derive the young's modulus of elasticity
Answers
Young’s modulus, numerical constant, named for the 18th-century English physician and physicist Thomas Young, that describes the elastic properties of a solid undergoing tension or compression in only one direction, as in the case of a metal rod that after being stretched or compressed lengthwise returns to its original length. Young’s modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Sometimes referred to as the modulus of elasticity, Young’s modulus is equal to the longitudinal stress divided by the strain. Stress and strain may be described as follows in the case of a metal bar under tension.
If a metal bar of cross-sectional area A is pulled by a force F at each end, the bar stretches from its original length L0 to a new length Ln. (Simultaneously the cross section decreases.) The stress is the quotient of the tensile force divided by the cross-sectional area, or F/A. The strain or relative deformation is the change in length, Ln − L0, divided by the original length, or (Ln − L0)/L0. (Strain is dimensionless.) Thus Young’s modulus may be expressed mathematically as
Young’s modulus = stress/strain = (FL0)/A(Ln − L0).
This is a specific form of Hooke’s law of elasticity. The units of Young’s modulus in the English system are pounds per square inch (psi), and in the metric system newtons per square metre (N/m2). The value of Young’s modulus for aluminum is about 1.0 × 107 psi, or 7.0 × 1010 N/m2. The value for steel is about three times greater, which means that it takes three times as much force to stretch a steel bar the same amount as a similarly shaped aluminum bar.
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