A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force F, its length increases by 5 cm. Another wire of the same material of length 4L and radius 4r is pulled by a force 4F under same conditions. The increase in length of this wire is ___ cm.
Correct Answer: 5
Solution and Explanation
\(\frac{\frac{F}{A}}{\frac{\Delta L}{L}}\)=Y ⇒△L=\(\frac{FL}{AY}\) \(\frac{△L_2}{△L_1}\)=(\(\frac{F_2}{F_1}\))×(\(\frac{L_2}{L_1}\))×(\(\frac{A_1}{A_2}\))
=\(\frac{4×4×1}{16}\)=1 △L2=△L1=5cm
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Concepts Used:
Elastic Moduli
Elastic moduli, also known as modulus of elasticity, are mechanical properties of materials that describe their ability to resist deformation under an applied load. They are used to quantify the stress-strain relationship of a material.
There are three main types of elastic moduli: Young's modulus (E), shear modulus (G), and bulk modulus (K). Young's modulus is a measure of a material's stiffness in tension or compression, while shear modulus is a measure of its resistance to shear deformation. Bulk modulus is a measure of its resistance to compression.
These moduli are important in many areas of engineering and physics, as they govern the behavior of materials under different types of loading conditions. For example, Young's modulus is used to design and analyze structures such as bridges and buildings, while shear modulus is important in the study of earthquakes and the behavior of soils.
The elastic moduli of a material are dependent on its microstructure, such as its crystal structure, grain size, and defects. They are also influenced by temperature, pressure, and other environmental factors.
The study and measurement of elastic moduli are essential for understanding the mechanical properties of materials, as well as for the design and analysis of structures and devices in engineering and physics.