensile stress and tensile strain. Hooke's law applies, relating tensile stress to tensile strain via Young's modulus \(E\).
• Compression: When a material is pushed or squeezed, it experiences compressive stress and compressive strain. Hooke's law also applies here, relating compressive stress to compressive strain via the same Young's modulus \(E\).
• Shear: When a material is subjected to forces parallel to a surface, it experiences shear stress and shear strain. Hooke's law applies, relating shear stress to shear strain via the shear modulus \(G\).
• Torsion: Torsion involves twisting, which induces shear stresses within the material. While the analysis of torsion uses principles derived from Hooke's law for shear, torsion itself is a specific type of loading causing varying shear stress.
• Bending: Bending involves a combination of tension on one side of the material and compression on the other, with shear stresses also often present. The stress distribution is not uniform. The material's elastic behavior under bending is governed by Hooke's law, but the analysis is more complex than simple tension or compression.

Tension and Compression Under Hooke's Law

The question asks for what Hooke's law "equally holds good". Tension and compression are axial loading conditions where the stress and strain are directly related by the same material property, Young's modulus \(E\), for most isotropic materials. This direct and uniform relationship is a primary application of the basic form of Hooke's law. While Hooke's law also holds for shear (with modulus \(G\)), the constant of proportionality is different from that for tension/compression. Bending and torsion are more complex loading scenarios that rely on the underlying Hooke's law behavior.

Therefore, Hooke's law is equally applicable and fundamental for both tension and compression, using the same Young's modulus \(E\) within the elastic limit.