Final answer:
To stretch a 2.0-mm-diameter steel wire by 1.09 mm, a force of A) 3.1 x 10^3 N is necessary.
Step-by-step explanation:
To calculate the force required to stretch a steel wire, we can use Hooke's Law which states that the force is equal to the product of the spring constant (Young's modulus) and the extension. First, we need to calculate the cross-sectional area of the wire using the formula A = πr^2. Given that the diameter of the wire is 2.0 mm, the radius is 1.0 mm or 0.001 m.
Therefore, the cross-sectional area is A = π(0.001)^2 = 3.14 x 10^-6 m^2.
Next, we can use the formula for tensile strain, which is the change in length divided by the original length. In this case, the change in length is 1.09 m and the original length is 2.0 mm or 0.002 m. So the tensile strain is ε = (1.09 - 0.002) / 0.002 = 0.5445. Now we can calculate the force using the formula F = YAε, where Y is Young's modulus and A is the cross-sectional area. Plugging in the values, we get F = (2.0 x 10^11 N/m^2)(3.14 x 10^-6 m^2)(0.5445) = 3.1 x 10^3 N.