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state-hookes-law

๐Ÿš€ Hooke's Law describes the relationship between stress and strain in elastic materials. It states that the strain in a material is proportional to the applied stress, provided the material's elastic limit is not exceeded. This principle is fundamental in understanding how materials deform under various forces and is crucial in fields such as engineering and materials science.

Theory Explanation

Understanding Stress and Strain

Stress is defined as the force applied per unit area of a material, while strain is the deformation experienced by the material in response to that stress. Mathematically, stress (ฯƒ) is given by ฯƒ = F/A, where F is the force applied and A is the area over which the force is applied. Strain (ฮต) is defined as the change in length (ฮ”L) divided by the original length (L0), or ฮต = ฮ”L/L0.

\[ \sigma = \frac{F}{A}, \quad \epsilon = \frac{\Delta L}{L_0} \]
Hooke's Law

According to Hooke's Law, the relationship between stress and strain can be expressed as ฯƒ = E * ฮต, where E is the modulus of elasticity (a material property). This law holds true as long as the material remains within its elastic limit, meaning it will return to its original shape once the stress is removed.

\[ \sigma = E \cdot \epsilon \]
Elastic Limit

The elastic limit is the maximum stress that a material can withstand without undergoing permanent deformation. Beyond this limit, the material will not return to its original shape, and plastic deformation occurs. Understanding the elastic limit is crucial for applications in engineering and construction.

Key Points

  • ๐ŸŽฏ Stress is the force per unit area applied to a material.
  • ๐ŸŽฏ Strain is the measure of deformation representing the displacement between particles in a material.
  • ๐ŸŽฏ Hooke's Law states that stress is directly proportional to strain within the elastic limit of a material.
  • ๐ŸŽฏ The modulus of elasticity (E) is a measure of a material's stiffness and is a key factor in Hooke's Law.
  • ๐ŸŽฏ Understanding the elastic limit is essential to prevent permanent deformation in materials.

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Examples:💡

A steel rod of length 2 m and cross-sectional area 0.01 mยฒ is subjected to a tensile force of 20,000 N. Calculate the stress and strain in the rod. Assume the modulus of elasticity for steel is 200 GPa.

Solution:

Step 1: Calculate the stress using the formula ฯƒ = F/A.

\[ \sigma = \frac{20000}{0.01} = 2000000 \, \text{Pa} = 2 \, \text{MPa} \]

Step 2: Calculate the strain using the formula ฮต = ฯƒ/E.

\[ \epsilon = \frac{2000000}{200 \times 10^9} = 0.00001 \]

A rubber band is stretched from its original length of 10 cm to 12 cm. Calculate the strain in the rubber band.

Solution:

Step 1: Determine the change in length: ฮ”L = 12 cm - 10 cm = 2 cm.

Step 2: Calculate the strain using the formula ฮต = ฮ”L/L0 = 2 cm / 10 cm = 0.2.

\[ \epsilon = \frac{2}{10} = 0.2 \]

Common Mistakes

  • Mistake: Confusing stress with strain; students often mix up the definitions of stress and strain.

    Correction: Stress is a measure of force per unit area, while strain is a measure of deformation. Remember that stress relates to the applied force, and strain relates to the change in shape.

  • Mistake: Ignoring the elastic limit when applying Hooke's Law; students may apply the law beyond the elastic limit.

    Correction: Always check if the material is within its elastic limit before applying Hooke's Law. If the material has exceeded this limit, the relationship between stress and strain is no longer linear.

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