To find the load $P$ that will result in an elongation (deformation) of 0.002 in. in the steel specimen, we can use Hooke's Law. Since the deformation is within the elastic range, the stress is proportional to strain.

Given dataTo find the load $P$ that will result in an elongation (deformation) of 0.002 in. in the steel specimen, we can use Hooke's Law. Since the deformation is within the elastic range, the stress is proportional to strain.

Given

• Diameter of the specimen: $d = 0.505 \, \text{in.}$
• Gage length: $L_0 = 2.00 \, \text{in.}$
• Deformation (elongation): $\Delta L = 0.002 \, \text{in.}$
• Proportional limit (yield stress): $\sigma = 34,000 \, \text{psi}$

Steps to solve:

1. Determine the strain: $\varepsilon = \frac{\Delta L}{L_0} = \frac{0.002 \, \text{in.}}{2.00 \, \text{in.}} = 0.001$ Strain is dimensionless.

2. Using Hooke's Law: Stress $\sigma$ is related to strain $\varepsilon$ through the proportional limit (in the elastic range): $\sigma = E \varepsilon$ where $E$ is the modulus of elasticity. However, since we know the stress is at the proportional limit of 34,000 psi, this is the stress causing the deformation.

3. Calculate the cross-sectional area of the specimen: The cross-sectional area $A$ of the specimen is the area of a circle: $A = \frac{\pi d^2}{4} = \frac{\pi (0.505 \, \text{in.})^2}{4} = 0.2003 \, \text{in}^2$

4. Calculate the load: Stress $\sigma$ is related to load $P$ and cross-sectional area $A$ by: $\sigma = \frac{P}{A}$ Rearranging for $P$: $P = \sigma \times A = 34,000 \, \text{psi} \times 0.2003 \, \text{in}^2 = 6,810.2 \, \text{lbs}$

Thus, the load $P$ required to cause a deformation of 0.002 in. is approximately 6,810 lbs.

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Here are some related questions:

1. How is the modulus of elasticity $E$ used in calculating deformation?
2. What happens to the material after it surpasses the proportional limit?
3. How do different diameters of specimens affect the load required for a given deformation?
4. Why is it important to know the gage length in tension tests?
5. What are the consequences of assuming an ideal elastic behavior in practical applications?

Tip: Always check that units are consistent when applying formulas, especially when dealing with stress and strain in different systems of measurement.