The elastic limit is the maximum stress or force that a material can withstand without undergoing permanent deformation. Beyond this limit the material will not return to its original shape when the applied force is removed. Understanding the elastic limit is crucial for designing safe and effective structures.
Class 11 Physics Chapter 8 examines the mechanical properties of solids focusing on stress strain and elasticity. The chapter explains how solids deform under applied forces and return to their original shape. Key topics include Hooke’s law different types of stress and strain and the relevance of moduli of elasticity in engineering applications.
Calculate thickness of metallic ropes in cranes.
The thickness of metallic ropes used in a crane is given to ensure that heavy loads can be safely picked up without exceeding the elastic limit of the material. The design process involves the following steps:
1. Elastic Limit: It is the maximum stress that can be applied to a material such that it comes back to its original shape after the removal of load. For stresses beyond the elastic limit, the material can suffer permanent deformation.
2. Computation of the Load: First, the load (W) that the rope needs to raise should be known. This refers not only to the weight but also to additional forces acting on the rope like dynamic loads and safety factors.
3. Factor of Safety: This is a measure of safety where uncertainties in both the load and material properties are accounted for. It is described as:
FOS = (Maximum Load) / (Allowable Load)
The allowable load can be determined from the following relationship:
P_allow = σₑ / FOS
4. Loading in Terms of Rope Thickness: The maximum tensile load in a rope may be expressed using the following expression:
P = A * σₑ
where P is the load, A is the cross-sectional area of the rope, and σₑ is the elastic limit of the material.
5. Determining the Cross-Sectional Area: The cross-sectional area, A, for a cylindrical rope can be found in terms of its diameter d as follows:
A = (π / 4) * d²
6. Combining Equations: By combining the equations, we can express the load in terms of the diameter:
P_allow = (π / 4) * d² * σₑ / FOS
Rearranging gives:
d² = (4 * P_allow * FOS) / (π * σₑ)
Thus, the required diameter (d) can be calculated as:
d = √[(4 * P_allow * FOS) / (π * σₑ)]
Conclusion: With these principles, engineers can determine the appropriate thickness of metallic ropes for cranes, ensuring safety and reliability while lifting heavy loads.
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