To solve this problem, we can use the equation for tensile strength:
Tensile strength = Force / Area
We know that we want the tensile strength to be at least 70,000 psi, and we can assume that the force required to achieve this will remain constant. Therefore, we can rearrange the equation to solve for the minimum area required:
Area = Force / Tensile strength
Next, we can use the equation for the area of a circle to relate the area of the original bar to its diameter:
Area = π * (diameter)^2 / 4
Substituting this into the previous equation, we get:
π * (diameter)^2 / 4 = Force / Tensile strength
Solving for the minimum diameter, we get:
diameter = √(4 * Force / (π * Tensile strength))
We don't know the force required to achieve the desired tensile strength, but we can use the equation for the ultimate tensile strength of copper (which is the maximum stress it can withstand before breaking) to estimate it:
Ultimate tensile strength = Yield strength / Safety factor
The yield strength of copper is around 30,000 psi, and a typical safety factor for engineering design is 2. Therefore, the estimated force required is:
Force = Ultimate tensile strength * Area * Safety factor
Force = 2 * 30,000 psi * π * (0.375 in / 2)^2
Plugging this into the equation for minimum diameter, we get:
diameter = √(4 * Force / (π * Tensile strength))
diameter = √(4 * 2 * 30,000 psi * π * (0.375 in / 2)^2 / (π * 70,000 psi))
diameter ≈ 0.564 in
Therefore, the minimum diameter of the original bar should be about 0.564 inches to achieve a final diameter of 0.375 inches with a tensile strength of at least 70,000 psi.