Certainly! Let's go through the problem-by-step with detailed explanations. --- ## ** the problem** - You have an **aluminum alloy** strengthened with **SiC particles**. - The **mean spacing** between SiC particles in the current alloy is **.9 μm**. - The **yield strength** of this composite is **9.4 MPa** **greater** than the unstrengthened alloy. - The question asks: **If a new composite is developed with a particle spacing of .4 μm, what would be the expected contribution to the strength?** --- ## **Step 1: Recognize the relationship** In composite materials reinforced with particles, the increase in strength (\(\Delta \sigma\)) due to the particles is often related to the particle spacing (\(L\)) by an inverse relationship: \[ \Delta \sigma \propto \frac{1}{L} \] This means that as the particle spacing decreases (particles are closer together), the strength contribution increases proportionally. --- ## **Step 2: Find the proportionality constant** From the given data: \[ \Delta \sigma_1 = 9.4\, \text{MPa} \] \[ L_1 = .9\, \mu\text{m} \] The proportionality constant \(k\) can be expressed as: \[ k = \Delta \sigma_1 \times L_1 \] Calculate \(k\): \[ k = 9.4\, \text{MPa} \times .9\, \mu\text{m} = 8.46\, \text{MPa} \cdot \mu\text{m} \] --- ## **Step 3: Use the proportionality to find the new strength contribution** For the new composite with: \[ L_2 = .4\, \mu\text{m} \] The expected strength contribution \(\Delta \sigma_2\) is: \[ \Delta \sigma_2 = \frac{k}{L_2} \] Substitute the known values: \[ \Delta \sigma_2 = \frac{8.46\, \text{MPa} \cdot \mu\text{m}}{.4\, \mu\text{m}} = 21.15\, \text{MPa} \] --- ## **Final Answer** The expected strength contribution from the particles in the new composite with a particle spacing of **.4 μm** is approximately **21.15 MPa**. --- ## **Summary** - The strength contribution is inversely proportional to particle spacing. - Using the initial data, we found a constant of proportionality. - Applying this constant to the new spacing, we calculated the expected strength increase. Let me know if you'd like further clarification!