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A cylindrical specimen of some metal alloy 6.6 mm in diameter is stressed in tension. A force of 7070 N produces an elastic reduction in specimen diameter of 0.0038 mm. Calculate the elastic modulus (in GPa) of this material if its Poisson's ratio is 0.34 .

User Shambolic
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Final answer:

To find the elastic modulus of the metal alloy, one must calculate the longitudinal strain using the change in the specimen's diameter and Poisson's ratio, find the stress applied using the force and cross-sectional area, and divide the stress by the longitudinal strain.

Step-by-step explanation:

To calculate the elastic modulus (Young's modulus, E) of the metal alloy, we can use the relation between stress, strain, and Poisson's ratio. The longitudinal strain can be found using the change in length divided by the original length, while the lateral strain is the change in diameter divided by the original diameter. Given that the lateral strain is the product of Poisson's ratio (v) and the longitudinal strain, we can determine the longitudinal strain:

  • Lateral strain = δD/D
  • Poisson's ratio x Longitudinal strain = Lateral strain
  • Longitudinal strain = Lateral strain / Poisson's ratio

Once we have the longitudinal strain, the stress applied can be calculated using the force divided by the cross-sectional area of the specimen, which for a cylinder is π*(D/2)². Finally, the elastic modulus is the stress divided by the longitudinal strain:

  • Stress = Force / Cross-sectional area
  • Elastic modulus (E) = Stress / Longitudinal strain

Using the given data:

  • Diameter (D) = 6.6 mm = 6.6 x 10⁻³ m
  • Change in diameter (δD) = 0.0038 mm = 3.8 x 10⁻¶ m
  • Force = 7070 N
  • Poisson's ratio (v) = 0.34

Date of Knowledge Cutoff: March 2021

Performing the calculations:

  • Stress = 7070 N / (π * (6.6 x 10⁻³ m / 2)²)
  • Lateral strain = 3.8 x 10⁻¶ m / 6.6 x 10⁻³ m
  • Longitudinal strain = Lateral strain / 0.34
  • Elastic modulus (E) = Stress / Longitudinal strain

By calculating the appropriate magnitude, we obtain the elastic modulus in Pascals (Pa), which can then be converted to gigapascals (GPa).

User Carinmeier
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