Final answer:
To find the minimum diameter of a steel wire that will not stretch more than 0.25 cm under a force of 400 N, Hooke's Law and the concepts of stress, strain, and Young's modulus are applied. The calculations involve finding the strain, calculating the stress, and finally determining the diameter using the known Young's modulus for steel.
Step-by-step explanation:
To determine the minimum diameter required for a circular steel wire to stretch no more than 0.25 cm when a tensile force of 400 N is applied, we need to use Hooke's Law and the formula for stress and strain in terms of Young's modulus. The tensile strain (ε) is the change in length (ΔL) divided by the original length (L), and stress (σ) is the force (F) divided by the area (A). Consequently, Young's modulus (E) is expressed as stress over strain (E = σ / ε).
First, we find the strain by using the maximum stretch and original length of the wire:
ε = ΔL / L = 0.0025 m / 2.00 m = 0.00125.
Next, we calculate the stress, which is the force divided by the area (which for a circle is πd^2/4, where d is the diameter):
σ = F / A = 400 N / (π(d^2)/4).
Now, we need to find the diameter of the wire using Young's modulus equation:
E = σ / ε = (400 N / (π(d^2)/4)) / 0.00125.
Since the value of Young's modulus for steel is known, we can rearrange this equation to solve for the diameter, d. Once we calculate the diameter, we obtain the minimum required diameter for the wire to ensure it does not stretch more than 0.25 cm under the given force.